id sid tid token lemma pos cana-3423 1 1 communications communication NOUN cana-3423 1 2 on on ADP cana-3423 1 3 applied apply VERB cana-3423 1 4 nonlinear nonlinear ADJ cana-3423 1 5 analysis analysis NOUN cana-3423 1 6 issn issn NOUN cana-3423 1 7 : : PUNCT cana-3423 1 8 1074 1074 NUM cana-3423 1 9 - - PUNCT cana-3423 1 10 133x 133x NUM cana-3423 1 11 vol vol NOUN cana-3423 1 12 32 32 NUM cana-3423 1 13 no no NOUN cana-3423 1 14 . . PUNCT cana-3423 2 1 7s 7 NOUN cana-3423 2 2 ( ( PUNCT cana-3423 2 3 2025 2025 NUM cana-3423 2 4 ) ) PUNCT cana-3423 2 5 https://internationalpubls.com https://internationalpubls.com X cana-3423 2 6 358 358 NUM cana-3423 2 7 advances advance NOUN cana-3423 2 8 in in ADP cana-3423 2 9 topological topological ADJ cana-3423 2 10 methods method NOUN cana-3423 2 11 for for ADP cana-3423 2 12 solving solve VERB cana-3423 2 13 nonlinear nonlinear ADJ cana-3423 2 14 partial partial ADJ cana-3423 2 15 differential differential NOUN cana-3423 2 16 equations equation NOUN cana-3423 2 17 dr dr PROPN cana-3423 2 18 . . PROPN cana-3423 2 19 k.s k.s PROPN cana-3423 2 20 . . PUNCT cana-3423 3 1 krishnamohan1 krishnamohan1 PROPN cana-3423 3 2 * * PROPN cana-3423 3 3 , , PUNCT cana-3423 3 4 dr dr PROPN cana-3423 3 5 . . PROPN cana-3423 3 6 nalini nalini PROPN cana-3423 3 7 suklal suklal PROPN cana-3423 3 8 patil2 patil2 PROPN cana-3423 3 9 , , PUNCT cana-3423 3 10 dr dr PROPN cana-3423 3 11 . . PROPN cana-3423 3 12 parvesh parvesh PROPN cana-3423 3 13 kumar3 kumar3 PROPN cana-3423 3 14 , , PUNCT cana-3423 3 15 robin robin PROPN cana-3423 3 16 singh4 singh4 PROPN cana-3423 3 17 , , PUNCT cana-3423 3 18 n. n. PROPN cana-3423 3 19 srimannarayana5 srimannarayana5 PROPN cana-3423 3 20 , , PUNCT cana-3423 3 21 dr dr PROPN cana-3423 3 22 . . PROPN cana-3423 3 23 nishank nishank PROPN cana-3423 3 24 sudhakar sudhakar PROPN cana-3423 3 25 pimple6 pimple6 PROPN cana-3423 4 1 1*associate 1*associate PROPN cana-3423 4 2 professor professor NOUN cana-3423 4 3 of of ADP cana-3423 4 4 mathematics mathematic NOUN cana-3423 4 5 , , PUNCT cana-3423 4 6 rajah rajah NOUN cana-3423 4 7 serfoji serfoji ADJ cana-3423 4 8 government government NOUN cana-3423 4 9 college college NOUN cana-3423 4 10 , , PUNCT cana-3423 4 11 thanjavur thanjavur PROPN cana-3423 4 12 , , PUNCT cana-3423 4 13 tamil tamil PROPN cana-3423 4 14 nadu nadu NOUN cana-3423 4 15 . . PUNCT cana-3423 5 1 kskrishnamohan@gmail.com kskrishnamohan@gmail.com PROPN cana-3423 5 2 , , PUNCT cana-3423 5 3 https https NOUN cana-3423 5 4 , , PUNCT cana-3423 5 5 : : PUNCT cana-3423 5 6 //orcid.org/0009 //orcid.org/0009 X cana-3423 5 7 - - PUNCT cana-3423 5 8 0004 0004 NUM cana-3423 5 9 - - PUNCT cana-3423 5 10 4421 4421 NUM cana-3423 5 11 - - PUNCT cana-3423 5 12 6177 6177 NUM cana-3423 5 13 . . PUNCT cana-3423 6 1 2head 2head NUM cana-3423 6 2 and and CCONJ cana-3423 6 3 associate associate ADJ cana-3423 6 4 professor professor NOUN cana-3423 6 5 , , PUNCT cana-3423 6 6 ( ( PUNCT cana-3423 6 7 department department NOUN cana-3423 6 8 of of ADP cana-3423 6 9 mathematics mathematic NOUN cana-3423 6 10 ) ) PUNCT cana-3423 6 11 , , PUNCT cana-3423 6 12 pratap pratap PROPN cana-3423 6 13 college college PROPN cana-3423 6 14 ( ( PUNCT cana-3423 6 15 autonomous autonomous ADJ cana-3423 6 16 ) ) PUNCT cana-3423 6 17 amalner amalner NOUN cana-3423 6 18 , , PUNCT cana-3423 6 19 jalgaon jalgaon PROPN cana-3423 6 20 , , PUNCT cana-3423 6 21 maharashtra maharashtra PROPN cana-3423 6 22 . . PROPN cana-3423 7 1 email email PROPN cana-3423 7 2 : : PUNCT cana-3423 7 3 nalinipatil2003@gmail.com nalinipatil2003@gmail.com X cana-3423 8 1 3professor 3professor NUM cana-3423 8 2 , , PUNCT cana-3423 8 3 pt pt PROPN cana-3423 8 4 . . PROPN cana-3423 8 5 n.r.s n.r.s PROPN cana-3423 8 6 . . PUNCT cana-3423 9 1 government government PROPN cana-3423 9 2 college college PROPN cana-3423 9 3 , , PUNCT cana-3423 9 4 rohtak rohtak PROPN cana-3423 9 5 ( ( PUNCT cana-3423 9 6 haryana haryana PROPN cana-3423 9 7 ) ) PUNCT cana-3423 9 8 . . PUNCT cana-3423 10 1 email email NOUN cana-3423 11 1 i i PROPN cana-3423 11 2 d d PROPN cana-3423 11 3 : : PUNCT cana-3423 12 1 parveshiitd@gmail.com parveshiitd@gmail.com X cana-3423 12 2 orcid orcid NOUN cana-3423 13 1 i i PRON cana-3423 13 2 d d NOUN cana-3423 13 3 : : PUNCT cana-3423 13 4 https://orcid.org/0000-0002-0815-6179 https://orcid.org/0000-0002-0815-6179 NOUN cana-3423 13 5 , , PUNCT cana-3423 13 6 4associate 4associate NUM cana-3423 13 7 professor professor NOUN cana-3423 13 8 , , PUNCT cana-3423 13 9 graphic graphic ADJ cana-3423 13 10 era era NOUN cana-3423 13 11 hill hill PROPN cana-3423 13 12 university university PROPN cana-3423 13 13 , , PUNCT cana-3423 13 14 dehradun dehradun PROPN cana-3423 13 15 . . PUNCT cana-3423 14 1 email email NOUN cana-3423 14 2 : : PUNCT cana-3423 15 1 robin1986sr@gmail.com robin1986sr@gmail.com PROPN cana-3423 15 2 orcid orcid PROPN cana-3423 16 1 i i PRON cana-3423 16 2 d d NOUN cana-3423 16 3 : : PUNCT cana-3423 16 4 0000 0000 NUM cana-3423 16 5 - - PUNCT cana-3423 16 6 0003 0003 NUM cana-3423 16 7 - - PUNCT cana-3423 16 8 1389 1389 NUM cana-3423 16 9 - - PUNCT cana-3423 16 10 7788 7788 NUM cana-3423 16 11 5professor 5professor NUM cana-3423 16 12 , , PUNCT cana-3423 16 13 department department NOUN cana-3423 16 14 of of ADP cana-3423 16 15 mathematics mathematic NOUN cana-3423 16 16 , , PUNCT cana-3423 16 17 koneru koneru PROPN cana-3423 16 18 lakshmaiah lakshmaiah PROPN cana-3423 16 19 education education PROPN cana-3423 16 20 foundation foundation PROPN cana-3423 16 21 , , PUNCT cana-3423 16 22 vaddeswaram vaddeswaram NOUN cana-3423 16 23 . . PUNCT cana-3423 17 1 email email NOUN cana-3423 17 2 : : PUNCT cana-3423 17 3 sriman72@gmail.com sriman72@gmail.com X cana-3423 17 4 , , PUNCT cana-3423 17 5 orcid orcid NOUN cana-3423 17 6 i i NOUN cana-3423 17 7 d d NOUN cana-3423 17 8 : : PUNCT cana-3423 17 9 0000 0000 NUM cana-3423 17 10 - - PUNCT cana-3423 17 11 0003 0003 NUM cana-3423 17 12 - - PUNCT cana-3423 17 13 4982 4982 NUM cana-3423 17 14 - - PUNCT cana-3423 17 15 5073 5073 NUM cana-3423 17 16 , , PUNCT cana-3423 17 17 scopus scopus PROPN cana-3423 17 18 i i PROPN cana-3423 17 19 d d PROPN cana-3423 17 20 : : PUNCT cana-3423 17 21 55899630100 55899630100 NUM cana-3423 17 22 6associate 6associate NUM cana-3423 17 23 professor professor NOUN cana-3423 17 24 , , PUNCT cana-3423 17 25 department department NOUN cana-3423 17 26 of of ADP cana-3423 17 27 mathematics mathematic NOUN cana-3423 17 28 , , PUNCT cana-3423 17 29 rajarshi rajarshi PROPN cana-3423 17 30 shahu shahu PROPN cana-3423 17 31 mahavidyalaya mahavidyalaya PROPN cana-3423 17 32 , , PUNCT cana-3423 17 33 latur latur PROPN cana-3423 17 34 – – PUNCT cana-3423 17 35 413512 413512 NUM cana-3423 17 36 ( ( PUNCT cana-3423 17 37 ms ms NOUN cana-3423 17 38 ) ) PUNCT cana-3423 17 39 ( ( PUNCT cana-3423 17 40 autonomous autonomous ADJ cana-3423 17 41 ) ) PUNCT cana-3423 17 42 . . PUNCT cana-3423 18 1 email email NOUN cana-3423 18 2 : : PUNCT cana-3423 19 1 pipnishank@gmail.com pipnishank@gmail.com X cana-3423 19 2 article article NOUN cana-3423 19 3 history history NOUN cana-3423 19 4 : : PUNCT cana-3423 19 5 received receive VERB cana-3423 19 6 : : PUNCT cana-3423 19 7 25 25 NUM cana-3423 19 8 - - SYM cana-3423 19 9 10 10 NUM cana-3423 19 10 - - PUNCT cana-3423 19 11 2024 2024 NUM cana-3423 19 12 revised:10 revised:10 NOUN cana-3423 19 13 - - PUNCT cana-3423 19 14 11 11 NUM cana-3423 19 15 - - PUNCT cana-3423 19 16 2024 2024 NUM cana-3423 19 17 accepted:18 accepted:18 PROPN cana-3423 19 18 - - PUNCT cana-3423 19 19 12 12 NUM cana-3423 19 20 - - PUNCT cana-3423 19 21 2024 2024 NUM cana-3423 19 22 abstract abstract NOUN cana-3423 19 23 : : PUNCT cana-3423 19 24 higher high ADJ cana-3423 19 25 order order NOUN cana-3423 19 26 nonlinear nonlinear ADJ cana-3423 19 27 partial partial ADJ cana-3423 19 28 differential differential NOUN cana-3423 19 29 equations equation NOUN cana-3423 19 30 ( ( PUNCT cana-3423 19 31 pdes pde NOUN cana-3423 19 32 ) ) PUNCT cana-3423 19 33 are be AUX cana-3423 19 34 very very ADV cana-3423 19 35 difficult difficult ADJ cana-3423 19 36 to to PART cana-3423 19 37 compare compare VERB cana-3423 19 38 or or CCONJ cana-3423 19 39 analyze analyze VERB cana-3423 19 40 and and CCONJ cana-3423 19 41 therefore therefore ADV cana-3423 19 42 there there PRON cana-3423 19 43 is be VERB cana-3423 19 44 restriction restriction NOUN cana-3423 19 45 towards towards ADP cana-3423 19 46 analytical analytical ADJ cana-3423 19 47 or or CCONJ cana-3423 19 48 numerical numerical ADJ cana-3423 19 49 appraisal appraisal NOUN cana-3423 19 50 . . PUNCT cana-3423 20 1 the the DET cana-3423 20 2 following follow VERB cana-3423 20 3 research research NOUN cana-3423 20 4 aims aim VERB cana-3423 20 5 at at ADP cana-3423 20 6 solving solve VERB cana-3423 20 7 the the DET cana-3423 20 8 problem problem NOUN cana-3423 20 9 of of ADP cana-3423 20 10 the the DET cana-3423 20 11 solutions solution NOUN cana-3423 20 12 of of ADP cana-3423 20 13 nonlinear nonlinear ADJ cana-3423 20 14 pdes pde NOUN cana-3423 20 15 through through ADP cana-3423 20 16 the the DET cana-3423 20 17 application application NOUN cana-3423 20 18 of of ADP cana-3423 20 19 topological topological ADJ cana-3423 20 20 fixed fix VERB cana-3423 20 21 - - PUNCT cana-3423 20 22 point point NOUN cana-3423 20 23 theory theory NOUN cana-3423 20 24 and and CCONJ cana-3423 20 25 degree degree NOUN cana-3423 20 26 theory theory NOUN cana-3423 20 27 , , PUNCT cana-3423 20 28 as as ADV cana-3423 20 29 well well ADV cana-3423 20 30 as as ADP cana-3423 20 31 the the DET cana-3423 20 32 application application NOUN cana-3423 20 33 of of ADP cana-3423 20 34 variational variational ADJ cana-3423 20 35 methods method NOUN cana-3423 20 36 towards towards ADP cana-3423 20 37 improving improve VERB cana-3423 20 38 the the DET cana-3423 20 39 existence existence NOUN cana-3423 20 40 , , PUNCT cana-3423 20 41 uniqueness uniqueness NOUN cana-3423 20 42 , , PUNCT cana-3423 20 43 and and CCONJ cana-3423 20 44 stability stability NOUN cana-3423 20 45 of of ADP cana-3423 20 46 solutions solution NOUN cana-3423 20 47 to to ADP cana-3423 20 48 pdes pde NOUN cana-3423 20 49 under under ADP cana-3423 20 50 all all DET cana-3423 20 51 possible possible ADJ cana-3423 20 52 boundary boundary ADJ cana-3423 20 53 conditions condition NOUN cana-3423 20 54 . . PUNCT cana-3423 21 1 the the DET cana-3423 21 2 purpose purpose NOUN cana-3423 21 3 of of ADP cana-3423 21 4 the the DET cana-3423 21 5 study study NOUN cana-3423 21 6 is be AUX cana-3423 21 7 to to PART cana-3423 21 8 investigate investigate VERB cana-3423 21 9 the the DET cana-3423 21 10 effectiveness effectiveness NOUN cana-3423 21 11 of of ADP cana-3423 21 12 these these DET cana-3423 21 13 methods method NOUN cana-3423 21 14 in in ADP cana-3423 21 15 multifaceted multifaceted ADJ cana-3423 21 16 and and CCONJ cana-3423 21 17 multi multi ADJ cana-3423 21 18 - - ADJ cana-3423 21 19 grained grained ADJ cana-3423 21 20 systems system NOUN cana-3423 21 21 . . PUNCT cana-3423 22 1 we we PRON cana-3423 22 2 used use VERB cana-3423 22 3 both both DET cana-3423 22 4 fixed fix VERB cana-3423 22 5 - - PUNCT cana-3423 22 6 point point NOUN cana-3423 22 7 analyses analysis NOUN cana-3423 22 8 to to PART cana-3423 22 9 check check VERB cana-3423 22 10 the the DET cana-3423 22 11 convergence convergence NOUN cana-3423 22 12 of of ADP cana-3423 22 13 solutions solution NOUN cana-3423 22 14 and and CCONJ cana-3423 22 15 degree degree NOUN cana-3423 22 16 - - PUNCT cana-3423 22 17 theoretic theoretic ADJ cana-3423 22 18 methods method NOUN cana-3423 22 19 for for ADP cana-3423 22 20 checking check VERB cana-3423 22 21 multiplicity multiplicity NOUN cana-3423 22 22 of of ADP cana-3423 22 23 solutions solution NOUN cana-3423 22 24 , , PUNCT cana-3423 22 25 and and CCONJ cana-3423 22 26 variational variational ADJ cana-3423 22 27 structure structure NOUN cana-3423 22 28 for for ADP cana-3423 22 29 critical critical ADJ cana-3423 22 30 points point NOUN cana-3423 22 31 of of ADP cana-3423 22 32 solutions solution NOUN cana-3423 22 33 . . PUNCT cana-3423 23 1 newton newton PROPN cana-3423 23 2 - - PUNCT cana-3423 23 3 raphson raphson PROPN cana-3423 23 4 and and CCONJ cana-3423 23 5 finite finite PROPN cana-3423 23 6 element element NOUN cana-3423 23 7 techniques technique NOUN cana-3423 23 8 were be AUX cana-3423 23 9 used use VERB cana-3423 23 10 in in ADP cana-3423 23 11 solving solve VERB cana-3423 23 12 solutions solution NOUN cana-3423 23 13 with with ADP cana-3423 23 14 much much ADJ cana-3423 23 15 ease ease NOUN cana-3423 23 16 for for ADP cana-3423 23 17 increased increase VERB cana-3423 23 18 efficiency efficiency NOUN cana-3423 23 19 . . PUNCT cana-3423 24 1 a a DET cana-3423 24 2 selection selection NOUN cana-3423 24 3 of of ADP cana-3423 24 4 findings finding NOUN cana-3423 24 5 shows show VERB cana-3423 24 6 that that SCONJ cana-3423 24 7 topological topological ADJ cana-3423 24 8 techniques technique NOUN cana-3423 24 9 are be AUX cana-3423 24 10 an an DET cana-3423 24 11 efficient efficient ADJ cana-3423 24 12 means mean NOUN cana-3423 24 13 of of ADP cana-3423 24 14 handling handle VERB cana-3423 24 15 nonlinearity nonlinearity NOUN cana-3423 24 16 in in ADP cana-3423 24 17 pdes pde NOUN cana-3423 24 18 ; ; PUNCT cana-3423 24 19 degree degree NOUN cana-3423 24 20 theory theory NOUN cana-3423 24 21 uncovers uncover VERB cana-3423 24 22 the the DET cana-3423 24 23 multiplicity multiplicity NOUN cana-3423 24 24 of of ADP cana-3423 24 25 solutions solution NOUN cana-3423 24 26 while while SCONJ cana-3423 24 27 variational variational ADJ cana-3423 24 28 techniques technique NOUN cana-3423 24 29 unveil unveil VERB cana-3423 24 30 the the DET cana-3423 24 31 stability stability NOUN cana-3423 24 32 characteristics characteristic NOUN cana-3423 24 33 . . PUNCT cana-3423 25 1 this this DET cana-3423 25 2 research research NOUN cana-3423 25 3 advances advance VERB cana-3423 25 4 the the DET cana-3423 25 5 field field NOUN cana-3423 25 6 of of ADP cana-3423 25 7 nonlinear nonlinear ADJ cana-3423 25 8 analysis analysis NOUN cana-3423 25 9 beyond beyond ADP cana-3423 25 10 the the DET cana-3423 25 11 prior prior ADJ cana-3423 25 12 art art NOUN cana-3423 25 13 by by ADP cana-3423 25 14 broadening broaden VERB cana-3423 25 15 the the DET cana-3423 25 16 scope scope NOUN cana-3423 25 17 of of ADP cana-3423 25 18 topology topology NOUN cana-3423 25 19 in in ADP cana-3423 25 20 new new ADJ cana-3423 25 21 domains domain NOUN cana-3423 25 22 and and CCONJ cana-3423 25 23 providing provide VERB cana-3423 25 24 enhanced enhanced ADJ cana-3423 25 25 methods method NOUN cana-3423 25 26 of of ADP cana-3423 25 27 computation computation NOUN cana-3423 25 28 of of ADP cana-3423 25 29 pdes pde NOUN cana-3423 25 30 where where SCONJ cana-3423 25 31 it it PRON cana-3423 25 32 matters matter VERB cana-3423 25 33 most most ADV cana-3423 25 34 : : PUNCT cana-3423 25 35 where where SCONJ cana-3423 25 36 traditional traditional ADJ cana-3423 25 37 linear linear ADJ cana-3423 25 38 methods method NOUN cana-3423 25 39 fail fail VERB cana-3423 25 40 to to PART cana-3423 25 41 apply apply VERB cana-3423 25 42 . . PUNCT cana-3423 26 1 in in ADP cana-3423 26 2 the the DET cana-3423 26 3 context context NOUN cana-3423 26 4 of of ADP cana-3423 26 5 physics physics NOUN cana-3423 26 6 , , PUNCT cana-3423 26 7 engineering engineering NOUN cana-3423 26 8 , , PUNCT cana-3423 26 9 and and CCONJ cana-3423 26 10 environmental environmental ADJ cana-3423 26 11 studies study NOUN cana-3423 26 12 nonlinear nonlinear ADJ cana-3423 26 13 modeling modeling NOUN cana-3423 26 14 is be AUX cana-3423 26 15 vital vital ADJ cana-3423 26 16 and and CCONJ cana-3423 26 17 the the DET cana-3423 26 18 practical practical ADJ cana-3423 26 19 implication implication NOUN cana-3423 26 20 highlighted highlight VERB cana-3423 26 21 herein herein NOUN cana-3423 26 22 is be AUX cana-3423 26 23 useful useful ADJ cana-3423 26 24 for for ADP cana-3423 26 25 all all PRON cana-3423 26 26 . . PUNCT cana-3423 27 1 possible possible ADJ cana-3423 27 2 future future ADJ cana-3423 27 3 work work NOUN cana-3423 27 4 might might AUX cana-3423 27 5 apply apply VERB cana-3423 27 6 these these DET cana-3423 27 7 methods method NOUN cana-3423 27 8 to to ADP cana-3423 27 9 more more ADV cana-3423 27 10 general general ADJ cana-3423 27 11 boundary boundary ADJ cana-3423 27 12 conditions condition NOUN cana-3423 27 13 as as ADV cana-3423 27 14 well well ADV cana-3423 27 15 as as ADP cana-3423 27 16 combined combine VERB cana-3423 27 17 analytical analytical ADJ cana-3423 27 18 - - PUNCT cana-3423 27 19 numerical numerical ADJ cana-3423 27 20 approaches approach NOUN cana-3423 27 21 . . PUNCT cana-3423 28 1 keywords keyword NOUN cana-3423 28 2 : : PUNCT cana-3423 28 3 nonlinear nonlinear ADJ cana-3423 28 4 partial partial ADJ cana-3423 28 5 differential differential ADJ cana-3423 28 6 equations equation NOUN cana-3423 28 7 ( ( PUNCT cana-3423 28 8 pdes pde NOUN cana-3423 28 9 ) ) PUNCT cana-3423 28 10 , , PUNCT cana-3423 28 11 topological topological ADJ cana-3423 28 12 methods method NOUN cana-3423 28 13 , , PUNCT cana-3423 28 14 fixed fix VERB cana-3423 28 15 - - PUNCT cana-3423 28 16 point point NOUN cana-3423 28 17 theory theory NOUN cana-3423 28 18 , , PUNCT cana-3423 28 19 degree degree NOUN cana-3423 28 20 theory theory NOUN cana-3423 28 21 , , PUNCT cana-3423 28 22 variational variational ADJ cana-3423 28 23 methods method NOUN cana-3423 28 24 . . PUNCT cana-3423 29 1 1 1 X cana-3423 29 2 . . X cana-3423 29 3 introduction introduction NOUN cana-3423 29 4 nonlinear nonlinear ADJ cana-3423 29 5 partial partial ADJ cana-3423 29 6 differential differential NOUN cana-3423 29 7 equations equation NOUN cana-3423 29 8 ( ( PUNCT cana-3423 29 9 pdes pde NOUN cana-3423 29 10 ) ) PUNCT cana-3423 29 11 appear appear VERB cana-3423 29 12 naturally naturally ADV cana-3423 29 13 in in ADP cana-3423 29 14 almost almost ADV cana-3423 29 15 all all PRON cana-3423 29 16 branches branch NOUN cana-3423 29 17 of of ADP cana-3423 29 18 science science NOUN cana-3423 29 19 and and CCONJ cana-3423 29 20 engineering engineering NOUN cana-3423 29 21 and and CCONJ cana-3423 29 22 are be AUX cana-3423 29 23 related relate VERB cana-3423 29 24 to to ADP cana-3423 29 25 problems problem NOUN cana-3423 29 26 in in ADP cana-3423 29 27 fluid fluid ADJ cana-3423 29 28 mechanics mechanic NOUN cana-3423 29 29 and and CCONJ cana-3423 29 30 dynamics dynamic NOUN cana-3423 29 31 of of ADP cana-3423 29 32 structures structure NOUN cana-3423 29 33 , , PUNCT cana-3423 29 34 electromagnetism electromagnetism NOUN cana-3423 29 35 and and CCONJ cana-3423 29 36 photonics photonic NOUN cana-3423 29 37 , , PUNCT cana-3423 29 38 heat heat NOUN cana-3423 29 39 and and CCONJ cana-3423 29 40 mass mass NOUN cana-3423 29 41 transfer transfer NOUN cana-3423 29 42 , , PUNCT cana-3423 29 43 fluid fluid ADJ cana-3423 29 44 dynamics dynamic NOUN cana-3423 29 45 and and CCONJ cana-3423 29 46 gas gas NOUN cana-3423 29 47 dynamics dynamic NOUN cana-3423 29 48 , , PUNCT cana-3423 29 49 material material NOUN cana-3423 29 50 science science NOUN cana-3423 29 51 and and CCONJ cana-3423 29 52 engineering engineering NOUN cana-3423 29 53 , , PUNCT cana-3423 29 54 financial financial ADJ cana-3423 29 55 mathematics mathematic NOUN cana-3423 29 56 and and CCONJ cana-3423 29 57 mathematical mathematical ADJ cana-3423 29 58 biology biology NOUN cana-3423 29 59 . . PUNCT cana-3423 30 1 in in ADP cana-3423 30 2 contrast contrast NOUN cana-3423 30 3 to to ADP cana-3423 30 4 l l NOUN cana-3423 30 5 - - NOUN cana-3423 30 6 pdes pde NOUN cana-3423 30 7 , , PUNCT cana-3423 30 8 which which PRON cana-3423 30 9 involve involve VERB cana-3423 30 10 well well ADV cana-3423 30 11 - - PUNCT cana-3423 30 12 studied study VERB cana-3423 30 13 analytical analytical ADJ cana-3423 30 14 and and CCONJ cana-3423 30 15 numerical numerical ADJ cana-3423 30 16 approaches approach NOUN cana-3423 30 17 , , PUNCT cana-3423 30 18 solving solve VERB cana-3423 30 19 nonlinear nonlinear ADJ cana-3423 30 20 pdes pde NOUN cana-3423 30 21 is be AUX cana-3423 30 22 challenging challenging ADJ cana-3423 30 23 because because SCONJ cana-3423 30 24 their their PRON cana-3423 30 25 behavior behavior NOUN cana-3423 30 26 is be AUX cana-3423 30 27 highly highly ADV cana-3423 30 28 nonlinear nonlinear ADJ cana-3423 30 29 , , PUNCT cana-3423 30 30 and and CCONJ cana-3423 30 31 no no DET cana-3423 30 32 universal universal ADJ cana-3423 30 33 solution solution NOUN cana-3423 30 34 methods method NOUN cana-3423 30 35 exist exist VERB cana-3423 30 36 ( ( PUNCT cana-3423 30 37 evans evans PROPN cana-3423 30 38 , , PUNCT cana-3423 30 39 2010 2010 NUM cana-3423 30 40 ) ) PUNCT cana-3423 30 41 . . PUNCT cana-3423 31 1 nonlinear nonlinear ADJ cana-3423 31 2 systems system NOUN cana-3423 31 3 are be AUX cana-3423 31 4 those those PRON cana-3423 31 5 that that PRON cana-3423 31 6 are be AUX cana-3423 31 7 described describe VERB cana-3423 31 8 by by ADP cana-3423 31 9 equations equation NOUN cana-3423 31 10 that that PRON cana-3423 31 11 are be AUX cana-3423 31 12 nonlinear nonlinear ADJ cana-3423 31 13 with with ADP cana-3423 31 14 synthesis synthesis NOUN cana-3423 31 15 , , PUNCT cana-3423 31 16 and and CCONJ cana-3423 31 17 the the DET cana-3423 31 18 mathematical mathematical ADJ cana-3423 31 19 complexity complexity NOUN cana-3423 31 20 of of ADP cana-3423 31 21 such such ADJ cana-3423 31 22 equations equation NOUN cana-3423 31 23 has have AUX cana-3423 31 24 prompted prompt VERB cana-3423 31 25 the the DET cana-3423 31 26 use use NOUN cana-3423 31 27 of of ADP cana-3423 31 28 sophisticated sophisticated ADJ cana-3423 31 29 methods method NOUN cana-3423 32 1 mailto:parveshiitd@gmail.com mailto:parveshiitd@gmail.com PROPN cana-3423 32 2 mailto:pipnishank@gmail.com mailto:pipnishank@gmail.com PROPN cana-3423 32 3 communications communication NOUN cana-3423 32 4 on on ADP cana-3423 32 5 applied apply VERB cana-3423 32 6 nonlinear nonlinear ADJ cana-3423 32 7 analysis analysis NOUN cana-3423 32 8 issn issn NOUN cana-3423 32 9 : : PUNCT cana-3423 32 10 1074 1074 NUM cana-3423 32 11 - - PUNCT cana-3423 32 12 133x 133x NUM cana-3423 32 13 vol vol NOUN cana-3423 32 14 32 32 NUM cana-3423 32 15 no no NOUN cana-3423 32 16 . . PUNCT cana-3423 33 1 7s 7 NOUN cana-3423 33 2 ( ( PUNCT cana-3423 33 3 2025 2025 NUM cana-3423 33 4 ) ) PUNCT cana-3423 34 1 https://internationalpubls.com https://internationalpubls.com X cana-3423 34 2 359 359 NUM cana-3423 34 3 when when SCONJ cana-3423 34 4 determining determine VERB cana-3423 34 5 solutions solution NOUN cana-3423 34 6 for for ADP cana-3423 34 7 such such ADJ cana-3423 34 8 systems system NOUN cana-3423 34 9 this this PRON cana-3423 34 10 may may AUX cana-3423 34 11 take take VERB cana-3423 34 12 learners learner NOUN cana-3423 34 13 far far ADV cana-3423 34 14 afield afield ADV cana-3423 34 15 when when SCONJ cana-3423 34 16 solving solve VERB cana-3423 34 17 textbook textbook NOUN cana-3423 34 18 problems problem NOUN cana-3423 34 19 ( ( PUNCT cana-3423 34 20 evans evans PROPN cana-3423 34 21 , , PUNCT cana-3423 34 22 2022 2022 NUM cana-3423 34 23 ) ) PUNCT cana-3423 34 24 . . PUNCT cana-3423 35 1 it it PRON cana-3423 35 2 has have AUX cana-3423 35 3 become become VERB cana-3423 35 4 clear clear ADJ cana-3423 35 5 that that SCONJ cana-3423 35 6 topological topological ADJ cana-3423 35 7 methods method NOUN cana-3423 35 8 represent represent VERB cana-3423 35 9 an an DET cana-3423 35 10 important important ADJ cana-3423 35 11 and and CCONJ cana-3423 35 12 promising promising ADJ cana-3423 35 13 approach approach NOUN cana-3423 35 14 to to ADP cana-3423 35 15 analyzing analyze VERB cana-3423 35 16 and and CCONJ cana-3423 35 17 solving solve VERB cana-3423 35 18 nonlinear nonlinear ADJ cana-3423 35 19 pdes pde NOUN cana-3423 35 20 as as SCONJ cana-3423 35 21 they they PRON cana-3423 35 22 provide provide VERB cana-3423 35 23 a a DET cana-3423 35 24 different different ADJ cana-3423 35 25 point point NOUN cana-3423 35 26 of of ADP cana-3423 35 27 view view NOUN cana-3423 35 28 on on ADP cana-3423 35 29 the the DET cana-3423 35 30 existence existence NOUN cana-3423 35 31 , , PUNCT cana-3423 35 32 regularity regularity NOUN cana-3423 35 33 , , PUNCT cana-3423 35 34 and and CCONJ cana-3423 35 35 stability stability NOUN cana-3423 35 36 of of ADP cana-3423 35 37 the the DET cana-3423 35 38 solutions solution NOUN cana-3423 35 39 . . PUNCT cana-3423 36 1 these these DET cana-3423 36 2 methods method NOUN cana-3423 36 3 take take VERB cana-3423 36 4 advantage advantage NOUN cana-3423 36 5 of of ADP cana-3423 36 6 the the DET cana-3423 36 7 aspect aspect NOUN cana-3423 36 8 of of ADP cana-3423 36 9 topological topological ADJ cana-3423 36 10 spaces space NOUN cana-3423 36 11 and and CCONJ cana-3423 36 12 mapping mapping NOUN cana-3423 36 13 ; ; PUNCT cana-3423 36 14 this this PRON cana-3423 36 15 makes make VERB cana-3423 36 16 it it PRON cana-3423 36 17 possible possible ADJ cana-3423 36 18 to to PART cana-3423 36 19 study study VERB cana-3423 36 20 nonlinearity nonlinearity NOUN cana-3423 36 21 in in ADP cana-3423 36 22 a a DET cana-3423 36 23 manner manner NOUN cana-3423 36 24 that that PRON cana-3423 36 25 might might AUX cana-3423 36 26 not not PART cana-3423 36 27 be be AUX cana-3423 36 28 achievable achievable ADJ cana-3423 36 29 by by ADP cana-3423 36 30 traditional traditional ADJ cana-3423 36 31 techniques technique NOUN cana-3423 36 32 ( ( PUNCT cana-3423 36 33 aubin aubin PROPN cana-3423 36 34 & & CCONJ cana-3423 36 35 frankowska frankowska PROPN cana-3423 36 36 , , PUNCT cana-3423 36 37 2009 2009 NUM cana-3423 36 38 ) ) PUNCT cana-3423 36 39 . . PUNCT cana-3423 37 1 with with ADP cana-3423 37 2 the the DET cana-3423 37 3 help help NOUN cana-3423 37 4 of of ADP cana-3423 37 5 fixed fix VERB cana-3423 37 6 - - PUNCT cana-3423 37 7 point point NOUN cana-3423 37 8 index index NOUN cana-3423 37 9 , , PUNCT cana-3423 37 10 homotopy homotopy NOUN cana-3423 37 11 , , PUNCT cana-3423 37 12 and and CCONJ cana-3423 37 13 topological topological ADJ cana-3423 37 14 degree degree NOUN cana-3423 37 15 theory theory NOUN cana-3423 37 16 mathematicians mathematician NOUN cana-3423 37 17 managed manage VERB cana-3423 37 18 to to PART cana-3423 37 19 solve solve VERB cana-3423 37 20 classes class NOUN cana-3423 37 21 of of ADP cana-3423 37 22 problems problem NOUN cana-3423 37 23 that that PRON cana-3423 37 24 could could AUX cana-3423 37 25 not not PART cana-3423 37 26 be be AUX cana-3423 37 27 solved solve VERB cana-3423 37 28 during during ADP cana-3423 37 29 earlier early ADJ cana-3423 37 30 days day NOUN cana-3423 37 31 ( ( PUNCT cana-3423 37 32 ambrosetti ambrosetti PROPN cana-3423 37 33 & & CCONJ cana-3423 37 34 prodi prodi PROPN cana-3423 37 35 , , PUNCT cana-3423 37 36 1995 1995 NUM cana-3423 37 37 ) ) PUNCT cana-3423 37 38 . . PUNCT cana-3423 38 1 these these DET cana-3423 38 2 topological topological ADJ cana-3423 38 3 techniques technique NOUN cana-3423 38 4 offer offer VERB cana-3423 38 5 not not PART cana-3423 38 6 only only ADV cana-3423 38 7 solutions solution NOUN cana-3423 38 8 with with ADP cana-3423 38 9 their their PRON cana-3423 38 10 inherent inherent ADJ cana-3423 38 11 qualitative qualitative ADJ cana-3423 38 12 behavior behavior NOUN cana-3423 38 13 , , PUNCT cana-3423 38 14 which which PRON cana-3423 38 15 is be AUX cana-3423 38 16 very very ADV cana-3423 38 17 useful useful ADJ cana-3423 38 18 in in ADP cana-3423 38 19 disciplines discipline NOUN cana-3423 38 20 where where SCONJ cana-3423 38 21 nonlinear nonlinear ADJ cana-3423 38 22 dynamics dynamic NOUN cana-3423 38 23 prevail prevail VERB cana-3423 38 24 considerably considerably ADV cana-3423 38 25 . . PUNCT cana-3423 39 1 thus thus ADV cana-3423 39 2 , , PUNCT cana-3423 39 3 it it PRON cana-3423 39 4 may may AUX cana-3423 39 5 be be AUX cana-3423 39 6 concluded conclude VERB cana-3423 39 7 that that SCONJ cana-3423 39 8 the the DET cana-3423 39 9 value value NOUN cana-3423 39 10 of of ADP cana-3423 39 11 the the DET cana-3423 39 12 present present ADJ cana-3423 39 13 study study NOUN cana-3423 39 14 is be AUX cana-3423 39 15 in in ADP cana-3423 39 16 the the DET cana-3423 39 17 development development NOUN cana-3423 39 18 of of ADP cana-3423 39 19 the the DET cana-3423 39 20 current current ADJ cana-3423 39 21 concepts concept NOUN cana-3423 39 22 and and CCONJ cana-3423 39 23 methods method NOUN cana-3423 39 24 for for ADP cana-3423 39 25 solving solve VERB cana-3423 39 26 nonlinear nonlinear ADJ cana-3423 39 27 pdes pde NOUN cana-3423 39 28 . . PUNCT cana-3423 40 1 however however ADV cana-3423 40 2 , , PUNCT cana-3423 40 3 due due ADP cana-3423 40 4 to to ADP cana-3423 40 5 the the DET cana-3423 40 6 topological topological ADJ cana-3423 40 7 techniques technique NOUN cana-3423 40 8 , , PUNCT cana-3423 40 9 in in ADP cana-3423 40 10 the the DET cana-3423 40 11 present present ADJ cana-3423 40 12 scenario scenario NOUN cana-3423 40 13 of of ADP cana-3423 40 14 research research NOUN cana-3423 40 15 based base VERB cana-3423 40 16 on on ADP cana-3423 40 17 numerical numerical ADJ cana-3423 40 18 methods method NOUN cana-3423 40 19 , , PUNCT cana-3423 40 20 the the DET cana-3423 40 21 theoretical theoretical ADJ cana-3423 40 22 background background NOUN cana-3423 40 23 provided provide VERB cana-3423 40 24 by by ADP cana-3423 40 25 topology topology NOUN cana-3423 40 26 - - PUNCT cana-3423 40 27 based base VERB cana-3423 40 28 methods method NOUN cana-3423 40 29 can can AUX cana-3423 40 30 not not PART cana-3423 40 31 be be AUX cana-3423 40 32 mounded mound VERB cana-3423 40 33 especially especially ADV cana-3423 40 34 for for ADP cana-3423 40 35 the the DET cana-3423 40 36 study study NOUN cana-3423 40 37 of of ADP cana-3423 40 38 the the DET cana-3423 40 39 existence existence NOUN cana-3423 40 40 and and CCONJ cana-3423 40 41 behavior behavior NOUN cana-3423 40 42 of of ADP cana-3423 40 43 solutions solution NOUN cana-3423 40 44 in in ADP cana-3423 40 45 infinite infinite ADJ cana-3423 40 46 dimensional dimensional ADJ cana-3423 40 47 solutions solution NOUN cana-3423 40 48 ( ( PUNCT cana-3423 40 49 rabinowitz rabinowitz NOUN cana-3423 40 50 , , PUNCT cana-3423 40 51 1986 1986 NUM cana-3423 40 52 ) ) PUNCT cana-3423 40 53 . . PUNCT cana-3423 41 1 the the DET cana-3423 41 2 purpose purpose NOUN cana-3423 41 3 of of ADP cana-3423 41 4 this this DET cana-3423 41 5 study study NOUN cana-3423 41 6 is be AUX cana-3423 41 7 to to PART cana-3423 41 8 fill fill VERB cana-3423 41 9 the the DET cana-3423 41 10 gap gap NOUN cana-3423 41 11 between between ADP cana-3423 41 12 abstract abstract ADJ cana-3423 41 13 topological topological ADJ cana-3423 41 14 theory theory NOUN cana-3423 41 15 and and CCONJ cana-3423 41 16 solving solve VERB cana-3423 41 17 concrete concrete ADJ cana-3423 41 18 problems problem NOUN cana-3423 41 19 in in ADP cana-3423 41 20 nonlinear nonlinear ADJ cana-3423 41 21 pdes pde NOUN cana-3423 41 22 , , PUNCT cana-3423 41 23 to to PART cana-3423 41 24 provide provide VERB cana-3423 41 25 both both DET cana-3423 41 26 theoretical theoretical ADJ cana-3423 41 27 promise promise NOUN cana-3423 41 28 and and CCONJ cana-3423 41 29 applications application NOUN cana-3423 41 30 for for ADP cana-3423 41 31 further further ADJ cana-3423 41 32 work work NOUN cana-3423 41 33 on on ADP cana-3423 41 34 the the DET cana-3423 41 35 topic topic NOUN cana-3423 41 36 . . PUNCT cana-3423 42 1 as as ADP cana-3423 42 2 such such ADJ cana-3423 42 3 , , PUNCT cana-3423 42 4 the the DET cana-3423 42 5 primary primary ADJ cana-3423 42 6 direction direction NOUN cana-3423 42 7 of of ADP cana-3423 42 8 the the DET cana-3423 42 9 presented present VERB cana-3423 42 10 investigation investigation NOUN cana-3423 42 11 extends extend VERB cana-3423 42 12 the the DET cana-3423 42 13 research research NOUN cana-3423 42 14 inquiries inquiry NOUN cana-3423 42 15 of of ADP cana-3423 42 16 this this DET cana-3423 42 17 article article NOUN cana-3423 42 18 toward toward ADP cana-3423 42 19 providing provide VERB cana-3423 42 20 an an DET cana-3423 42 21 analysis analysis NOUN cana-3423 42 22 of of ADP cana-3423 42 23 the the DET cana-3423 42 24 development development NOUN cana-3423 42 25 and and CCONJ cana-3423 42 26 modern modern ADJ cana-3423 42 27 state state NOUN cana-3423 42 28 of of ADP cana-3423 42 29 the the DET cana-3423 42 30 application application NOUN cana-3423 42 31 of of ADP cana-3423 42 32 topological topological ADJ cana-3423 42 33 methods method NOUN cana-3423 42 34 for for ADP cana-3423 42 35 the the DET cana-3423 42 36 solution solution NOUN cana-3423 42 37 of of ADP cana-3423 42 38 nonlinear nonlinear ADJ cana-3423 42 39 pdes pde NOUN cana-3423 42 40 . . PUNCT cana-3423 43 1 in in ADP cana-3423 43 2 particular particular ADJ cana-3423 43 3 , , PUNCT cana-3423 43 4 the the DET cana-3423 43 5 study study NOUN cana-3423 43 6 aims aim VERB cana-3423 43 7 to to PART cana-3423 43 8 assess assess VERB cana-3423 43 9 the the DET cana-3423 43 10 efficiency efficiency NOUN cana-3423 43 11 of of ADP cana-3423 43 12 these these DET cana-3423 43 13 methods method NOUN cana-3423 43 14 in in ADP cana-3423 43 15 furnishing furnish VERB cana-3423 43 16 sound sound ADJ cana-3423 43 17 solutions solution NOUN cana-3423 43 18 to to ADP cana-3423 43 19 various various ADJ cana-3423 43 20 nonlinear nonlinear ADJ cana-3423 43 21 systems system NOUN cana-3423 43 22 , , PUNCT cana-3423 43 23 which which PRON cana-3423 43 24 constitutes constitute VERB cana-3423 43 25 a a DET cana-3423 43 26 critical critical ADJ cana-3423 43 27 research research NOUN cana-3423 43 28 niche niche NOUN cana-3423 43 29 in in ADP cana-3423 43 30 contemporary contemporary ADJ cana-3423 43 31 mathematics mathematic NOUN cana-3423 43 32 and and CCONJ cana-3423 43 33 applications application NOUN cana-3423 43 34 . . PUNCT cana-3423 44 1 2 2 X cana-3423 44 2 . . X cana-3423 44 3 literature literature NOUN cana-3423 44 4 review review PROPN cana-3423 44 5 recently recently ADV cana-3423 44 6 much much ADJ cana-3423 44 7 has have AUX cana-3423 44 8 been be AUX cana-3423 44 9 written write VERB cana-3423 44 10 in in ADP cana-3423 44 11 literature literature NOUN cana-3423 44 12 about about ADP cana-3423 44 13 nonlinear nonlinear ADJ cana-3423 44 14 partial partial ADJ cana-3423 44 15 differential differential NOUN cana-3423 44 16 equations equation NOUN cana-3423 44 17 ( ( PUNCT cana-3423 44 18 pdes pde NOUN cana-3423 44 19 ) ) PUNCT cana-3423 44 20 with with ADP cana-3423 44 21 various various ADJ cana-3423 44 22 fields field NOUN cana-3423 44 23 of of ADP cana-3423 44 24 knowledge knowledge NOUN cana-3423 44 25 showing show VERB cana-3423 44 26 interest interest NOUN cana-3423 44 27 in in ADP cana-3423 44 28 different different ADJ cana-3423 44 29 aspects aspect NOUN cana-3423 44 30 of of ADP cana-3423 44 31 the the DET cana-3423 44 32 methodologies methodology NOUN cana-3423 44 33 developed develop VERB cana-3423 44 34 in in ADP cana-3423 44 35 this this DET cana-3423 44 36 area area NOUN cana-3423 44 37 aiming aim VERB cana-3423 44 38 at at ADP cana-3423 44 39 providing provide VERB cana-3423 44 40 theoretical theoretical ADJ cana-3423 44 41 as as ADV cana-3423 44 42 well well ADV cana-3423 44 43 as as ADP cana-3423 44 44 practical practical ADJ cana-3423 44 45 approaches approach NOUN cana-3423 44 46 to to PART cana-3423 44 47 solve solve VERB cana-3423 44 48 nonlinear nonlinear ADJ cana-3423 44 49 systems system NOUN cana-3423 44 50 . . PUNCT cana-3423 45 1 perturbation perturbation NOUN cana-3423 45 2 techniques technique NOUN cana-3423 45 3 and and CCONJ cana-3423 45 4 variational variational ADJ cana-3423 45 5 methods method NOUN cana-3423 45 6 have have AUX cana-3423 45 7 been be AUX cana-3423 45 8 partially partially ADV cana-3423 45 9 used use VERB cana-3423 45 10 as as ADP cana-3423 45 11 a a DET cana-3423 45 12 method method NOUN cana-3423 45 13 to to PART cana-3423 45 14 solve solve VERB cana-3423 45 15 the the DET cana-3423 45 16 problem problem NOUN cana-3423 45 17 of of ADP cana-3423 45 18 complexity complexity NOUN cana-3423 45 19 and and CCONJ cana-3423 45 20 uncertainty uncertainty NOUN cana-3423 45 21 of of ADP cana-3423 45 22 nonlinear nonlinear ADJ cana-3423 45 23 pdes pde NOUN cana-3423 45 24 ( ( PUNCT cana-3423 45 25 brezis brezis NOUN cana-3423 45 26 , , PUNCT cana-3423 45 27 2010 2010 NUM cana-3423 45 28 ) ) PUNCT cana-3423 45 29 . . PUNCT cana-3423 46 1 the the DET cana-3423 46 2 development development NOUN cana-3423 46 3 of of ADP cana-3423 46 4 topological topological ADJ cana-3423 46 5 techniques technique NOUN cana-3423 46 6 especially especially ADV cana-3423 46 7 fixed fix VERB cana-3423 46 8 - - PUNCT cana-3423 46 9 point point NOUN cana-3423 46 10 index index NOUN cana-3423 46 11 topology topology NOUN cana-3423 46 12 and and CCONJ cana-3423 46 13 homotopy homotopy NOUN cana-3423 46 14 represents represent VERB cana-3423 46 15 a a DET cana-3423 46 16 new new ADJ cana-3423 46 17 direction direction NOUN cana-3423 46 18 in in ADP cana-3423 46 19 this this DET cana-3423 46 20 context context NOUN cana-3423 46 21 , , PUNCT cana-3423 46 22 which which PRON cana-3423 46 23 seems seem VERB cana-3423 46 24 to to PART cana-3423 46 25 be be AUX cana-3423 46 26 more more ADV cana-3423 46 27 reliable reliable ADJ cana-3423 46 28 for for ADP cana-3423 46 29 proving prove VERB cana-3423 46 30 the the DET cana-3423 46 31 existence existence NOUN cana-3423 46 32 and and CCONJ cana-3423 46 33 uniqueness uniqueness NOUN cana-3423 46 34 of of ADP cana-3423 46 35 solutions solution NOUN cana-3423 46 36 . . PUNCT cana-3423 47 1 2.1 2.1 NUM cana-3423 47 2 key key ADJ cana-3423 47 3 findings finding NOUN cana-3423 47 4 from from ADP cana-3423 47 5 recent recent ADJ cana-3423 47 6 studies study NOUN cana-3423 47 7 several several ADJ cana-3423 47 8 prior prior ADJ cana-3423 47 9 works work NOUN cana-3423 47 10 have have AUX cana-3423 47 11 also also ADV cana-3423 47 12 shown show VERB cana-3423 47 13 that that SCONJ cana-3423 47 14 topological topological ADJ cana-3423 47 15 methods method NOUN cana-3423 47 16 can can AUX cana-3423 47 17 be be AUX cana-3423 47 18 used use VERB cana-3423 47 19 to to PART cana-3423 47 20 solve solve VERB cana-3423 47 21 nonlinear nonlinear ADJ cana-3423 47 22 pdes pde NOUN cana-3423 47 23 by by ADP cana-3423 47 24 using use VERB cana-3423 47 25 degree degree NOUN cana-3423 47 26 theory theory NOUN cana-3423 47 27 and and CCONJ cana-3423 47 28 variational variational ADJ cana-3423 47 29 topological topological ADJ cana-3423 47 30 methods method NOUN cana-3423 47 31 . . PUNCT cana-3423 48 1 for for ADP cana-3423 48 2 example example NOUN cana-3423 48 3 , , PUNCT cana-3423 48 4 berestycki berestycki NOUN cana-3423 48 5 and and CCONJ cana-3423 48 6 lions lion NOUN cana-3423 48 7 ( ( PUNCT cana-3423 48 8 2013 2013 NUM cana-3423 48 9 ) ) PUNCT cana-3423 48 10 showed show VERB cana-3423 48 11 that that SCONJ cana-3423 48 12 one one PRON cana-3423 48 13 can can AUX cana-3423 48 14 use use VERB cana-3423 48 15 degree degree NOUN cana-3423 48 16 - - PUNCT cana-3423 48 17 theoretic theoretic NOUN cana-3423 48 18 techniques technique NOUN cana-3423 48 19 to to PART cana-3423 48 20 establish establish VERB cana-3423 48 21 the the DET cana-3423 48 22 existence existence NOUN cana-3423 48 23 of of ADP cana-3423 48 24 solutions solution NOUN cana-3423 48 25 in in ADP cana-3423 48 26 semilinear semilinear ADJ cana-3423 48 27 pdes pde NOUN cana-3423 48 28 , , PUNCT cana-3423 48 29 a a DET cana-3423 48 30 result result NOUN cana-3423 48 31 that that PRON cana-3423 48 32 is be AUX cana-3423 48 33 replete replete ADJ cana-3423 48 34 with with ADP cana-3423 48 35 exciting exciting ADJ cana-3423 48 36 applications application NOUN cana-3423 48 37 ranging range VERB cana-3423 48 38 from from ADP cana-3423 48 39 fluid fluid ADJ cana-3423 48 40 mechanics mechanic NOUN cana-3423 48 41 to to ADP cana-3423 48 42 chemical chemical ADJ cana-3423 48 43 kinetics kinetic NOUN cana-3423 48 44 . . PUNCT cana-3423 49 1 in in ADP cana-3423 49 2 addition addition NOUN cana-3423 49 3 , , PUNCT cana-3423 49 4 chang chang PROPN cana-3423 49 5 ( ( PUNCT cana-3423 49 6 2013 2013 NUM cana-3423 49 7 ) ) PUNCT cana-3423 49 8 discussed discuss VERB cana-3423 49 9 the the DET cana-3423 49 10 works work NOUN cana-3423 49 11 done do VERB cana-3423 49 12 using use VERB cana-3423 49 13 critical critical ADJ cana-3423 49 14 point point NOUN cana-3423 49 15 theory theory NOUN cana-3423 49 16 where where SCONJ cana-3423 49 17 topological topological ADJ cana-3423 49 18 invariants invariant NOUN cana-3423 49 19 have have AUX cana-3423 49 20 been be AUX cana-3423 49 21 used use VERB cana-3423 49 22 when when SCONJ cana-3423 49 23 explaining explain VERB cana-3423 49 24 the the DET cana-3423 49 25 qualitative qualitative ADJ cana-3423 49 26 features feature NOUN cana-3423 49 27 of of ADP cana-3423 49 28 solutions solution NOUN cana-3423 49 29 , , PUNCT cana-3423 49 30 which which PRON cana-3423 49 31 has have AUX cana-3423 49 32 led lead VERB cana-3423 49 33 to to ADP cana-3423 49 34 the the DET cana-3423 49 35 creation creation NOUN cana-3423 49 36 of of ADP cana-3423 49 37 new new ADJ cana-3423 49 38 models model NOUN cana-3423 49 39 that that PRON cana-3423 49 40 solve solve VERB cana-3423 49 41 physical physical ADJ cana-3423 49 42 and and CCONJ cana-3423 49 43 engineering engineering NOUN cana-3423 49 44 problems problem NOUN cana-3423 49 45 . . PUNCT cana-3423 50 1 however however ADV cana-3423 50 2 , , PUNCT cana-3423 50 3 new new ADJ cana-3423 50 4 and and CCONJ cana-3423 50 5 related related ADJ cana-3423 50 6 work work NOUN cana-3423 50 7 by by ADP cana-3423 50 8 mawhin mawhin NOUN cana-3423 50 9 in in ADP cana-3423 50 10 2017 2017 NUM cana-3423 50 11 has have AUX cana-3423 50 12 also also ADV cana-3423 50 13 contributed contribute VERB cana-3423 50 14 to to ADP cana-3423 50 15 reviewing review VERB cana-3423 50 16 the the DET cana-3423 50 17 way way NOUN cana-3423 50 18 fixed fix VERB cana-3423 50 19 point point NOUN cana-3423 50 20 theorems theorem NOUN cana-3423 50 21 apply apply VERB cana-3423 50 22 in in ADP cana-3423 50 23 boundary boundary ADJ cana-3423 50 24 value value NOUN cana-3423 50 25 problems problem NOUN cana-3423 50 26 to to PART cana-3423 50 27 which which PRON cana-3423 50 28 linear linear ADJ cana-3423 50 29 methods method NOUN cana-3423 50 30 can can AUX cana-3423 50 31 not not PART cana-3423 50 32 adequately adequately ADV cana-3423 50 33 respond respond VERB cana-3423 50 34 to to ADP cana-3423 50 35 nonlinear nonlinear ADJ cana-3423 50 36 characteristics characteristic NOUN cana-3423 50 37 . . PUNCT cana-3423 51 1 rabinowitz rabinowitz NOUN cana-3423 51 2 ( ( PUNCT cana-3423 51 3 2015 2015 NUM cana-3423 51 4 ) ) PUNCT cana-3423 51 5 in in ADP cana-3423 51 6 his his PRON cana-3423 51 7 study study NOUN cana-3423 51 8 also also ADV cana-3423 51 9 revealed reveal VERB cana-3423 51 10 that that SCONJ cana-3423 51 11 the the DET cana-3423 51 12 bifurcation bifurcation NOUN cana-3423 51 13 theory theory NOUN cana-3423 51 14 which which PRON cana-3423 51 15 is be AUX cana-3423 51 16 a a DET cana-3423 51 17 subtopic subtopic NOUN cana-3423 51 18 within within ADP cana-3423 51 19 topology topology NOUN cana-3423 51 20 is be AUX cana-3423 51 21 useful useful ADJ cana-3423 51 22 when when SCONJ cana-3423 51 23 identifying identify VERB cana-3423 51 24 stability stability NOUN cana-3423 51 25 and and CCONJ cana-3423 51 26 multiplicity multiplicity NOUN cana-3423 51 27 of of ADP cana-3423 51 28 solutions solution NOUN cana-3423 51 29 in in ADP cana-3423 51 30 the the DET cana-3423 51 31 nonlinear nonlinear ADJ cana-3423 51 32 pde pde NOUN cana-3423 51 33 when when SCONJ cana-3423 51 34 the the DET cana-3423 51 35 space space NOUN cana-3423 51 36 dimension dimension NOUN cana-3423 51 37 is be AUX cana-3423 51 38 greater great ADJ cana-3423 51 39 than than ADP cana-3423 51 40 1 1 NUM cana-3423 51 41 and and CCONJ cana-3423 51 42 even even ADV cana-3423 51 43 approaches approach VERB cana-3423 51 44 infinity infinity NOUN cana-3423 51 45 . . PUNCT cana-3423 52 1 communications communication NOUN cana-3423 52 2 on on ADP cana-3423 52 3 applied apply VERB cana-3423 52 4 nonlinear nonlinear ADJ cana-3423 52 5 analysis analysis NOUN cana-3423 52 6 issn issn NOUN cana-3423 52 7 : : PUNCT cana-3423 52 8 1074 1074 NUM cana-3423 52 9 - - PUNCT cana-3423 52 10 133x 133x NUM cana-3423 52 11 vol vol NOUN cana-3423 52 12 32 32 NUM cana-3423 52 13 no no NOUN cana-3423 52 14 . . PUNCT cana-3423 53 1 7s 7 NOUN cana-3423 53 2 ( ( PUNCT cana-3423 53 3 2025 2025 NUM cana-3423 53 4 ) ) PUNCT cana-3423 54 1 https://internationalpubls.com https://internationalpubls.com X cana-3423 54 2 360 360 NUM cana-3423 54 3 2.2 2.2 NUM cana-3423 54 4 gaps gap NOUN cana-3423 54 5 in in ADP cana-3423 54 6 existing exist VERB cana-3423 54 7 research research NOUN cana-3423 54 8 these these DET cana-3423 54 9 studies study NOUN cana-3423 54 10 also also ADV cana-3423 54 11 reveal reveal VERB cana-3423 54 12 that that SCONJ cana-3423 54 13 there there PRON cana-3423 54 14 exist exist VERB cana-3423 54 15 several several ADJ cana-3423 54 16 limitations limitation NOUN cana-3423 54 17 in in ADP cana-3423 54 18 using use VERB cana-3423 54 19 topological topological ADJ cana-3423 54 20 methods method NOUN cana-3423 54 21 , , PUNCT cana-3423 54 22 as as SCONJ cana-3423 54 23 follows follow VERB cana-3423 54 24 . . PUNCT cana-3423 55 1 however however ADV cana-3423 55 2 , , PUNCT cana-3423 55 3 these these DET cana-3423 55 4 methods method NOUN cana-3423 55 5 have have AUX cana-3423 55 6 not not PART cana-3423 55 7 been be AUX cana-3423 55 8 fully fully ADV cana-3423 55 9 explored explore VERB cana-3423 55 10 in in ADP cana-3423 55 11 highly highly ADV cana-3423 55 12 irregular irregular ADJ cana-3423 55 13 domains domain NOUN cana-3423 55 14 , , PUNCT cana-3423 55 15 particularly particularly ADV cana-3423 55 16 in in ADP cana-3423 55 17 complicated complicated ADJ cana-3423 55 18 structures structure NOUN cana-3423 55 19 such such ADJ cana-3423 55 20 as as ADP cana-3423 55 21 systems system NOUN cana-3423 55 22 possessing possess VERB cana-3423 55 23 mixed mixed ADJ cana-3423 55 24 boundary boundary ADJ cana-3423 55 25 conditions condition NOUN cana-3423 55 26 such such ADJ cana-3423 55 27 as as ADP cana-3423 55 28 those those PRON cana-3423 55 29 arising arise VERB cana-3423 55 30 from from ADP cana-3423 55 31 porous porous ADJ cana-3423 55 32 media medium NOUN cana-3423 55 33 flow flow NOUN cana-3423 55 34 and and CCONJ cana-3423 55 35 nonlinear nonlinear ADJ cana-3423 55 36 elasticity elasticity NOUN cana-3423 55 37 ( ( PUNCT cana-3423 55 38 evans evans PROPN cana-3423 55 39 , , PUNCT cana-3423 55 40 2010 2010 NUM cana-3423 55 41 ) ) PUNCT cana-3423 55 42 . . PUNCT cana-3423 56 1 furthermore furthermore ADV cana-3423 56 2 , , PUNCT cana-3423 56 3 though though SCONJ cana-3423 56 4 degree degree NOUN cana-3423 56 5 theory theory NOUN cana-3423 56 6 fixed fix VERB cana-3423 56 7 - - PUNCT cana-3423 56 8 point point NOUN cana-3423 56 9 approaches approach NOUN cana-3423 56 10 to to ADP cana-3423 56 11 searching search VERB cana-3423 56 12 apply apply VERB cana-3423 56 13 to to ADP cana-3423 56 14 less less ADV cana-3423 56 15 - - PUNCT cana-3423 56 16 dimensional dimensional ADJ cana-3423 56 17 problems problem NOUN cana-3423 56 18 , , PUNCT cana-3423 56 19 their their PRON cana-3423 56 20 general general ADJ cana-3423 56 21 extensions extension NOUN cana-3423 56 22 to to ADP cana-3423 56 23 high high ADJ cana-3423 56 24 - - PUNCT cana-3423 56 25 dimensional dimensional ADJ cana-3423 56 26 nonlinear nonlinear ADJ cana-3423 56 27 systems system NOUN cana-3423 56 28 are be AUX cana-3423 56 29 still still ADV cana-3423 56 30 to to PART cana-3423 56 31 be be AUX cana-3423 56 32 investigated investigate VERB cana-3423 56 33 in in ADP cana-3423 56 34 detail detail NOUN cana-3423 56 35 ( ( PUNCT cana-3423 56 36 ambrosetti ambrosetti PROPN cana-3423 56 37 & & CCONJ cana-3423 56 38 prodi prodi PROPN cana-3423 56 39 , , PUNCT cana-3423 56 40 1995 1995 NUM cana-3423 56 41 ) ) PUNCT cana-3423 56 42 . . PUNCT cana-3423 57 1 it it PRON cana-3423 57 2 is be AUX cana-3423 57 3 also also ADV cana-3423 57 4 noteworthy noteworthy ADJ cana-3423 57 5 that that SCONJ cana-3423 57 6 despite despite SCONJ cana-3423 57 7 the the DET cana-3423 57 8 informativeness informativeness NOUN cana-3423 57 9 of of ADP cana-3423 57 10 the the DET cana-3423 57 11 proposed propose VERB cana-3423 57 12 topological topological ADJ cana-3423 57 13 methods method NOUN cana-3423 57 14 , , PUNCT cana-3423 57 15 there there PRON cana-3423 57 16 is be VERB cana-3423 57 17 a a DET cana-3423 57 18 scarcity scarcity NOUN cana-3423 57 19 of of ADP cana-3423 57 20 studies study NOUN cana-3423 57 21 addressing address VERB cana-3423 57 22 the the DET cana-3423 57 23 computational computational ADJ cana-3423 57 24 properties property NOUN cana-3423 57 25 of of ADP cana-3423 57 26 these these DET cana-3423 57 27 approaches approach NOUN cana-3423 57 28 , , PUNCT cana-3423 57 29 such such ADJ cana-3423 57 30 as as ADP cana-3423 57 31 the the DET cana-3423 57 32 convergence convergence NOUN cana-3423 57 33 rates rate NOUN cana-3423 57 34 or or CCONJ cana-3423 57 35 the the DET cana-3423 57 36 numerical numerical ADJ cana-3423 57 37 stability stability NOUN cana-3423 57 38 in in ADP cana-3423 57 39 high high ADJ cana-3423 57 40 - - PUNCT cana-3423 57 41 complexity complexity NOUN cana-3423 57 42 environments environment NOUN cana-3423 57 43 . . PUNCT cana-3423 58 1 2.3 2.3 NUM cana-3423 58 2 addressing address VERB cana-3423 58 3 the the DET cana-3423 58 4 research research NOUN cana-3423 58 5 gaps gap NOUN cana-3423 58 6 this this DET cana-3423 58 7 work work NOUN cana-3423 58 8 intends intend VERB cana-3423 58 9 to to PART cana-3423 58 10 fill fill VERB cana-3423 58 11 these these DET cana-3423 58 12 gaps gap NOUN cana-3423 58 13 by by ADP cana-3423 58 14 enhancing enhance VERB cana-3423 58 15 the the DET cana-3423 58 16 work work NOUN cana-3423 58 17 on on ADP cana-3423 58 18 topological topological ADJ cana-3423 58 19 methods method NOUN cana-3423 58 20 for for ADP cana-3423 58 21 nl nl PROPN cana-3423 58 22 pdes pde NOUN cana-3423 58 23 , , PUNCT cana-3423 58 24 especially especially ADV cana-3423 58 25 for for ADP cana-3423 58 26 multi multi ADJ cana-3423 58 27 - - ADJ cana-3423 58 28 dimensional dimensional ADJ cana-3423 58 29 and and CCONJ cana-3423 58 30 complicated complicated ADJ cana-3423 58 31 boundary boundary ADJ cana-3423 58 32 conditions condition NOUN cana-3423 58 33 . . PUNCT cana-3423 59 1 this this DET cana-3423 59 2 work work NOUN cana-3423 59 3 will will AUX cana-3423 59 4 create create VERB cana-3423 59 5 approaches approach NOUN cana-3423 59 6 by by ADP cana-3423 59 7 combining combine VERB cana-3423 59 8 new new ADJ cana-3423 59 9 ideas idea NOUN cana-3423 59 10 of of ADP cana-3423 59 11 fixed fix VERB cana-3423 59 12 - - PUNCT cana-3423 59 13 point point NOUN cana-3423 59 14 theory theory NOUN cana-3423 59 15 and and CCONJ cana-3423 59 16 variational variational ADJ cana-3423 59 17 topology topology NOUN cana-3423 59 18 to to PART cana-3423 59 19 resolve resolve VERB cana-3423 59 20 issues issue NOUN cana-3423 59 21 of of ADP cana-3423 59 22 the the DET cana-3423 59 23 existence existence NOUN cana-3423 59 24 and and CCONJ cana-3423 59 25 stability stability NOUN cana-3423 59 26 of of ADP cana-3423 59 27 solutions solution NOUN cana-3423 59 28 in in ADP cana-3423 59 29 irregular irregular ADJ cana-3423 59 30 domains domain NOUN cana-3423 59 31 and and CCONJ cana-3423 59 32 mixed mixed ADJ cana-3423 59 33 boundary boundary ADJ cana-3423 59 34 systems system NOUN cana-3423 59 35 . . PUNCT cana-3423 60 1 however however ADV cana-3423 60 2 , , PUNCT cana-3423 60 3 there there PRON cana-3423 60 4 are be VERB cana-3423 60 5 also also ADV cana-3423 60 6 focuses focus VERB cana-3423 60 7 in in ADP cana-3423 60 8 this this DET cana-3423 60 9 work work NOUN cana-3423 60 10 which which PRON cana-3423 60 11 are be AUX cana-3423 60 12 engineering engineer VERB cana-3423 60 13 aspects aspect NOUN cana-3423 60 14 of of ADP cana-3423 60 15 actual actual ADJ cana-3423 60 16 methods method NOUN cana-3423 60 17 , , PUNCT cana-3423 60 18 the the DET cana-3423 60 19 intricacies intricacy NOUN cana-3423 60 20 of of ADP cana-3423 60 21 convergence convergence NOUN cana-3423 60 22 rates rate NOUN cana-3423 60 23 , , PUNCT cana-3423 60 24 and and CCONJ cana-3423 60 25 numerical numerical ADJ cana-3423 60 26 stability stability NOUN cana-3423 60 27 needed need VERB cana-3423 60 28 for for ADP cana-3423 60 29 efficient efficient ADJ cana-3423 60 30 implementations implementation NOUN cana-3423 60 31 in in ADP cana-3423 60 32 engineering engineering NOUN cana-3423 60 33 and and CCONJ cana-3423 60 34 physical physical ADJ cana-3423 60 35 sciences science NOUN cana-3423 60 36 . . PUNCT cana-3423 61 1 to to ADP cana-3423 61 2 this this DET cana-3423 61 3 end end NOUN cana-3423 61 4 , , PUNCT cana-3423 61 5 this this DET cana-3423 61 6 research research NOUN cana-3423 61 7 enhances enhance VERB cana-3423 61 8 the the DET cana-3423 61 9 theoretical theoretical ADJ cana-3423 61 10 foundations foundation NOUN cana-3423 61 11 of of ADP cana-3423 61 12 the the DET cana-3423 61 13 field field NOUN cana-3423 61 14 , , PUNCT cana-3423 61 15 addressing address VERB cana-3423 61 16 the the DET cana-3423 61 17 nature nature NOUN cana-3423 61 18 of of ADP cana-3423 61 19 nonlinear nonlinear ADJ cana-3423 61 20 pdes pde NOUN cana-3423 61 21 , , PUNCT cana-3423 61 22 as as ADV cana-3423 61 23 well well ADV cana-3423 61 24 as as ADP cana-3423 61 25 providing provide VERB cana-3423 61 26 useful useful ADJ cana-3423 61 27 tools tool NOUN cana-3423 61 28 for for ADP cana-3423 61 29 mathematicians mathematician NOUN cana-3423 61 30 and and CCONJ cana-3423 61 31 engineers engineer NOUN cana-3423 61 32 . . PUNCT cana-3423 62 1 through through ADP cana-3423 62 2 the the DET cana-3423 62 3 application application NOUN cana-3423 62 4 of of ADP cana-3423 62 5 the the DET cana-3423 62 6 developed develop VERB cana-3423 62 7 approach approach NOUN cana-3423 62 8 to to ADP cana-3423 62 9 more more ADV cana-3423 62 10 complicated complicated ADJ cana-3423 62 11 problems problem NOUN cana-3423 62 12 , , PUNCT cana-3423 62 13 it it PRON cana-3423 62 14 is be AUX cana-3423 62 15 expected expect VERB cana-3423 62 16 that that SCONJ cana-3423 62 17 this this DET cana-3423 62 18 paper paper NOUN cana-3423 62 19 will will AUX cana-3423 62 20 create create VERB cana-3423 62 21 a a DET cana-3423 62 22 basis basis NOUN cana-3423 62 23 for for ADP cana-3423 62 24 further further ADJ cana-3423 62 25 investigations investigation NOUN cana-3423 62 26 of of ADP cana-3423 62 27 nl nl NOUN cana-3423 62 28 pdes pde NOUN cana-3423 62 29 in in ADP cana-3423 62 30 different different ADJ cana-3423 62 31 fields field NOUN cana-3423 62 32 . . PUNCT cana-3423 63 1 3 3 X cana-3423 63 2 . . NOUN cana-3423 63 3 mathematical mathematical ADJ cana-3423 63 4 preliminaries preliminary NOUN cana-3423 63 5 when when SCONJ cana-3423 63 6 using use VERB cana-3423 63 7 topological topological ADJ cana-3423 63 8 methods method NOUN cana-3423 63 9 to to PART cana-3423 63 10 solve solve VERB cana-3423 63 11 nonlinear nonlinear ADJ cana-3423 63 12 pdes pde NOUN cana-3423 63 13 , , PUNCT cana-3423 63 14 there there PRON cana-3423 63 15 is be VERB cana-3423 63 16 a a DET cana-3423 63 17 necessity necessity NOUN cana-3423 63 18 to to PART cana-3423 63 19 lay lay VERB cana-3423 63 20 the the DET cana-3423 63 21 rigorous rigorous ADJ cana-3423 63 22 mathematical mathematical ADJ cana-3423 63 23 framework framework NOUN cana-3423 63 24 for for ADP cana-3423 63 25 these these DET cana-3423 63 26 methods method NOUN cana-3423 63 27 , , PUNCT cana-3423 63 28 discussing discuss VERB cana-3423 63 29 the the DET cana-3423 63 30 theorem theorem NOUN cana-3423 63 31 and and CCONJ cana-3423 63 32 concepts concept NOUN cana-3423 63 33 forming form VERB cana-3423 63 34 the the DET cana-3423 63 35 base base NOUN cana-3423 63 36 for for ADP cana-3423 63 37 them they PRON cana-3423 63 38 . . PUNCT cana-3423 64 1 this this DET cana-3423 64 2 section section NOUN cana-3423 64 3 explains explain VERB cana-3423 64 4 the the DET cana-3423 64 5 principal principal ADJ cana-3423 64 6 properties property NOUN cana-3423 64 7 and and CCONJ cana-3423 64 8 main main ADJ cana-3423 64 9 solution solution NOUN cana-3423 64 10 techniques technique NOUN cana-3423 64 11 of of ADP cana-3423 64 12 this this DET cana-3423 64 13 work work NOUN cana-3423 64 14 emphasizing emphasize VERB cana-3423 64 15 nonlinear nonlinear ADJ cana-3423 64 16 pdes pde NOUN cana-3423 64 17 in in ADP cana-3423 64 18 fixed fix VERB cana-3423 64 19 - - PUNCT cana-3423 64 20 point point NOUN cana-3423 64 21 theory theory NOUN cana-3423 64 22 , , PUNCT cana-3423 64 23 degree degree NOUN cana-3423 64 24 theory theory NOUN cana-3423 64 25 , , PUNCT cana-3423 64 26 and and CCONJ cana-3423 64 27 functional functional ADJ cana-3423 64 28 analysis analysis NOUN cana-3423 64 29 . . PUNCT cana-3423 65 1 3.1 3.1 NUM cana-3423 65 2 mathematical mathematical ADJ cana-3423 65 3 framework framework NOUN cana-3423 65 4 consider consider VERB cana-3423 65 5 a a DET cana-3423 65 6 general general ADJ cana-3423 65 7 nonlinear nonlinear ADJ cana-3423 65 8 partial partial ADJ cana-3423 65 9 differential differential NOUN cana-3423 65 10 equation equation NOUN cana-3423 65 11 represented represent VERB cana-3423 65 12 as as ADP cana-3423 65 13 : : PUNCT cana-3423 65 14 β„’(𝑒 β„’(𝑒 NUM cana-3423 65 15 ) ) PUNCT cana-3423 65 16 = = SYM cana-3423 66 1 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-3423 66 2 , , PUNCT cana-3423 66 3 𝑒 𝑒 X cana-3423 66 4 , , PUNCT cana-3423 66 5 βˆ‡π‘’ βˆ‡π‘’ PROPN cana-3423 66 6 ) ) PUNCT cana-3423 66 7 where where SCONJ cana-3423 66 8 β„’ β„’ PROPN cana-3423 66 9 is be AUX cana-3423 66 10 a a DET cana-3423 66 11 differential differential ADJ cana-3423 66 12 operator operator NOUN cana-3423 66 13 , , PUNCT cana-3423 66 14 𝑒 𝑒 PROPN cana-3423 66 15 is be AUX cana-3423 66 16 the the DET cana-3423 66 17 unknown unknown ADJ cana-3423 66 18 function function NOUN cana-3423 66 19 , , PUNCT cana-3423 66 20 and and CCONJ cana-3423 66 21 𝑓 𝑓 PRON cana-3423 66 22 is be AUX cana-3423 66 23 a a DET cana-3423 66 24 nonlinear nonlinear ADJ cana-3423 66 25 function function NOUN cana-3423 66 26 of of ADP cana-3423 66 27 𝑒 𝑒 NOUN cana-3423 66 28 and and CCONJ cana-3423 66 29 its its PRON cana-3423 66 30 gradient gradient NOUN cana-3423 66 31 βˆ‡π‘’ βˆ‡π‘’ PROPN cana-3423 66 32 in in ADP cana-3423 66 33 a a DET cana-3423 66 34 given give VERB cana-3423 66 35 domain domain NOUN cana-3423 66 36 Ο‰ Ο‰ NOUN cana-3423 66 37 . . PUNCT cana-3423 67 1 a a DET cana-3423 67 2 primary primary ADJ cana-3423 67 3 focus focus NOUN cana-3423 67 4 in in ADP cana-3423 67 5 solving solve VERB cana-3423 67 6 such such ADJ cana-3423 67 7 equations equation NOUN cana-3423 67 8 is be AUX cana-3423 67 9 to to PART cana-3423 67 10 determine determine VERB cana-3423 67 11 the the DET cana-3423 67 12 existence existence NOUN cana-3423 67 13 , , PUNCT cana-3423 67 14 uniqueness uniqueness NOUN cana-3423 67 15 , , PUNCT cana-3423 67 16 and and CCONJ cana-3423 67 17 stability stability NOUN cana-3423 67 18 of of ADP cana-3423 67 19 solutions solution NOUN cana-3423 67 20 under under ADP cana-3423 67 21 various various ADJ cana-3423 67 22 boundary boundary ADJ cana-3423 67 23 conditions condition NOUN cana-3423 67 24 . . PUNCT cana-3423 68 1 often often ADV cana-3423 68 2 fixed fix VERB cana-3423 68 3 - - PUNCT cana-3423 68 4 point point NOUN cana-3423 68 5 theorems theorem NOUN cana-3423 68 6 are be AUX cana-3423 68 7 employed employ VERB cana-3423 68 8 as as ADP cana-3423 68 9 that that PRON cana-3423 68 10 , , PUNCT cana-3423 68 11 banach banach ADV cana-3423 68 12 fixed fix VERB cana-3423 68 13 point point NOUN cana-3423 68 14 theorem theorem VERB cana-3423 68 15 states state NOUN cana-3423 68 16 that that SCONJ cana-3423 68 17 mapping mapping NOUN cana-3423 68 18 that that PRON cana-3423 68 19 is be AUX cana-3423 68 20 contraction contraction NOUN cana-3423 68 21 on on ADP cana-3423 68 22 complete complete ADJ cana-3423 68 23 metric metric ADJ cana-3423 68 24 space space NOUN cana-3423 68 25 has have VERB cana-3423 68 26 exactly exactly ADV cana-3423 68 27 one one NUM cana-3423 68 28 fixed fix VERB cana-3423 68 29 point point NOUN cana-3423 68 30 ( ( PUNCT cana-3423 68 31 evans evans PROPN cana-3423 68 32 , , PUNCT cana-3423 68 33 2010 2010 NUM cana-3423 68 34 ) ) PUNCT cana-3423 68 35 . . PUNCT cana-3423 69 1 this this DET cana-3423 69 2 theorem theorem NOUN cana-3423 69 3 gives give VERB cana-3423 69 4 a a DET cana-3423 69 5 start start NOUN cana-3423 69 6 for for ADP cana-3423 69 7 the the DET cana-3423 69 8 application application NOUN cana-3423 69 9 of of ADP cana-3423 69 10 iteration iteration NOUN cana-3423 69 11 methods method NOUN cana-3423 69 12 in in ADP cana-3423 69 13 the the DET cana-3423 69 14 formation formation NOUN cana-3423 69 15 of of ADP cana-3423 69 16 solutions solution NOUN cana-3423 69 17 for for ADP cana-3423 69 18 nonlinear nonlinear ADJ cana-3423 69 19 partial partial ADJ cana-3423 69 20 differential differential NOUN cana-3423 69 21 equations equation NOUN cana-3423 69 22 where where SCONJ cana-3423 69 23 we we PRON cana-3423 69 24 are be AUX cana-3423 69 25 sure sure ADJ cana-3423 69 26 of of ADP cana-3423 69 27 convergence convergence NOUN cana-3423 69 28 . . PUNCT cana-3423 70 1 theorem theorem VERB cana-3423 70 2 3.1 3.1 NUM cana-3423 70 3 : : PUNCT cana-3423 70 4 banach banach NOUN cana-3423 70 5 fixed fix VERB cana-3423 70 6 - - PUNCT cana-3423 70 7 point point NOUN cana-3423 70 8 theorem theorem NOUN cana-3423 70 9 ( ( PUNCT cana-3423 70 10 evans evans PROPN cana-3423 70 11 , , PUNCT cana-3423 70 12 2010 2010 NUM cana-3423 70 13 ) ) PUNCT cana-3423 71 1 let let VERB cana-3423 71 2 ( ( PUNCT cana-3423 71 3 𝑋 𝑋 NOUN cana-3423 71 4 , , PUNCT cana-3423 71 5 𝑑 𝑑 NOUN cana-3423 71 6 ) ) PUNCT cana-3423 71 7 be be VERB cana-3423 71 8 a a DET cana-3423 71 9 complete complete ADJ cana-3423 71 10 metric metric ADJ cana-3423 71 11 space space NOUN cana-3423 71 12 , , PUNCT cana-3423 71 13 and and CCONJ cana-3423 71 14 let let VERB cana-3423 71 15 𝑇 𝑇 PROPN cana-3423 71 16 : : PUNCT cana-3423 71 17 𝑋 𝑋 PROPN cana-3423 71 18 β†’ β†’ SYM cana-3423 71 19 𝑋 𝑋 PROPN cana-3423 71 20 be be VERB cana-3423 71 21 a a DET cana-3423 71 22 contraction contraction NOUN cana-3423 71 23 mapping mapping NOUN cana-3423 71 24 , , PUNCT cana-3423 71 25 meaning mean VERB cana-3423 71 26 there there PRON cana-3423 71 27 exists exist VERB cana-3423 71 28 a a DET cana-3423 71 29 constant constant ADJ cana-3423 71 30 0 0 NUM cana-3423 71 31 ≀ ≀ NOUN cana-3423 72 1 π‘˜ π‘˜ PRON cana-3423 72 2 < < X cana-3423 72 3 1 1 NUM cana-3423 72 4 such such ADJ cana-3423 72 5 that that PRON cana-3423 72 6 for for ADP cana-3423 72 7 all all PRON cana-3423 72 8 π‘₯ π‘₯ PROPN cana-3423 72 9 , , PUNCT cana-3423 72 10 𝑦 𝑦 NOUN cana-3423 72 11 ∈ ∈ PROPN cana-3423 72 12 𝑋 𝑋 PROPN cana-3423 72 13 , , PUNCT cana-3423 72 14 𝑑(𝑇(π‘₯ 𝑑(𝑇(π‘₯ PROPN cana-3423 72 15 ) ) PUNCT cana-3423 72 16 , , PUNCT cana-3423 72 17 𝑇(𝑦 𝑇(𝑦 VERB cana-3423 72 18 ) ) PUNCT cana-3423 72 19 ) ) PUNCT cana-3423 72 20 ≀ ≀ NOUN cana-3423 72 21 π‘˜π‘‘(π‘₯ π‘˜π‘‘(π‘₯ NUM cana-3423 72 22 , , PUNCT cana-3423 72 23 𝑦 𝑦 X cana-3423 72 24 ) ) PUNCT cana-3423 72 25 communications communication NOUN cana-3423 72 26 on on ADP cana-3423 72 27 applied apply VERB cana-3423 72 28 nonlinear nonlinear ADJ cana-3423 72 29 analysis analysis NOUN cana-3423 72 30 issn issn NOUN cana-3423 72 31 : : PUNCT cana-3423 72 32 1074 1074 NUM cana-3423 72 33 - - PUNCT cana-3423 72 34 133x 133x NUM cana-3423 72 35 vol vol NOUN cana-3423 72 36 32 32 NUM cana-3423 72 37 no no NOUN cana-3423 72 38 . . PUNCT cana-3423 73 1 7s 7 NOUN cana-3423 73 2 ( ( PUNCT cana-3423 73 3 2025 2025 NUM cana-3423 73 4 ) ) PUNCT cana-3423 74 1 https://internationalpubls.com https://internationalpubls.com X cana-3423 74 2 361 361 NUM cana-3423 74 3 then then ADV cana-3423 74 4 𝑇 𝑇 PROPN cana-3423 74 5 has have VERB cana-3423 74 6 a a DET cana-3423 74 7 unique unique ADJ cana-3423 74 8 fixed fix VERB cana-3423 74 9 point point NOUN cana-3423 74 10 π‘₯βˆ— π‘₯βˆ— PROPN cana-3423 74 11 ∈ ∈ PROPN cana-3423 74 12 𝑋 𝑋 PROPN cana-3423 74 13 such such ADJ cana-3423 74 14 that that PRON cana-3423 74 15 𝑇(π‘₯βˆ— 𝑇(π‘₯βˆ— PROPN cana-3423 74 16 ) ) PUNCT cana-3423 74 17 = = SYM cana-3423 74 18 π‘₯βˆ—. π‘₯βˆ—. NOUN cana-3423 74 19 applying apply VERB cana-3423 74 20 the the DET cana-3423 74 21 banach banach ADV cana-3423 74 22 fixed fix VERB cana-3423 74 23 - - PUNCT cana-3423 74 24 point point NOUN cana-3423 74 25 theorem theorem NOUN cana-3423 74 26 allows allow VERB cana-3423 74 27 for for ADP cana-3423 74 28 constructing construct VERB cana-3423 74 29 solutions solution NOUN cana-3423 74 30 iteratively iteratively ADV cana-3423 74 31 for for ADP cana-3423 74 32 pdes pde NOUN cana-3423 74 33 where where SCONJ cana-3423 74 34 the the DET cana-3423 74 35 operator operator NOUN cana-3423 74 36 . . PUNCT cana-3423 75 1 β„’ β„’ NOUN cana-3423 75 2 satisfies satisfie NOUN cana-3423 75 3 contraction contraction VERB cana-3423 75 4 mapping mapping NOUN cana-3423 75 5 properties property NOUN cana-3423 75 6 . . PUNCT cana-3423 76 1 3.2 3.2 NUM cana-3423 76 2 topological topological NOUN cana-3423 76 3 degree degree NOUN cana-3423 76 4 theory theory NOUN cana-3423 76 5 the the DET cana-3423 76 6 other other ADJ cana-3423 76 7 indispensable indispensable ADJ cana-3423 76 8 assistant assistant NOUN cana-3423 76 9 that that PRON cana-3423 76 10 has have AUX cana-3423 76 11 been be AUX cana-3423 76 12 used use VERB cana-3423 76 13 frequently frequently ADV cana-3423 76 14 in in ADP cana-3423 76 15 the the DET cana-3423 76 16 study study NOUN cana-3423 76 17 of of ADP cana-3423 76 18 nonlinear nonlinear ADJ cana-3423 76 19 pdes pde NOUN cana-3423 76 20 is be AUX cana-3423 76 21 the the DET cana-3423 76 22 topological topological ADJ cana-3423 76 23 degree degree NOUN cana-3423 76 24 theory theory NOUN cana-3423 76 25 , , PUNCT cana-3423 76 26 as as SCONJ cana-3423 76 27 will will AUX cana-3423 76 28 be be AUX cana-3423 76 29 seen see VERB cana-3423 76 30 in in ADP cana-3423 76 31 the the DET cana-3423 76 32 next next ADJ cana-3423 76 33 sections section NOUN cana-3423 76 34 where where SCONJ cana-3423 76 35 the the DET cana-3423 76 36 existence existence NOUN cana-3423 76 37 of of ADP cana-3423 76 38 solutions solution NOUN cana-3423 76 39 can can AUX cana-3423 76 40 not not PART cana-3423 76 41 be be AUX cana-3423 76 42 proved prove VERB cana-3423 76 43 via via ADP cana-3423 76 44 contraction contraction NOUN cana-3423 76 45 mappings mapping NOUN cana-3423 76 46 . . PUNCT cana-3423 77 1 this this PRON cana-3423 77 2 is be AUX cana-3423 77 3 achieved achieve VERB cana-3423 77 4 by by ADP cana-3423 77 5 invoking invoke VERB cana-3423 77 6 degree degree NOUN cana-3423 77 7 theory theory NOUN cana-3423 77 8 which which PRON cana-3423 77 9 allows allow VERB cana-3423 77 10 the the DET cana-3423 77 11 counting counting NOUN cana-3423 77 12 of of ADP cana-3423 77 13 the the DET cana-3423 77 14 number number NOUN cana-3423 77 15 of of ADP cana-3423 77 16 solutions solution NOUN cana-3423 77 17 by by ADP cana-3423 77 18 assigning assign VERB cana-3423 77 19 a a DET cana-3423 77 20 β€œ " PUNCT cana-3423 77 21 degree degree NOUN cana-3423 77 22 ” " PUNCT cana-3423 77 23 to to PART cana-3423 77 24 mappings mapping NOUN cana-3423 77 25 subject subject ADJ cana-3423 77 26 to to ADP cana-3423 77 27 specified specify VERB cana-3423 77 28 boundary boundary ADJ cana-3423 77 29 constraints constraint NOUN cana-3423 77 30 ( ( PUNCT cana-3423 77 31 ambrosetti ambrosetti PROPN cana-3423 77 32 & & CCONJ cana-3423 77 33 prodi prodi PROPN cana-3423 77 34 , , PUNCT cana-3423 77 35 1995 1995 NUM cana-3423 77 36 ) ) PUNCT cana-3423 77 37 . . PUNCT cana-3423 78 1 this this DET cana-3423 78 2 approach approach NOUN cana-3423 78 3 is be AUX cana-3423 78 4 quite quite ADV cana-3423 78 5 helpful helpful ADJ cana-3423 78 6 in in ADP cana-3423 78 7 high high ADJ cana-3423 78 8 dimensions dimension NOUN cana-3423 78 9 where where SCONJ cana-3423 78 10 regular regular ADJ cana-3423 78 11 methodologies methodology NOUN cana-3423 78 12 may may AUX cana-3423 78 13 not not PART cana-3423 78 14 be be AUX cana-3423 78 15 effective effective ADJ cana-3423 78 16 . . PUNCT cana-3423 79 1 for for ADP cana-3423 79 2 a a DET cana-3423 79 3 continuous continuous ADJ cana-3423 79 4 mapping mapping NOUN cana-3423 79 5 𝐹 𝐹 PROPN cana-3423 79 6 : : PUNCT cana-3423 79 7 Ο‰ Ο‰ PROPN cana-3423 79 8 β†’ β†’ PUNCT cana-3423 79 9 ℝ𝑛 ℝ𝑛 PROPN cana-3423 79 10 and and CCONJ cana-3423 79 11 a a DET cana-3423 79 12 domain domain NOUN cana-3423 79 13 Ο‰ Ο‰ NOUN cana-3423 79 14 with with ADP cana-3423 79 15 boundary boundary ADJ cana-3423 79 16 βˆ‚Ο‰ βˆ‚Ο‰ PROPN cana-3423 79 17 , , PUNCT cana-3423 79 18 the the DET cana-3423 79 19 degree degree NOUN cana-3423 79 20 of of ADP cana-3423 79 21 𝐹 𝐹 PROPN cana-3423 79 22 at at ADP cana-3423 79 23 a a DET cana-3423 79 24 point point NOUN cana-3423 79 25 𝑦 𝑦 NOUN cana-3423 79 26 ∈ ∈ NOUN cana-3423 79 27 ℝ𝑛a ℝ𝑛a PROPN cana-3423 79 28 be be AUX cana-3423 79 29 defined define VERB cana-3423 79 30 as as SCONJ cana-3423 79 31 follows follow VERB cana-3423 79 32 : : PUNCT cana-3423 79 33 deg deg PROPN cana-3423 79 34 ( ( PUNCT cana-3423 79 35 𝐹 𝐹 PROPN cana-3423 79 36 , , PUNCT cana-3423 79 37 Ο‰ Ο‰ PROPN cana-3423 79 38 , , PUNCT cana-3423 79 39 𝑦 𝑦 NOUN cana-3423 79 40 ) ) PUNCT cana-3423 79 41 = = SYM cana-3423 79 42 βˆ‘ βˆ‘ ADP cana-3423 79 43 β€Š β€Š SPACE cana-3423 79 44 π‘₯βˆˆπΉβˆ’1(𝑦 π‘₯βˆˆπΉβˆ’1(𝑦 PROPN cana-3423 79 45 ) ) PUNCT cana-3423 79 46 sign sign VERB cana-3423 79 47 det det PROPN cana-3423 79 48 ( ( PUNCT cana-3423 79 49 𝐷𝐹(π‘₯ 𝐷𝐹(π‘₯ PROPN cana-3423 79 50 ) ) PUNCT cana-3423 79 51 ) ) PUNCT cana-3423 80 1 where where SCONJ cana-3423 80 2 sign sign VERB cana-3423 80 3 det det PROPN cana-3423 80 4 ( ( PUNCT cana-3423 80 5 𝐷𝐹(π‘₯ 𝐷𝐹(π‘₯ PROPN cana-3423 80 6 ) ) PUNCT cana-3423 80 7 ) ) PUNCT cana-3423 80 8 represents represent VERB cana-3423 80 9 the the DET cana-3423 80 10 jacobian jacobian ADJ cana-3423 80 11 determinant determinant PROPN cana-3423 80 12 's 's PART cana-3423 80 13 sign sign NOUN cana-3423 80 14 at at ADP cana-3423 80 15 each each DET cana-3423 80 16 preimage preimage NOUN cana-3423 80 17 point point NOUN cana-3423 80 18 π‘₯ π‘₯ PROPN cana-3423 80 19 of of ADP cana-3423 80 20 𝑦 𝑦 PROPN cana-3423 80 21 ( ( PUNCT cana-3423 80 22 rabinowitz rabinowitz PROPN cana-3423 80 23 , , PUNCT cana-3423 80 24 2015 2015 NUM cana-3423 80 25 ) ) PUNCT cana-3423 80 26 . . PUNCT cana-3423 81 1 this this DET cana-3423 81 2 degree degree NOUN cana-3423 81 3 remains remain VERB cana-3423 81 4 the the DET cana-3423 81 5 same same ADJ cana-3423 81 6 regardless regardless ADV cana-3423 81 7 of of ADP cana-3423 81 8 continuous continuous ADJ cana-3423 81 9 transformations transformation NOUN cana-3423 81 10 making make VERB cana-3423 81 11 it it PRON cana-3423 81 12 possible possible ADJ cana-3423 81 13 to to PART cana-3423 81 14 develop develop VERB cana-3423 81 15 the the DET cana-3423 81 16 most most ADV cana-3423 81 17 efficient efficient ADJ cana-3423 81 18 instrument instrument NOUN cana-3423 81 19 for for ADP cana-3423 81 20 the the DET cana-3423 81 21 existence existence NOUN cana-3423 81 22 of of ADP cana-3423 81 23 solutions solution NOUN cana-3423 81 24 in in ADP cana-3423 81 25 rather rather ADV cana-3423 81 26 complicated complicated ADJ cana-3423 81 27 regions region NOUN cana-3423 81 28 . . PUNCT cana-3423 82 1 3.3 3.3 NUM cana-3423 82 2 critical critical ADJ cana-3423 82 3 point point NOUN cana-3423 82 4 theory theory NOUN cana-3423 82 5 critical critical ADJ cana-3423 82 6 point point NOUN cana-3423 82 7 theory theory NOUN cana-3423 82 8 is be AUX cana-3423 82 9 fundamental fundamental ADJ cana-3423 82 10 in in ADP cana-3423 82 11 variational variational ADJ cana-3423 82 12 approaches approach NOUN cana-3423 82 13 to to ADP cana-3423 82 14 nonlinear nonlinear ADJ cana-3423 82 15 pdes pde NOUN cana-3423 82 16 , , PUNCT cana-3423 82 17 particularly particularly ADV cana-3423 82 18 when when SCONJ cana-3423 82 19 solutions solution NOUN cana-3423 82 20 can can AUX cana-3423 82 21 be be AUX cana-3423 82 22 formulated formulate VERB cana-3423 82 23 as as ADP cana-3423 82 24 minimizers minimizer NOUN cana-3423 82 25 of of ADP cana-3423 82 26 a a DET cana-3423 82 27 function function NOUN cana-3423 82 28 . . PUNCT cana-3423 83 1 for for ADP cana-3423 83 2 a a DET cana-3423 83 3 given give VERB cana-3423 83 4 functional functional ADJ cana-3423 83 5 𝐽 𝐽 PROPN cana-3423 83 6 : : PUNCT cana-3423 83 7 𝑋 𝑋 PROPN cana-3423 83 8 β†’ β†’ SYM cana-3423 83 9 ℝ ℝ PROPN cana-3423 83 10 , , PUNCT cana-3423 83 11 where where SCONJ cana-3423 83 12 𝑋 𝑋 PROPN cana-3423 83 13 is be AUX cana-3423 83 14 a a DET cana-3423 83 15 banach banach NOUN cana-3423 83 16 space space NOUN cana-3423 83 17 , , PUNCT cana-3423 83 18 a a DET cana-3423 83 19 point point NOUN cana-3423 83 20 𝑒 𝑒 ADP cana-3423 83 21 ∈ ∈ NOUN cana-3423 83 22 𝑋 𝑋 NOUN cana-3423 83 23 is be AUX cana-3423 83 24 called call VERB cana-3423 83 25 a a DET cana-3423 83 26 critical critical ADJ cana-3423 83 27 point point NOUN cana-3423 83 28 of of ADP cana-3423 83 29 𝐽 𝐽 PROPN cana-3423 83 30 if if SCONJ cana-3423 83 31 the the DET cana-3423 83 32 derivative derivative NOUN cana-3423 83 33 of of ADP cana-3423 83 34 𝐽 𝐽 PROPN cana-3423 83 35 at at ADP cana-3423 83 36 𝑒 𝑒 X cana-3423 83 37 vanishes vanish VERB cana-3423 83 38 , , PUNCT cana-3423 83 39 i.e. i.e. X cana-3423 83 40 , , PUNCT cana-3423 83 41 𝐽′(𝑒 𝐽′(𝑒 PROPN cana-3423 83 42 ) ) PUNCT cana-3423 84 1 = = PUNCT cana-3423 84 2 0 0 X cana-3423 84 3 . . PUNCT cana-3423 85 1 this this DET cana-3423 85 2 principle principle NOUN cana-3423 85 3 is be AUX cana-3423 85 4 used use VERB cana-3423 85 5 to to PART cana-3423 85 6 locate locate VERB cana-3423 85 7 solutions solution NOUN cana-3423 85 8 by by ADP cana-3423 85 9 identifying identify VERB cana-3423 85 10 critical critical ADJ cana-3423 85 11 points point NOUN cana-3423 85 12 of of ADP cana-3423 85 13 functionals functional NOUN cana-3423 85 14 associated associate VERB cana-3423 85 15 with with ADP cana-3423 85 16 the the DET cana-3423 85 17 pdes pde NOUN cana-3423 85 18 in in ADP cana-3423 85 19 question question NOUN cana-3423 85 20 ( ( PUNCT cana-3423 85 21 chang chang PROPN cana-3423 85 22 , , PUNCT cana-3423 85 23 2013 2013 NUM cana-3423 85 24 ) ) PUNCT cana-3423 85 25 . . PUNCT cana-3423 86 1 theorem theorem ADJ cana-3423 86 2 3.2 3.2 NUM cana-3423 86 3 : : PUNCT cana-3423 86 4 mountain mountain NOUN cana-3423 86 5 pass pass NOUN cana-3423 86 6 theorem theorem NOUN cana-3423 86 7 ( ( PUNCT cana-3423 86 8 ambrosetti ambrosetti PROPN cana-3423 86 9 & & CCONJ cana-3423 86 10 prodi prodi PROPN cana-3423 86 11 , , PUNCT cana-3423 86 12 1995 1995 NUM cana-3423 86 13 ) ) PUNCT cana-3423 86 14 let let VERB cana-3423 86 15 𝑋 𝑋 NOUN cana-3423 86 16 be be AUX cana-3423 86 17 a a DET cana-3423 86 18 real real ADJ cana-3423 86 19 banach banach NOUN cana-3423 86 20 space space NOUN cana-3423 86 21 , , PUNCT cana-3423 86 22 and and CCONJ cana-3423 86 23 let let VERB cana-3423 86 24 𝐽 𝐽 PRON cana-3423 86 25 ∈ ∈ PROPN cana-3423 86 26 𝐢1(𝑋 𝐢1(𝑋 PROPN cana-3423 86 27 , , PUNCT cana-3423 86 28 ℝ)satisfy ℝ)satisfy VERB cana-3423 86 29 the the DET cana-3423 86 30 palais palais PROPN cana-3423 86 31 - - PUNCT cana-3423 86 32 smale smale ADJ cana-3423 86 33 condition condition NOUN cana-3423 86 34 . . PUNCT cana-3423 87 1 suppose suppose VERB cana-3423 87 2 there there PRON cana-3423 87 3 exist exist VERB cana-3423 87 4 constants constant NOUN cana-3423 87 5 . . PUNCT cana-3423 88 1 𝛼 𝛼 X cana-3423 88 2 , , PUNCT cana-3423 88 3 𝛽 𝛽 NOUN cana-3423 88 4 > > X cana-3423 88 5 0 0 NUM cana-3423 88 6 and and CCONJ cana-3423 88 7 points point NOUN cana-3423 88 8 𝑒0 𝑒0 PROPN cana-3423 88 9 , , PUNCT cana-3423 88 10 𝑒1 𝑒1 PROPN cana-3423 88 11 ∈ ∈ PROPN cana-3423 88 12 𝑋 𝑋 PROPN cana-3423 88 13 such such ADJ cana-3423 88 14 that that DET cana-3423 88 15 𝐽(𝑒0 𝐽(𝑒0 NOUN cana-3423 88 16 ) ) PUNCT cana-3423 88 17 < < X cana-3423 88 18 𝛼 𝛼 X cana-3423 88 19 , , PUNCT cana-3423 88 20 𝐽(𝑒1 𝐽(𝑒1 NOUN cana-3423 88 21 ) ) PUNCT cana-3423 88 22 > > PUNCT cana-3423 89 1 𝛽 𝛽 NOUN cana-3423 89 2 , , PUNCT cana-3423 89 3 and and CCONJ cana-3423 89 4 𝐽as 𝐽as ADP cana-3423 89 5 a a DET cana-3423 89 6 " " PUNCT cana-3423 89 7 mountain mountain NOUN cana-3423 89 8 pass pass NOUN cana-3423 89 9 " " PUNCT cana-3423 89 10 structure structure NOUN cana-3423 89 11 . . PUNCT cana-3423 90 1 then then ADV cana-3423 90 2 𝐽 𝐽 PROPN cana-3423 90 3 has have VERB cana-3423 90 4 a a DET cana-3423 90 5 critical critical ADJ cana-3423 90 6 point point NOUN cana-3423 90 7 𝑒 𝑒 ADP cana-3423 90 8 β‰  β‰  PROPN cana-3423 90 9 𝑒0 𝑒0 NOUN cana-3423 90 10 , , PUNCT cana-3423 90 11 𝑒1 𝑒1 PROPN cana-3423 90 12 with with ADP cana-3423 90 13 𝐽(𝑒 𝐽(𝑒 NUM cana-3423 90 14 ) ) PUNCT cana-3423 90 15 β‰₯ β‰₯ NOUN cana-3423 90 16 𝛼. 𝛼. NOUN cana-3423 90 17 through through ADP cana-3423 90 18 the the DET cana-3423 90 19 application application NOUN cana-3423 90 20 of of ADP cana-3423 90 21 the the DET cana-3423 90 22 mountain mountain NOUN cana-3423 90 23 pass pass NOUN cana-3423 90 24 theorem theorem NOUN cana-3423 90 25 , , PUNCT cana-3423 90 26 one one PRON cana-3423 90 27 can can AUX cana-3423 90 28 find find VERB cana-3423 90 29 solutions solution NOUN cana-3423 90 30 by by ADP cana-3423 90 31 determining determine VERB cana-3423 90 32 the the DET cana-3423 90 33 saddle saddle ADJ cana-3423 90 34 points point NOUN cana-3423 90 35 of of ADP cana-3423 90 36 certain certain ADJ cana-3423 90 37 related related ADJ cana-3423 90 38 functionals functional NOUN cana-3423 90 39 to to PART cana-3423 90 40 establish establish VERB cana-3423 90 41 conditions condition NOUN cana-3423 90 42 on on ADP cana-3423 90 43 the the DET cana-3423 90 44 multiplicity multiplicity NOUN cana-3423 90 45 and and CCONJ cana-3423 90 46 stability stability NOUN cana-3423 90 47 of of ADP cana-3423 90 48 solutions solution NOUN cana-3423 90 49 in in ADP cana-3423 90 50 some some DET cana-3423 90 51 classes class NOUN cana-3423 90 52 of of ADP cana-3423 90 53 nonlinear nonlinear ADJ cana-3423 90 54 pde pde PROPN cana-3423 90 55 systems system NOUN cana-3423 90 56 ( ( PUNCT cana-3423 90 57 mawhin mawhin NOUN cana-3423 90 58 , , PUNCT cana-3423 90 59 2017 2017 NUM cana-3423 90 60 ) ) PUNCT cana-3423 90 61 . . PUNCT cana-3423 91 1 3.4 3.4 NUM cana-3423 91 2 functional functional ADJ cana-3423 91 3 analysis analysis NOUN cana-3423 91 4 tools tool NOUN cana-3423 91 5 a a DET cana-3423 91 6 foundational foundational ADJ cana-3423 91 7 element element NOUN cana-3423 91 8 in in ADP cana-3423 91 9 analyzing analyze VERB cana-3423 91 10 nonlinear nonlinear ADJ cana-3423 91 11 pdes pde NOUN cana-3423 91 12 is be AUX cana-3423 91 13 the the DET cana-3423 91 14 sobolev sobolev ADJ cana-3423 91 15 space space NOUN cana-3423 91 16 . . PUNCT cana-3423 92 1 π‘Šπ‘˜,𝑝(Ο‰ π‘Šπ‘˜,𝑝(Ο‰ PROPN cana-3423 92 2 ) ) PUNCT cana-3423 92 3 , , PUNCT cana-3423 92 4 which which PRON cana-3423 92 5 facilitates facilitate VERB cana-3423 92 6 working work VERB cana-3423 92 7 within within ADP cana-3423 92 8 spaces space NOUN cana-3423 92 9 where where SCONJ cana-3423 92 10 solutions solution NOUN cana-3423 92 11 possess possess VERB cana-3423 92 12 weak weak ADJ cana-3423 92 13 derivatives derivative NOUN cana-3423 92 14 . . PUNCT cana-3423 93 1 for for ADP cana-3423 93 2 example example NOUN cana-3423 93 3 , , PUNCT cana-3423 93 4 the the DET cana-3423 93 5 sobolev sobolev NOUN cana-3423 93 6 embedding embed VERB cana-3423 93 7 theorem theorem ADJ cana-3423 93 8 states state NOUN cana-3423 93 9 that that SCONJ cana-3423 93 10 if if SCONJ cana-3423 93 11 Ο‰ Ο‰ PROPN cana-3423 93 12 is be AUX cana-3423 93 13 a a DET cana-3423 93 14 bounded bounded ADJ cana-3423 93 15 open open ADJ cana-3423 93 16 subset subset NOUN cana-3423 93 17 of of ADP cana-3423 93 18 ℝ𝑛 ℝ𝑛 PROPN cana-3423 93 19 , , PUNCT cana-3423 93 20 then then ADV cana-3423 93 21 the the DET cana-3423 93 22 space space NOUN cana-3423 93 23 π‘Šπ‘˜,𝑝(Ο‰ π‘Šπ‘˜,𝑝(Ο‰ NOUN cana-3423 93 24 ) ) PUNCT cana-3423 93 25 can can AUX cana-3423 93 26 be be AUX cana-3423 93 27 continuously continuously ADV cana-3423 93 28 added add VERB cana-3423 93 29 in in ADP cana-3423 93 30 πΏπ‘ž(Ο‰ πΏπ‘ž(Ο‰ NOUN cana-3423 93 31 ) ) PUNCT cana-3423 93 32 for for ADP cana-3423 93 33 appropriate appropriate ADJ cana-3423 93 34 values value NOUN cana-3423 93 35 of of ADP cana-3423 93 36 𝑝 𝑝 NOUN cana-3423 93 37 and and CCONJ cana-3423 93 38 π‘ž π‘ž X cana-3423 93 39 , , PUNCT cana-3423 93 40 depending depend VERB cana-3423 93 41 on on ADP cana-3423 93 42 the the DET cana-3423 93 43 dimensions dimension NOUN cana-3423 93 44 and and CCONJ cana-3423 93 45 regularity regularity NOUN cana-3423 93 46 of of ADP cana-3423 93 47 Ο‰ Ο‰ PROPN cana-3423 93 48 ( ( PUNCT cana-3423 93 49 brezis brezis NOUN cana-3423 93 50 , , PUNCT cana-3423 93 51 2010 2010 NUM cana-3423 93 52 ) ) PUNCT cana-3423 93 53 . . PUNCT cana-3423 94 1 such such ADJ cana-3423 94 2 embeddings embedding NOUN cana-3423 94 3 are be AUX cana-3423 94 4 essential essential ADJ cana-3423 94 5 for for ADP cana-3423 94 6 ensuring ensure VERB cana-3423 94 7 the the DET cana-3423 94 8 integrability integrability NOUN cana-3423 94 9 and and CCONJ cana-3423 94 10 boundedness boundedness NOUN cana-3423 94 11 of of ADP cana-3423 94 12 solutions solution NOUN cana-3423 94 13 , , PUNCT cana-3423 94 14 enabling enable VERB cana-3423 94 15 a a DET cana-3423 94 16 rigorous rigorous ADJ cana-3423 94 17 analytical analytical ADJ cana-3423 94 18 framework framework NOUN cana-3423 94 19 for for ADP cana-3423 94 20 handling handle VERB cana-3423 94 21 nonlinear nonlinear ADJ cana-3423 94 22 pdes pde NOUN cana-3423 94 23 . . PUNCT cana-3423 95 1 4 4 X cana-3423 95 2 . . X cana-3423 95 3 methodology methodology NOUN cana-3423 95 4 this this DET cana-3423 95 5 research research NOUN cana-3423 95 6 aims aim VERB cana-3423 95 7 to to PART cana-3423 95 8 utilize utilize VERB cana-3423 95 9 a a DET cana-3423 95 10 structured structured ADJ cana-3423 95 11 rigorous rigorous ADJ cana-3423 95 12 work work NOUN cana-3423 95 13 plan plan NOUN cana-3423 95 14 which which PRON cana-3423 95 15 incorporates incorporate VERB cana-3423 95 16 topological topological ADJ cana-3423 95 17 methods method NOUN cana-3423 95 18 to to PART cana-3423 95 19 provide provide VERB cana-3423 95 20 methods method NOUN cana-3423 95 21 of of ADP cana-3423 95 22 solution solution NOUN cana-3423 95 23 to to ADP cana-3423 95 24 the the DET cana-3423 95 25 given give VERB cana-3423 95 26 nonlinear nonlinear ADJ cana-3423 95 27 partial partial ADJ cana-3423 95 28 differential differential NOUN cana-3423 95 29 equations equation NOUN cana-3423 95 30 ( ( PUNCT cana-3423 95 31 pdes pde NOUN cana-3423 95 32 ) ) PUNCT cana-3423 95 33 with with ADP cana-3423 95 34 theoretical theoretical ADJ cana-3423 95 35 analysis analysis NOUN cana-3423 95 36 backed back VERB cana-3423 95 37 by by ADP cana-3423 95 38 computational computational ADJ cana-3423 95 39 checks check NOUN cana-3423 95 40 ( ( PUNCT cana-3423 95 41 ciarlet,2013 ciarlet,2013 NOUN cana-3423 95 42 ) ) PUNCT cana-3423 95 43 . . PUNCT cana-3423 96 1 these these DET cana-3423 96 2 components component NOUN cana-3423 96 3 of of ADP cana-3423 96 4 the the DET cana-3423 96 5 communications communication NOUN cana-3423 96 6 on on ADP cana-3423 96 7 applied apply VERB cana-3423 96 8 nonlinear nonlinear ADJ cana-3423 96 9 analysis analysis NOUN cana-3423 96 10 issn issn NOUN cana-3423 96 11 : : PUNCT cana-3423 96 12 1074 1074 NUM cana-3423 96 13 - - PUNCT cana-3423 96 14 133x 133x NUM cana-3423 96 15 vol vol NOUN cana-3423 96 16 32 32 NUM cana-3423 96 17 no no NOUN cana-3423 96 18 . . PUNCT cana-3423 97 1 7s 7 NOUN cana-3423 97 2 ( ( PUNCT cana-3423 97 3 2025 2025 NUM cana-3423 97 4 ) ) PUNCT cana-3423 97 5 https://internationalpubls.com https://internationalpubls.com X cana-3423 97 6 362 362 NUM cana-3423 97 7 methodology methodology NOUN cana-3423 97 8 include include VERB cana-3423 97 9 research research NOUN cana-3423 97 10 design design NOUN cana-3423 97 11 , , PUNCT cana-3423 97 12 formulation formulation NOUN cana-3423 97 13 of of ADP cana-3423 97 14 nonlinear nonlinear ADJ cana-3423 97 15 pdes pde NOUN cana-3423 97 16 , , PUNCT cana-3423 97 17 selection selection NOUN cana-3423 97 18 of of ADP cana-3423 97 19 topological topological ADJ cana-3423 97 20 concepts concept NOUN cana-3423 97 21 , , PUNCT cana-3423 97 22 analytical analytical ADJ cana-3423 97 23 methods method NOUN cana-3423 97 24 , , PUNCT cana-3423 97 25 computational computational ADJ cana-3423 97 26 methods method NOUN cana-3423 97 27 , , PUNCT cana-3423 97 28 and and CCONJ cana-3423 97 29 validation validation NOUN cana-3423 97 30 methods method NOUN cana-3423 97 31 . . PUNCT cana-3423 98 1 4.1 4.1 NUM cana-3423 98 2 research research NOUN cana-3423 98 3 design design NOUN cana-3423 98 4 it it PRON cana-3423 98 5 is be AUX cana-3423 98 6 an an DET cana-3423 98 7 exploratory exploratory ADJ cana-3423 98 8 nature nature NOUN cana-3423 98 9 that that PRON cana-3423 98 10 the the DET cana-3423 98 11 study study NOUN cana-3423 98 12 assumes assume VERB cana-3423 98 13 as as ADP cana-3423 98 14 the the DET cana-3423 98 15 main main ADJ cana-3423 98 16 objective objective NOUN cana-3423 98 17 of of ADP cana-3423 98 18 the the DET cana-3423 98 19 study study NOUN cana-3423 98 20 which which PRON cana-3423 98 21 is be AUX cana-3423 98 22 to to PART cana-3423 98 23 formulate formulate VERB cana-3423 98 24 and and CCONJ cana-3423 98 25 prove prove VERB cana-3423 98 26 topological topological ADJ cana-3423 98 27 methods method NOUN cana-3423 98 28 for for ADP cana-3423 98 29 nonlinear nonlinear ADJ cana-3423 98 30 pdes pde NOUN cana-3423 98 31 . . PUNCT cana-3423 99 1 it it PRON cana-3423 99 2 is be AUX cana-3423 99 3 organized organize VERB cana-3423 99 4 in in ADP cana-3423 99 5 a a DET cana-3423 99 6 form form NOUN cana-3423 99 7 that that PRON cana-3423 99 8 directly directly ADV cana-3423 99 9 deals deal VERB cana-3423 99 10 with with ADP cana-3423 99 11 problems problem NOUN cana-3423 99 12 of of ADP cana-3423 99 13 existence existence NOUN cana-3423 99 14 and and CCONJ cana-3423 99 15 stability stability NOUN cana-3423 99 16 of of ADP cana-3423 99 17 solutions solution NOUN cana-3423 99 18 as as ADV cana-3423 99 19 well well ADV cana-3423 99 20 as as ADP cana-3423 99 21 with with ADP cana-3423 99 22 the the DET cana-3423 99 23 computational computational ADJ cana-3423 99 24 implementation implementation NOUN cana-3423 99 25 of of ADP cana-3423 99 26 the the DET cana-3423 99 27 approach approach NOUN cana-3423 99 28 . . PUNCT cana-3423 100 1 the the DET cana-3423 100 2 goal goal NOUN cana-3423 100 3 is be AUX cana-3423 100 4 to to PART cana-3423 100 5 explore explore VERB cana-3423 100 6 different different ADJ cana-3423 100 7 sorts sort NOUN cana-3423 100 8 of of ADP cana-3423 100 9 nonlinear nonlinear ADJ cana-3423 100 10 pdes pde NOUN cana-3423 100 11 with with ADP cana-3423 100 12 different different ADJ cana-3423 100 13 boundaries boundary NOUN cana-3423 100 14 and and CCONJ cana-3423 100 15 determine determine VERB cana-3423 100 16 the the DET cana-3423 100 17 feasibility feasibility NOUN cana-3423 100 18 of of ADP cana-3423 100 19 topological topological ADJ cana-3423 100 20 approaches approach NOUN cana-3423 100 21 , , PUNCT cana-3423 100 22 especially especially ADV cana-3423 100 23 in in ADP cana-3423 100 24 high high ADJ cana-3423 100 25 - - PUNCT cana-3423 100 26 dimensional dimensional ADJ cana-3423 100 27 spaces space NOUN cana-3423 100 28 and and CCONJ cana-3423 100 29 complicated complicated ADJ cana-3423 100 30 domains domain NOUN cana-3423 100 31 ( ( PUNCT cana-3423 100 32 evans evan NOUN cana-3423 100 33 , , PUNCT cana-3423 100 34 2010 2010 NUM cana-3423 100 35 ) ) PUNCT cana-3423 100 36 . . PUNCT cana-3423 101 1 4.2 4.2 NUM cana-3423 101 2 topological topological ADJ cana-3423 101 3 concepts concept NOUN cana-3423 101 4 the the DET cana-3423 101 5 base base NOUN cana-3423 101 6 fundamental fundamental ADJ cana-3423 101 7 concepts concept NOUN cana-3423 101 8 of of ADP cana-3423 101 9 topology topology NOUN cana-3423 101 10 used use VERB cana-3423 101 11 in in ADP cana-3423 101 12 this this DET cana-3423 101 13 research research NOUN cana-3423 101 14 are be AUX cana-3423 101 15 fixed fix VERB cana-3423 101 16 point point NOUN cana-3423 101 17 theory theory NOUN cana-3423 101 18 , , PUNCT cana-3423 101 19 degree degree NOUN cana-3423 101 20 theory theory NOUN cana-3423 101 21 as as ADV cana-3423 101 22 well well ADV cana-3423 101 23 as as ADP cana-3423 101 24 critical critical ADJ cana-3423 101 25 point point NOUN cana-3423 101 26 theory theory NOUN cana-3423 101 27 . . PUNCT cana-3423 102 1 these these DET cana-3423 102 2 concepts concept NOUN cana-3423 102 3 provide provide VERB cana-3423 102 4 the the DET cana-3423 102 5 mathematical mathematical ADJ cana-3423 102 6 basis basis NOUN cana-3423 102 7 for for ADP cana-3423 102 8 addressing address VERB cana-3423 102 9 the the DET cana-3423 102 10 nonlinearity nonlinearity NOUN cana-3423 102 11 inherent inherent ADJ cana-3423 102 12 in in ADP cana-3423 102 13 pdes pde NOUN cana-3423 102 14 : : PUNCT cana-3423 102 15 β€’ β€’ NUM cana-3423 102 16 fixed fix VERB cana-3423 102 17 - - PUNCT cana-3423 102 18 point point NOUN cana-3423 102 19 theory theory NOUN cana-3423 102 20 : : PUNCT cana-3423 102 21 the the DET cana-3423 102 22 existence existence NOUN cana-3423 102 23 of of ADP cana-3423 102 24 unique unique ADJ cana-3423 102 25 solutions solution NOUN cana-3423 102 26 is be AUX cana-3423 102 27 shown show VERB cana-3423 102 28 using use VERB cana-3423 102 29 the the DET cana-3423 102 30 banach banach ADV cana-3423 102 31 fixed fix VERB cana-3423 102 32 - - PUNCT cana-3423 102 33 point point NOUN cana-3423 102 34 theorem theorem NOUN cana-3423 102 35 where where SCONJ cana-3423 102 36 contraction contraction NOUN cana-3423 102 37 mappings mapping NOUN cana-3423 102 38 are be AUX cana-3423 102 39 applicable applicable ADJ cana-3423 102 40 . . PUNCT cana-3423 103 1 there there PRON cana-3423 103 2 are be VERB cana-3423 103 3 numerous numerous ADJ cana-3423 103 4 theorems theorem NOUN cana-3423 103 5 , , PUNCT cana-3423 103 6 among among ADP cana-3423 103 7 which which PRON cana-3423 103 8 this this DET cana-3423 103 9 one one NOUN cana-3423 103 10 is be AUX cana-3423 103 11 a a DET cana-3423 103 12 primary primary ADJ cana-3423 103 13 tool tool NOUN cana-3423 103 14 of of ADP cana-3423 103 15 iterative iterative NOUN cana-3423 103 16 solution solution NOUN cana-3423 103 17 methods method NOUN cana-3423 103 18 ( ( PUNCT cana-3423 103 19 aubin aubin PROPN cana-3423 103 20 & & CCONJ cana-3423 103 21 frankowska frankowska PROPN cana-3423 103 22 , , PUNCT cana-3423 103 23 2009 2009 NUM cana-3423 103 24 ) ) PUNCT cana-3423 103 25 . . PUNCT cana-3423 104 1 β€’ β€’ NUM cana-3423 104 2 degree degree NOUN cana-3423 104 3 theory theory NOUN cana-3423 104 4 : : PUNCT cana-3423 104 5 degree degree NOUN cana-3423 104 6 theory theory NOUN cana-3423 104 7 enables enable VERB cana-3423 104 8 case case NOUN cana-3423 104 9 discussions discussion NOUN cana-3423 104 10 where where SCONJ cana-3423 104 11 the the DET cana-3423 104 12 fixed fix VERB cana-3423 104 13 point point NOUN cana-3423 104 14 conditions condition NOUN cana-3423 104 15 are be AUX cana-3423 104 16 too too ADV cana-3423 104 17 limiting limit VERB cana-3423 104 18 toward toward ADP cana-3423 104 19 finding find VERB cana-3423 104 20 solutions solution NOUN cana-3423 104 21 . . PUNCT cana-3423 105 1 it it PRON cana-3423 105 2 allows allow VERB cana-3423 105 3 analyzing analyze VERB cana-3423 105 4 solution solution NOUN cana-3423 105 5 multiplication multiplication NOUN cana-3423 105 6 and and CCONJ cana-3423 105 7 assists assist NOUN cana-3423 105 8 in in ADP cana-3423 105 9 counting count VERB cana-3423 105 10 solutions solution NOUN cana-3423 105 11 in in ADP cana-3423 105 12 conditions condition NOUN cana-3423 105 13 that that PRON cana-3423 105 14 are be AUX cana-3423 105 15 higher high ADJ cana-3423 105 16 than than ADP cana-3423 105 17 three three NUM cana-3423 105 18 dimensions dimension NOUN cana-3423 105 19 ( ( PUNCT cana-3423 105 20 rabinowitz rabinowitz NOUN cana-3423 105 21 , , PUNCT cana-3423 105 22 2015 2015 NUM cana-3423 105 23 ) ) PUNCT cana-3423 105 24 . . PUNCT cana-3423 106 1 β€’ β€’ NUM cana-3423 106 2 critical critical ADJ cana-3423 106 3 point point NOUN cana-3423 106 4 theory theory NOUN cana-3423 106 5 : : PUNCT cana-3423 106 6 the the DET cana-3423 106 7 critical critical ADJ cana-3423 106 8 point point NOUN cana-3423 106 9 theory theory NOUN cana-3423 106 10 helps help VERB cana-3423 106 11 to to PART cana-3423 106 12 find find VERB cana-3423 106 13 minimizers minimizer NOUN cana-3423 106 14 and and CCONJ cana-3423 106 15 saddle saddle NOUN cana-3423 106 16 points point NOUN cana-3423 106 17 due due ADP cana-3423 106 18 to to ADP cana-3423 106 19 higher high ADJ cana-3423 106 20 scrutiny scrutiny NOUN cana-3423 106 21 in in ADP cana-3423 106 22 the the DET cana-3423 106 23 associated associated ADJ cana-3423 106 24 gradients gradient NOUN cana-3423 106 25 ; ; PUNCT cana-3423 106 26 this this PRON cana-3423 106 27 broadens broaden VERB cana-3423 106 28 the the DET cana-3423 106 29 scope scope NOUN cana-3423 106 30 of of ADP cana-3423 106 31 analyzing analyze VERB cana-3423 106 32 the the DET cana-3423 106 33 solution solution NOUN cana-3423 106 34 space space NOUN cana-3423 106 35 ( ( PUNCT cana-3423 106 36 chang chang PROPN cana-3423 106 37 , , PUNCT cana-3423 106 38 2013 2013 NUM cana-3423 106 39 ) ) PUNCT cana-3423 106 40 . . PUNCT cana-3423 107 1 4.3 4.3 NUM cana-3423 107 2 formulation formulation NOUN cana-3423 107 3 of of ADP cana-3423 107 4 nonlinear nonlinear ADJ cana-3423 107 5 pdes pde NOUN cana-3423 107 6 the the DET cana-3423 107 7 pdes pde NOUN cana-3423 107 8 investigated investigate VERB cana-3423 107 9 in in ADP cana-3423 107 10 this this DET cana-3423 107 11 work work NOUN cana-3423 107 12 depend depend VERB cana-3423 107 13 on on ADP cana-3423 107 14 several several ADJ cana-3423 107 15 inputs input NOUN cana-3423 107 16 and and CCONJ cana-3423 107 17 are be AUX cana-3423 107 18 designed design VERB cana-3423 107 19 to to PART cana-3423 107 20 take take VERB cana-3423 107 21 into into ADP cana-3423 107 22 account account NOUN cana-3423 107 23 various various ADJ cana-3423 107 24 forms form NOUN cana-3423 107 25 of of ADP cana-3423 107 26 boundary boundary ADJ cana-3423 107 27 conditions condition NOUN cana-3423 107 28 and and CCONJ cana-3423 107 29 various various ADJ cana-3423 107 30 types type NOUN cana-3423 107 31 of of ADP cana-3423 107 32 non non ADJ cana-3423 107 33 - - NOUN cana-3423 107 34 linearity linearity NOUN cana-3423 107 35 . . PUNCT cana-3423 108 1 a a DET cana-3423 108 2 general general ADJ cana-3423 108 3 nonlinear nonlinear ADJ cana-3423 108 4 pde pde NOUN cana-3423 108 5 for for ADP cana-3423 108 6 this this DET cana-3423 108 7 research research NOUN cana-3423 108 8 can can AUX cana-3423 108 9 be be AUX cana-3423 108 10 expressed express VERB cana-3423 108 11 as as ADP cana-3423 108 12 : : PUNCT cana-3423 108 13 β„’(𝑒 β„’(𝑒 NUM cana-3423 108 14 ) ) PUNCT cana-3423 108 15 = = SYM cana-3423 108 16 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-3423 108 17 , , PUNCT cana-3423 108 18 𝑒 𝑒 X cana-3423 108 19 , , PUNCT cana-3423 108 20 βˆ‡π‘’ βˆ‡π‘’ PROPN cana-3423 108 21 ) ) PUNCT cana-3423 108 22 where where SCONJ cana-3423 108 23 β„’ β„’ PROPN cana-3423 108 24 represents represent VERB cana-3423 108 25 a a DET cana-3423 108 26 differential differential ADJ cana-3423 108 27 operator operator NOUN cana-3423 108 28 acting act VERB cana-3423 108 29 on on ADP cana-3423 108 30 the the DET cana-3423 108 31 unknown unknown ADJ cana-3423 108 32 function function NOUN cana-3423 108 33 𝑒 𝑒 PROPN cana-3423 108 34 , , PUNCT cana-3423 108 35 and and CCONJ cana-3423 108 36 𝑓denotes 𝑓denote VERB cana-3423 108 37 a a DET cana-3423 108 38 nonlinear nonlinear ADJ cana-3423 108 39 function function NOUN cana-3423 108 40 that that PRON cana-3423 108 41 incorporates incorporate VERB cana-3423 108 42 spatial spatial ADJ cana-3423 108 43 variables variable NOUN cana-3423 108 44 π‘₯ π‘₯ PROPN cana-3423 108 45 , , PUNCT cana-3423 108 46 as as ADV cana-3423 108 47 well well ADV cana-3423 108 48 as as ADP cana-3423 108 49 𝑒 𝑒 NOUN cana-3423 108 50 and and CCONJ cana-3423 108 51 its its PRON cana-3423 108 52 gradient gradient NOUN cana-3423 108 53 βˆ‡π‘’. βˆ‡π‘’. NOUN cana-3423 108 54 the the DET cana-3423 108 55 equations equation NOUN cana-3423 108 56 are be AUX cana-3423 108 57 further far ADV cana-3423 108 58 categorized categorize VERB cana-3423 108 59 using use VERB cana-3423 108 60 the the DET cana-3423 108 61 characteristics characteristic NOUN cana-3423 108 62 of of ADP cana-3423 108 63 nonlinearity nonlinearity NOUN cana-3423 108 64 for for ADP cana-3423 108 65 instance instance NOUN cana-3423 108 66 quasilinear quasilinear NOUN cana-3423 108 67 and and CCONJ cana-3423 108 68 semilinear semilinear NOUN cana-3423 108 69 and and CCONJ cana-3423 108 70 boundary boundary ADJ cana-3423 108 71 conditions condition NOUN cana-3423 108 72 to to PART cana-3423 108 73 include include VERB cana-3423 108 74 amongst amongst ADP cana-3423 108 75 others others PROPN cana-3423 108 76 dirichlet dirichlet PROPN cana-3423 108 77 neumann neumann PROPN cana-3423 108 78 and and CCONJ cana-3423 108 79 mixed mixed ADJ cana-3423 108 80 conditions condition NOUN cana-3423 108 81 ( ( PUNCT cana-3423 108 82 brezis brezis NOUN cana-3423 108 83 , , PUNCT cana-3423 108 84 2010 2010 NUM cana-3423 108 85 ) ) PUNCT cana-3423 108 86 . . PUNCT cana-3423 109 1 4.4 4.4 NUM cana-3423 109 2 analytical analytical ADJ cana-3423 109 3 techniques technique NOUN cana-3423 109 4 analytical analytical ADJ cana-3423 109 5 techniques technique NOUN cana-3423 109 6 form form VERB cana-3423 109 7 a a DET cana-3423 109 8 core core NOUN cana-3423 109 9 component component NOUN cana-3423 109 10 of of ADP cana-3423 109 11 this this DET cana-3423 109 12 research research NOUN cana-3423 109 13 , , PUNCT cana-3423 109 14 providing provide VERB cana-3423 109 15 theoretical theoretical ADJ cana-3423 109 16 foundations foundation NOUN cana-3423 109 17 for for ADP cana-3423 109 18 solution solution NOUN cana-3423 109 19 analysis analysis NOUN cana-3423 109 20 : : PUNCT cana-3423 109 21 β€’ β€’ NUM cana-3423 109 22 fixed fix VERB cana-3423 109 23 - - PUNCT cana-3423 109 24 point point NOUN cana-3423 109 25 applications application NOUN cana-3423 109 26 : : PUNCT cana-3423 109 27 where where SCONJ cana-3423 109 28 pdes pde NOUN cana-3423 109 29 exhibit exhibit VERB cana-3423 109 30 contraction contraction NOUN cana-3423 109 31 characteristics characteristic NOUN cana-3423 109 32 , , PUNCT cana-3423 109 33 iterative iterative NOUN cana-3423 109 34 solution solution NOUN cana-3423 109 35 techniques technique NOUN cana-3423 109 36 are be AUX cana-3423 109 37 developed develop VERB cana-3423 109 38 using use VERB cana-3423 109 39 banach banach NOUN cana-3423 109 40 ’s ’s PART cana-3423 109 41 fixed fix VERB cana-3423 109 42 point point NOUN cana-3423 109 43 theorem theorem ADJ cana-3423 109 44 and and CCONJ cana-3423 109 45 converge converge VERB cana-3423 109 46 uniquely uniquely ADV cana-3423 109 47 in in ADP cana-3423 109 48 suitable suitable ADJ cana-3423 109 49 function function NOUN cana-3423 109 50 spaces space NOUN cana-3423 109 51 ( ( PUNCT cana-3423 109 52 evans evans PROPN cana-3423 109 53 , , PUNCT cana-3423 109 54 2010 2010 NUM cana-3423 109 55 ) ) PUNCT cana-3423 109 56 . . PUNCT cana-3423 110 1 β€’ β€’ NUM cana-3423 110 2 degree degree NOUN cana-3423 110 3 - - PUNCT cana-3423 110 4 theoretic theoretic NOUN cana-3423 110 5 analysis analysis NOUN cana-3423 110 6 : : PUNCT cana-3423 110 7 degree degree NOUN cana-3423 110 8 theory theory NOUN cana-3423 110 9 is be AUX cana-3423 110 10 used use VERB cana-3423 110 11 to to PART cana-3423 110 12 prove prove VERB cana-3423 110 13 the the DET cana-3423 110 14 existence existence NOUN cana-3423 110 15 of of ADP cana-3423 110 16 solutions solution NOUN cana-3423 110 17 in in ADP cana-3423 110 18 locations location NOUN cana-3423 110 19 where where SCONJ cana-3423 110 20 fixed fix VERB cana-3423 110 21 - - PUNCT cana-3423 110 22 point point NOUN cana-3423 110 23 techniques technique NOUN cana-3423 110 24 are be AUX cana-3423 110 25 inapplicable inapplicable ADJ cana-3423 110 26 . . PUNCT cana-3423 111 1 particularly particularly ADV cana-3423 111 2 , , PUNCT cana-3423 111 3 the the DET cana-3423 111 4 leray leray ADJ cana-3423 111 5 - - PUNCT cana-3423 111 6 schauder schauder NOUN cana-3423 111 7 degree degree NOUN cana-3423 111 8 is be AUX cana-3423 111 9 employed employ VERB cana-3423 111 10 to to PART cana-3423 111 11 apply apply VERB cana-3423 111 12 to to ADP cana-3423 111 13 problems problem NOUN cana-3423 111 14 of of ADP cana-3423 111 15 solvability solvability NOUN cana-3423 111 16 of of ADP cana-3423 111 17 high high ADJ cana-3423 111 18 - - PUNCT cana-3423 111 19 dimensional dimensional ADJ cana-3423 111 20 nonlinear nonlinear ADJ cana-3423 111 21 pdes pde NOUN cana-3423 111 22 ( ( PUNCT cana-3423 111 23 ambrosetti ambrosetti PROPN cana-3423 111 24 & & CCONJ cana-3423 111 25 prodi prodi PROPN cana-3423 111 26 , , PUNCT cana-3423 111 27 1995 1995 NUM cana-3423 111 28 ) ) PUNCT cana-3423 111 29 . . PUNCT cana-3423 112 1 communications communication NOUN cana-3423 112 2 on on ADP cana-3423 112 3 applied apply VERB cana-3423 112 4 nonlinear nonlinear ADJ cana-3423 112 5 analysis analysis NOUN cana-3423 112 6 issn issn NOUN cana-3423 112 7 : : PUNCT cana-3423 112 8 1074 1074 NUM cana-3423 112 9 - - PUNCT cana-3423 112 10 133x 133x NUM cana-3423 112 11 vol vol NOUN cana-3423 112 12 32 32 NUM cana-3423 112 13 no no NOUN cana-3423 112 14 . . PUNCT cana-3423 113 1 7s 7 NOUN cana-3423 113 2 ( ( PUNCT cana-3423 113 3 2025 2025 NUM cana-3423 113 4 ) ) PUNCT cana-3423 113 5 https://internationalpubls.com https://internationalpubls.com X cana-3423 113 6 363 363 NUM cana-3423 113 7 β€’ β€’ NUM cana-3423 113 8 variational variational ADJ cana-3423 113 9 and and CCONJ cana-3423 113 10 critical critical ADJ cana-3423 113 11 point point NOUN cana-3423 113 12 methods method NOUN cana-3423 113 13 : : PUNCT cana-3423 113 14 topological topological ADJ cana-3423 113 15 and and CCONJ cana-3423 113 16 critical critical ADJ cana-3423 113 17 point point NOUN cana-3423 113 18 methods method NOUN cana-3423 113 19 using use VERB cana-3423 113 20 the the DET cana-3423 113 21 variational variational ADJ cana-3423 113 22 methods method NOUN cana-3423 113 23 and and CCONJ cana-3423 113 24 the the DET cana-3423 113 25 mountain mountain NOUN cana-3423 113 26 pass pass NOUN cana-3423 113 27 theorem theorem NOUN cana-3423 113 28 serve serve VERB cana-3423 113 29 to to PART cana-3423 113 30 find find VERB cana-3423 113 31 critical critical ADJ cana-3423 113 32 points point NOUN cana-3423 113 33 of of ADP cana-3423 113 34 functionals functional NOUN cana-3423 113 35 related relate VERB cana-3423 113 36 to to ADP cana-3423 113 37 nonlinear nonlinear ADJ cana-3423 113 38 pdes pde NOUN cana-3423 113 39 . . PUNCT cana-3423 114 1 this this DET cana-3423 114 2 approach approach NOUN cana-3423 114 3 gives give VERB cana-3423 114 4 a a DET cana-3423 114 5 glimpse glimpse NOUN cana-3423 114 6 of of ADP cana-3423 114 7 the the DET cana-3423 114 8 stability stability NOUN cana-3423 114 9 of of ADP cana-3423 114 10 solutions solution NOUN cana-3423 114 11 and and CCONJ cana-3423 114 12 a a DET cana-3423 114 13 better well ADJ cana-3423 114 14 view view NOUN cana-3423 114 15 of of ADP cana-3423 114 16 solution solution NOUN cana-3423 114 17 multiplicity multiplicity NOUN cana-3423 114 18 ( ( PUNCT cana-3423 114 19 mawhin mawhin NOUN cana-3423 114 20 , , PUNCT cana-3423 114 21 2017 2017 NUM cana-3423 114 22 ) ) PUNCT cana-3423 114 23 . . PUNCT cana-3423 115 1 4.5 4.5 NUM cana-3423 115 2 computational computational ADJ cana-3423 115 3 methods method NOUN cana-3423 115 4 the the DET cana-3423 115 5 computational computational ADJ cana-3423 115 6 part part NOUN cana-3423 115 7 of of ADP cana-3423 115 8 this this DET cana-3423 115 9 work work NOUN cana-3423 115 10 is be AUX cana-3423 115 11 to to PART cana-3423 115 12 algebraise algebraise VERB cana-3423 115 13 these these DET cana-3423 115 14 topologies topology NOUN cana-3423 115 15 numerically numerically ADV cana-3423 115 16 over over ADP cana-3423 115 17 the the DET cana-3423 115 18 coming come VERB cana-3423 115 19 sections section NOUN cana-3423 115 20 and and CCONJ cana-3423 115 21 chapters chapter NOUN cana-3423 115 22 . . PUNCT cana-3423 116 1 the the DET cana-3423 116 2 finite finite PROPN cana-3423 116 3 element element NOUN cana-3423 116 4 and and CCONJ cana-3423 116 5 finite finite ADJ cana-3423 116 6 difference difference NOUN cana-3423 116 7 techniques technique NOUN cana-3423 116 8 are be AUX cana-3423 116 9 predominantly predominantly ADV cana-3423 116 10 applied apply VERB cana-3423 116 11 to to ADP cana-3423 116 12 the the DET cana-3423 116 13 discretization discretization NOUN cana-3423 116 14 process process NOUN cana-3423 116 15 of of ADP cana-3423 116 16 the the DET cana-3423 116 17 nlpdes nlpde NOUN cana-3423 116 18 . . PUNCT cana-3423 117 1 the the DET cana-3423 117 2 discretized discretized ADJ cana-3423 117 3 equations equation NOUN cana-3423 117 4 are be AUX cana-3423 117 5 then then ADV cana-3423 117 6 solved solve VERB cana-3423 117 7 iteratively iteratively ADV cana-3423 117 8 using use VERB cana-3423 117 9 the the DET cana-3423 117 10 following follow VERB cana-3423 117 11 computational computational ADJ cana-3423 117 12 techniques technique NOUN cana-3423 117 13 : : PUNCT cana-3423 117 14 β€’ β€’ NUM cana-3423 117 15 newton newton PROPN cana-3423 117 16 - - PUNCT cana-3423 117 17 raphson raphson PROPN cana-3423 117 18 iterative iterative NOUN cana-3423 117 19 method method NOUN cana-3423 117 20 : : PUNCT cana-3423 117 21 used use VERB cana-3423 117 22 in in ADP cana-3423 117 23 nonlinear nonlinear ADJ cana-3423 117 24 system system NOUN cana-3423 117 25 control control NOUN cana-3423 117 26 problems problem NOUN cana-3423 117 27 for for ADP cana-3423 117 28 which which PRON cana-3423 117 29 it it PRON cana-3423 117 30 is be AUX cana-3423 117 31 impossible impossible ADJ cana-3423 117 32 to to PART cana-3423 117 33 find find VERB cana-3423 117 34 definite definite ADJ cana-3423 117 35 solutions solution NOUN cana-3423 117 36 . . PUNCT cana-3423 118 1 this this PRON cana-3423 118 2 together together ADV cana-3423 118 3 with with ADP cana-3423 118 4 the the DET cana-3423 118 5 degree degree NOUN cana-3423 118 6 - - PUNCT cana-3423 118 7 theoretic theoretic NOUN cana-3423 118 8 observations observation NOUN cana-3423 118 9 allows allow VERB cana-3423 118 10 for for ADP cana-3423 118 11 computing compute VERB cana-3423 118 12 the the DET cana-3423 118 13 approximate approximate ADJ cana-3423 118 14 solutions solution NOUN cana-3423 118 15 ( ( PUNCT cana-3423 118 16 sΓΌli sΓΌli PROPN cana-3423 118 17 & & CCONJ cana-3423 118 18 mayers mayers PROPN cana-3423 118 19 , , PUNCT cana-3423 118 20 2003 2003 NUM cana-3423 118 21 ) ) PUNCT cana-3423 118 22 . . PUNCT cana-3423 119 1 β€’ β€’ NUM cana-3423 119 2 fixed fix VERB cana-3423 119 3 - - PUNCT cana-3423 119 4 point point NOUN cana-3423 119 5 iterative iterative NOUN cana-3423 119 6 scheme scheme NOUN cana-3423 119 7 : : PUNCT cana-3423 119 8 applied apply VERB cana-3423 119 9 to to ADP cana-3423 119 10 solutions solution NOUN cana-3423 119 11 that that PRON cana-3423 119 12 meet meet VERB cana-3423 119 13 the the DET cana-3423 119 14 contraction contraction NOUN cana-3423 119 15 property property NOUN cana-3423 119 16 for for ADP cana-3423 119 17 the the DET cana-3423 119 18 achievement achievement NOUN cana-3423 119 19 of of ADP cana-3423 119 20 a a DET cana-3423 119 21 unique unique ADJ cana-3423 119 22 solution solution NOUN cana-3423 119 23 as as SCONJ cana-3423 119 24 postulated postulate VERB cana-3423 119 25 by by ADP cana-3423 119 26 banach banach NOUN cana-3423 119 27 . . PUNCT cana-3423 120 1 this this DET cana-3423 120 2 scheme scheme NOUN cana-3423 120 3 is be AUX cana-3423 120 4 particularly particularly ADV cana-3423 120 5 useful useful ADJ cana-3423 120 6 for for ADP cana-3423 120 7 semi semi ADJ cana-3423 120 8 - - ADJ cana-3423 120 9 linear linear ADJ cana-3423 120 10 pdes pde NOUN cana-3423 120 11 where where SCONJ cana-3423 120 12 some some DET cana-3423 120 13 standard standard ADJ cana-3423 120 14 boundary boundary ADJ cana-3423 120 15 value value NOUN cana-3423 120 16 conditions condition NOUN cana-3423 120 17 are be AUX cana-3423 120 18 met meet VERB cana-3423 120 19 ( ( PUNCT cana-3423 120 20 schechter schechter NOUN cana-3423 120 21 , , PUNCT cana-3423 120 22 2004 2004 NUM cana-3423 120 23 ) ) PUNCT cana-3423 120 24 . . PUNCT cana-3423 121 1 β€’ β€’ NUM cana-3423 121 2 multigrid multigrid NOUN cana-3423 121 3 and and CCONJ cana-3423 121 4 adaptive adaptive ADJ cana-3423 121 5 mesh mesh NOUN cana-3423 121 6 refinement refinement NOUN cana-3423 121 7 : : PUNCT cana-3423 121 8 these these DET cana-3423 121 9 methods method NOUN cana-3423 121 10 are be AUX cana-3423 121 11 used use VERB cana-3423 121 12 to to PART cana-3423 121 13 enhance enhance VERB cana-3423 121 14 the the DET cana-3423 121 15 computational computational ADJ cana-3423 121 16 escalation escalation NOUN cana-3423 121 17 and and CCONJ cana-3423 121 18 precision precision NOUN cana-3423 121 19 , , PUNCT cana-3423 121 20 specifically specifically ADV cana-3423 121 21 for for ADP cana-3423 121 22 high high ADJ cana-3423 121 23 dimensional dimensional ADJ cana-3423 121 24 problems problem NOUN cana-3423 121 25 where where SCONJ cana-3423 121 26 solution solution NOUN cana-3423 121 27 smoothness smoothness NOUN cana-3423 121 28 is be AUX cana-3423 121 29 not not PART cana-3423 121 30 constant constant ADJ cana-3423 121 31 in in ADP cana-3423 121 32 the the DET cana-3423 121 33 domain domain NOUN cana-3423 121 34 ( ( PUNCT cana-3423 121 35 brezis brezis NOUN cana-3423 121 36 , , PUNCT cana-3423 121 37 2010 2010 NUM cana-3423 121 38 ) ) PUNCT cana-3423 121 39 . . PUNCT cana-3423 122 1 4.6 4.6 NUM cana-3423 122 2 validation validation NOUN cana-3423 122 3 techniques technique NOUN cana-3423 122 4 validation validation NOUN cana-3423 122 5 techniques technique NOUN cana-3423 122 6 are be AUX cana-3423 122 7 critical critical ADJ cana-3423 122 8 for for ADP cana-3423 122 9 ensuring ensure VERB cana-3423 122 10 the the DET cana-3423 122 11 reliability reliability NOUN cana-3423 122 12 and and CCONJ cana-3423 122 13 accuracy accuracy NOUN cana-3423 122 14 of of ADP cana-3423 122 15 the the DET cana-3423 122 16 solutions solution NOUN cana-3423 122 17 obtained obtain VERB cana-3423 122 18 . . PUNCT cana-3423 123 1 this this DET cana-3423 123 2 study study NOUN cana-3423 123 3 applies apply VERB cana-3423 123 4 the the DET cana-3423 123 5 following follow VERB cana-3423 123 6 validation validation NOUN cana-3423 123 7 methods method NOUN cana-3423 123 8 : : PUNCT cana-3423 123 9 β€’ β€’ NUM cana-3423 123 10 analytical analytical ADJ cana-3423 123 11 benchmarking benchmarking NOUN cana-3423 123 12 : : PUNCT cana-3423 123 13 results result NOUN cana-3423 123 14 of of ADP cana-3423 123 15 the the DET cana-3423 123 16 topological topological ADJ cana-3423 123 17 methods method NOUN cana-3423 123 18 are be AUX cana-3423 123 19 compared compare VERB cana-3423 123 20 to to ADP cana-3423 123 21 simple simple ADJ cana-3423 123 22 analytical analytical ADJ cana-3423 123 23 solutions solution NOUN cana-3423 123 24 for for ADP cana-3423 123 25 similar similar ADJ cana-3423 123 26 problems problem NOUN cana-3423 123 27 . . PUNCT cana-3423 124 1 of of ADP cana-3423 124 2 importance importance NOUN cana-3423 124 3 , , PUNCT cana-3423 124 4 this this DET cana-3423 124 5 benchmarking benchmarking NOUN cana-3423 124 6 offers offer VERB cana-3423 124 7 some some DET cana-3423 124 8 proof proof NOUN cana-3423 124 9 for for ADP cana-3423 124 10 checking check VERB cana-3423 124 11 accuracy accuracy NOUN cana-3423 124 12 ( ( PUNCT cana-3423 124 13 rabinowitz rabinowitz NOUN cana-3423 124 14 , , PUNCT cana-3423 124 15 2015 2015 NUM cana-3423 124 16 ) ) PUNCT cana-3423 124 17 . . PUNCT cana-3423 125 1 β€’ β€’ NUM cana-3423 125 2 convergence convergence NOUN cana-3423 125 3 and and CCONJ cana-3423 125 4 stability stability NOUN cana-3423 125 5 analysis analysis NOUN cana-3423 125 6 : : PUNCT cana-3423 125 7 the the DET cana-3423 125 8 stability stability NOUN cana-3423 125 9 of of ADP cana-3423 125 10 the the DET cana-3423 125 11 iterative iterative NOUN cana-3423 125 12 schemes scheme NOUN cana-3423 125 13 is be AUX cana-3423 125 14 checked check VERB cana-3423 125 15 with with ADP cana-3423 125 16 the the DET cana-3423 125 17 help help NOUN cana-3423 125 18 of of ADP cana-3423 125 19 convergence convergence NOUN cana-3423 125 20 tests test NOUN cana-3423 125 21 . . PUNCT cana-3423 126 1 the the DET cana-3423 126 2 computational computational ADJ cana-3423 126 3 schemes scheme NOUN cana-3423 126 4 are be AUX cana-3423 126 5 also also ADV cana-3423 126 6 checked check VERB cana-3423 126 7 under under ADP cana-3423 126 8 other other ADJ cana-3423 126 9 values value NOUN cana-3423 126 10 of of ADP cana-3423 126 11 boundary boundary ADJ cana-3423 126 12 conditions condition NOUN cana-3423 126 13 to to PART cana-3423 126 14 determine determine VERB cana-3423 126 15 stability stability NOUN cana-3423 126 16 ( ( PUNCT cana-3423 126 17 mawhin mawhin NOUN cana-3423 126 18 , , PUNCT cana-3423 126 19 2017 2017 NUM cana-3423 126 20 ) ) PUNCT cana-3423 126 21 . . PUNCT cana-3423 127 1 β€’ β€’ NUM cana-3423 127 2 error error NOUN cana-3423 127 3 analysis analysis NOUN cana-3423 127 4 : : PUNCT cana-3423 127 5 a a DET cana-3423 127 6 posteriori posteriori NOUN cana-3423 127 7 error error NOUN cana-3423 127 8 estimates estimate NOUN cana-3423 127 9 are be AUX cana-3423 127 10 calculated calculate VERB cana-3423 127 11 to to PART cana-3423 127 12 evaluate evaluate VERB cana-3423 127 13 the the DET cana-3423 127 14 accuracy accuracy NOUN cana-3423 127 15 of of ADP cana-3423 127 16 numerical numerical ADJ cana-3423 127 17 solutions solution NOUN cana-3423 127 18 relative relative ADJ cana-3423 127 19 to to ADP cana-3423 127 20 approximate approximate ADJ cana-3423 127 21 analytical analytical ADJ cana-3423 127 22 results result NOUN cana-3423 127 23 . . PUNCT cana-3423 128 1 this this DET cana-3423 128 2 analysis analysis NOUN cana-3423 128 3 is be AUX cana-3423 128 4 performed perform VERB cana-3423 128 5 across across ADP cana-3423 128 6 various various ADJ cana-3423 128 7 mesh mesh NOUN cana-3423 128 8 densities density NOUN cana-3423 128 9 to to PART cana-3423 128 10 ensure ensure VERB cana-3423 128 11 reliability reliability NOUN cana-3423 128 12 ( ( PUNCT cana-3423 128 13 aubin aubin PROPN cana-3423 128 14 & & CCONJ cana-3423 128 15 frankowska frankowska PROPN cana-3423 128 16 , , PUNCT cana-3423 128 17 2009 2009 NUM cana-3423 128 18 ) ) PUNCT cana-3423 128 19 . . PUNCT cana-3423 129 1 5 5 X cana-3423 129 2 . . X cana-3423 129 3 results result NOUN cana-3423 129 4 in in ADP cana-3423 129 5 this this DET cana-3423 129 6 section section NOUN cana-3423 129 7 , , PUNCT cana-3423 129 8 the the DET cana-3423 129 9 results result NOUN cana-3423 129 10 derived derive VERB cana-3423 129 11 from from ADP cana-3423 129 12 the the DET cana-3423 129 13 application application NOUN cana-3423 129 14 of of ADP cana-3423 129 15 the the DET cana-3423 129 16 topological topological ADJ cana-3423 129 17 methods method NOUN cana-3423 129 18 for for ADP cana-3423 129 19 the the DET cana-3423 129 20 analysis analysis NOUN cana-3423 129 21 of of ADP cana-3423 129 22 nonlinear nonlinear ADJ cana-3423 129 23 pdes pde NOUN cana-3423 129 24 using use VERB cana-3423 129 25 analytical analytical ADJ cana-3423 129 26 as as ADV cana-3423 129 27 well well ADV cana-3423 129 28 as as ADP cana-3423 129 29 numerical numerical ADJ cana-3423 129 30 techniques technique NOUN cana-3423 129 31 are be AUX cana-3423 129 32 described describe VERB cana-3423 129 33 . . PUNCT cana-3423 130 1 the the DET cana-3423 130 2 results result NOUN cana-3423 130 3 are be AUX cana-3423 130 4 grouped group VERB cana-3423 130 5 and and CCONJ cana-3423 130 6 arranged arrange VERB cana-3423 130 7 to to PART cana-3423 130 8 exhibit exhibit VERB cana-3423 130 9 the the DET cana-3423 130 10 ability ability NOUN cana-3423 130 11 of of ADP cana-3423 130 12 fixed fix VERB cana-3423 130 13 - - PUNCT cana-3423 130 14 point point NOUN cana-3423 130 15 , , PUNCT cana-3423 130 16 degree degree NOUN cana-3423 130 17 theory theory NOUN cana-3423 130 18 , , PUNCT cana-3423 130 19 and and CCONJ cana-3423 130 20 variational variational ADJ cana-3423 130 21 methods method NOUN cana-3423 130 22 to to PART cana-3423 130 23 provide provide VERB cana-3423 130 24 solutions solution NOUN cana-3423 130 25 to to ADP cana-3423 130 26 different different ADJ cana-3423 130 27 nonlinear nonlinear ADJ cana-3423 130 28 pdes pde NOUN cana-3423 130 29 . . PUNCT cana-3423 131 1 these these DET cana-3423 131 2 comprise comprise VERB cana-3423 131 3 convergence convergence NOUN cana-3423 131 4 rates rate NOUN cana-3423 131 5 , , PUNCT cana-3423 131 6 the the DET cana-3423 131 7 stability stability NOUN cana-3423 131 8 of of ADP cana-3423 131 9 the the DET cana-3423 131 10 solutions solution NOUN cana-3423 131 11 , , PUNCT cana-3423 131 12 computation computation NOUN cana-3423 131 13 time time NOUN cana-3423 131 14 , , PUNCT cana-3423 131 15 and and CCONJ cana-3423 131 16 validation validation NOUN cana-3423 131 17 results result NOUN cana-3423 131 18 . . PUNCT cana-3423 132 1 the the DET cana-3423 132 2 next next ADJ cana-3423 132 3 parts part NOUN cana-3423 132 4 of of ADP cana-3423 132 5 the the DET cana-3423 132 6 subsections subsection NOUN cana-3423 132 7 explicate explicate VERB cana-3423 132 8 each each PRON cana-3423 132 9 of of ADP cana-3423 132 10 those those DET cana-3423 132 11 components component NOUN cana-3423 132 12 . . PUNCT cana-3423 133 1 5.1 5.1 NUM cana-3423 133 2 existence existence NOUN cana-3423 133 3 and and CCONJ cana-3423 133 4 uniqueness uniqueness NOUN cana-3423 133 5 of of ADP cana-3423 133 6 solutions solution NOUN cana-3423 133 7 here here ADV cana-3423 133 8 by by ADP cana-3423 133 9 application application NOUN cana-3423 133 10 of of ADP cana-3423 133 11 the the DET cana-3423 133 12 banach banach ADV cana-3423 133 13 fixed fix VERB cana-3423 133 14 - - PUNCT cana-3423 133 15 point point NOUN cana-3423 133 16 theorem theorem NOUN cana-3423 133 17 , , PUNCT cana-3423 133 18 we we PRON cana-3423 133 19 proved prove VERB cana-3423 133 20 semi semi ADJ cana-3423 133 21 - - ADJ cana-3423 133 22 linear linear ADJ cana-3423 133 23 pdes pde NOUN cana-3423 133 24 have have VERB cana-3423 133 25 the the DET cana-3423 133 26 existence existence NOUN cana-3423 133 27 and and CCONJ cana-3423 133 28 uniqueness uniqueness NOUN cana-3423 133 29 of of ADP cana-3423 133 30 solutions solution NOUN cana-3423 133 31 under under ADP cana-3423 133 32 contraction contraction NOUN cana-3423 133 33 conditions condition NOUN cana-3423 133 34 . . PUNCT cana-3423 134 1 in in ADP cana-3423 134 2 table table NOUN cana-3423 134 3 1 1 NUM cana-3423 134 4 we we PRON cana-3423 134 5 present present VERB cana-3423 134 6 a a DET cana-3423 134 7 comparison comparison NOUN cana-3423 134 8 of of ADP cana-3423 134 9 the the DET cana-3423 134 10 results result NOUN cana-3423 134 11 of of ADP cana-3423 134 12 the the DET cana-3423 134 13 fixed fix VERB cana-3423 134 14 - - PUNCT cana-3423 134 15 point point NOUN cana-3423 134 16 iterative iterative NOUN cana-3423 134 17 scheme scheme NOUN cana-3423 134 18 and and CCONJ cana-3423 134 19 different different ADJ cana-3423 134 20 initial initial ADJ cana-3423 134 21 conditions condition NOUN cana-3423 134 22 as as ADV cana-3423 134 23 well well ADV cana-3423 134 24 as as ADP cana-3423 134 25 boundary boundary ADJ cana-3423 134 26 values value NOUN cana-3423 134 27 in in ADP cana-3423 134 28 terms term NOUN cana-3423 134 29 of of ADP cana-3423 134 30 the the DET cana-3423 134 31 convergence convergence NOUN cana-3423 134 32 rates rate NOUN cana-3423 134 33 . . PUNCT cana-3423 135 1 communications communication NOUN cana-3423 135 2 on on ADP cana-3423 135 3 applied apply VERB cana-3423 135 4 nonlinear nonlinear ADJ cana-3423 135 5 analysis analysis NOUN cana-3423 135 6 issn issn NOUN cana-3423 135 7 : : PUNCT cana-3423 135 8 1074 1074 NUM cana-3423 135 9 - - PUNCT cana-3423 135 10 133x 133x NUM cana-3423 135 11 vol vol NOUN cana-3423 135 12 32 32 NUM cana-3423 135 13 no no NOUN cana-3423 135 14 . . PUNCT cana-3423 136 1 7s 7 NOUN cana-3423 136 2 ( ( PUNCT cana-3423 136 3 2025 2025 NUM cana-3423 136 4 ) ) PUNCT cana-3423 136 5 https://internationalpubls.com https://internationalpubls.com X cana-3423 136 6 364 364 NUM cana-3423 136 7 table table NOUN cana-3423 136 8 1 1 NUM cana-3423 136 9 : : PUNCT cana-3423 136 10 convergence convergence NOUN cana-3423 136 11 rates rate NOUN cana-3423 136 12 of of ADP cana-3423 136 13 fixed fix VERB cana-3423 136 14 - - PUNCT cana-3423 136 15 point point NOUN cana-3423 136 16 iterative iterative NOUN cana-3423 136 17 scheme scheme NOUN cana-3423 136 18 initial initial ADJ cana-3423 136 19 condition condition NOUN cana-3423 136 20 boundary boundary ADJ cana-3423 136 21 type type NOUN cana-3423 136 22 convergence convergence NOUN cana-3423 136 23 rate rate NOUN cana-3423 136 24 ( ( PUNCT cana-3423 136 25 iterations iteration NOUN cana-3423 136 26 ) ) PUNCT cana-3423 136 27 final final ADJ cana-3423 136 28 error error NOUN cana-3423 136 29 ( ( PUNCT cana-3423 136 30 10 10 NUM cana-3423 136 31 ^ ^ SYM cana-3423 136 32 -6 -6 NOUN cana-3423 136 33 ) ) PUNCT cana-3423 136 34 condition condition NOUN cana-3423 136 35 1 1 NUM cana-3423 136 36 dirichlet dirichlet PROPN cana-3423 136 37 10 10 NUM cana-3423 136 38 1.2 1.2 NUM cana-3423 136 39 condition condition NOUN cana-3423 136 40 2 2 NUM cana-3423 136 41 neumann neumann PROPN cana-3423 136 42 15 15 NUM cana-3423 136 43 0.9 0.9 NUM cana-3423 136 44 condition condition NOUN cana-3423 136 45 3 3 NUM cana-3423 136 46 mixed mixed ADJ cana-3423 136 47 12 12 NUM cana-3423 136 48 1.0 1.0 NUM cana-3423 136 49 the the DET cana-3423 136 50 findings finding NOUN cana-3423 136 51 show show VERB cana-3423 136 52 that that SCONJ cana-3423 136 53 the the DET cana-3423 136 54 use use NOUN cana-3423 136 55 of of ADP cana-3423 136 56 the the DET cana-3423 136 57 fixed fix VERB cana-3423 136 58 - - PUNCT cana-3423 136 59 point point NOUN cana-3423 136 60 scheme scheme NOUN cana-3423 136 61 is be AUX cana-3423 136 62 accurate accurate ADJ cana-3423 136 63 in in ADP cana-3423 136 64 achieving achieve VERB cana-3423 136 65 a a DET cana-3423 136 66 fast fast ADJ cana-3423 136 67 convergence convergence NOUN cana-3423 136 68 with with ADP cana-3423 136 69 dirichlet dirichlet PROPN cana-3423 136 70 and and CCONJ cana-3423 136 71 mixed mixed ADJ cana-3423 136 72 boundary boundary ADJ cana-3423 136 73 conditions condition NOUN cana-3423 136 74 with with ADP cana-3423 136 75 almost almost ADV cana-3423 136 76 negligible negligible ADJ cana-3423 136 77 error error NOUN cana-3423 136 78 , , PUNCT cana-3423 136 79 thus thus ADV cana-3423 136 80 proving prove VERB cana-3423 136 81 the the DET cana-3423 136 82 applicability applicability NOUN cana-3423 136 83 of of ADP cana-3423 136 84 the the DET cana-3423 136 85 method method NOUN cana-3423 136 86 for for ADP cana-3423 136 87 the the DET cana-3423 136 88 mentioned mention VERB cana-3423 136 89 nonlinearities nonlinearitie NOUN cana-3423 136 90 in in ADP cana-3423 136 91 pdes pde NOUN cana-3423 136 92 . . PUNCT cana-3423 137 1 5.2 5.2 NUM cana-3423 137 2 degree degree NOUN cana-3423 137 3 theory theory NOUN cana-3423 137 4 application application NOUN cana-3423 137 5 and and CCONJ cana-3423 137 6 solution solution NOUN cana-3423 137 7 multiplicity multiplicity NOUN cana-3423 137 8 to to PART cana-3423 137 9 assess assess VERB cana-3423 137 10 the the DET cana-3423 137 11 solution solution NOUN cana-3423 137 12 multiplicity multiplicity NOUN cana-3423 137 13 in in ADP cana-3423 137 14 higher high ADJ cana-3423 137 15 dimensional dimensional ADJ cana-3423 137 16 systems system NOUN cana-3423 137 17 the the DET cana-3423 137 18 topological topological ADJ cana-3423 137 19 degree degree NOUN cana-3423 137 20 theory theory NOUN cana-3423 137 21 using use VERB cana-3423 137 22 leray leray ADJ cana-3423 137 23 - - PUNCT cana-3423 137 24 schauder schauder NOUN cana-3423 137 25 degree degree NOUN cana-3423 137 26 was be AUX cana-3423 137 27 used use VERB cana-3423 137 28 . . PUNCT cana-3423 138 1 regarding regard VERB cana-3423 138 2 the the DET cana-3423 138 3 degree degree NOUN cana-3423 138 4 distribution distribution NOUN cana-3423 138 5 , , PUNCT cana-3423 138 6 potential potential ADJ cana-3423 138 7 boundary boundary ADJ cana-3423 138 8 conditions condition NOUN cana-3423 138 9 and and CCONJ cana-3423 138 10 their their PRON cana-3423 138 11 relation relation NOUN cana-3423 138 12 to to ADP cana-3423 138 13 the the DET cana-3423 138 14 solution solution NOUN cana-3423 138 15 count count NOUN cana-3423 138 16 are be AUX cana-3423 138 17 depicted depict VERB cana-3423 138 18 in in ADP cana-3423 138 19 figure figure NOUN cana-3423 138 20 1 1 NUM cana-3423 138 21 . . PUNCT cana-3423 139 1 figure figure VERB cana-3423 139 2 1 1 NUM cana-3423 139 3 : : PUNCT cana-3423 139 4 degree degree NOUN cana-3423 139 5 distribution distribution NOUN cana-3423 139 6 for for ADP cana-3423 139 7 different different ADJ cana-3423 139 8 boundary boundary ADJ cana-3423 139 9 conditions condition NOUN cana-3423 139 10 graph graph NOUN cana-3423 139 11 description description NOUN cana-3423 139 12 : : PUNCT cana-3423 139 13 a a DET cana-3423 139 14 multi multi ADJ cana-3423 139 15 - - ADJ cana-3423 139 16 line line ADJ cana-3423 139 17 plot plot NOUN cana-3423 139 18 showing showing NOUN cana-3423 139 19 degree degree NOUN cana-3423 139 20 values value NOUN cana-3423 139 21 across across ADP cana-3423 139 22 boundary boundary ADJ cana-3423 139 23 types type NOUN cana-3423 139 24 ( ( PUNCT cana-3423 139 25 dirichlet dirichlet PROPN cana-3423 139 26 , , PUNCT cana-3423 139 27 neumann neumann PROPN cana-3423 139 28 , , PUNCT cana-3423 139 29 and and CCONJ cana-3423 139 30 mixed mixed ADJ cana-3423 139 31 ) ) PUNCT cana-3423 139 32 , , PUNCT cana-3423 139 33 with with ADP cana-3423 139 34 solution solution NOUN cana-3423 139 35 multiplicity multiplicity NOUN cana-3423 139 36 increasing increase VERB cana-3423 139 37 under under ADP cana-3423 139 38 complex complex ADJ cana-3423 139 39 mixed mixed ADJ cana-3423 139 40 conditions condition NOUN cana-3423 139 41 . . PUNCT cana-3423 140 1 this this DET cana-3423 140 2 analysis analysis NOUN cana-3423 140 3 shows show VERB cana-3423 140 4 that that SCONJ cana-3423 140 5 degree degree NOUN cana-3423 140 6 theory theory NOUN cana-3423 140 7 not not PART cana-3423 140 8 only only ADV cana-3423 140 9 confirms confirm VERB cana-3423 140 10 solution solution NOUN cana-3423 140 11 existence existence NOUN cana-3423 140 12 but but CCONJ cana-3423 140 13 also also ADV cana-3423 140 14 highlights highlight VERB cana-3423 140 15 cases case NOUN cana-3423 140 16 with with ADP cana-3423 140 17 multiple multiple ADJ cana-3423 140 18 solutions solution NOUN cana-3423 140 19 , , PUNCT cana-3423 140 20 particularly particularly ADV cana-3423 140 21 in in ADP cana-3423 140 22 nonlinear nonlinear ADJ cana-3423 140 23 systems system NOUN cana-3423 140 24 with with ADP cana-3423 140 25 mixed mixed ADJ cana-3423 140 26 boundaries boundary NOUN cana-3423 140 27 . . PUNCT cana-3423 141 1 5.3 5.3 NUM cana-3423 141 2 critical critical ADJ cana-3423 141 3 point point NOUN cana-3423 141 4 analysis analysis NOUN cana-3423 141 5 using use VERB cana-3423 141 6 variational variational ADJ cana-3423 141 7 methods method NOUN cana-3423 141 8 it it PRON cana-3423 141 9 was be AUX cana-3423 141 10 possible possible ADJ cana-3423 141 11 to to PART cana-3423 141 12 apply apply VERB cana-3423 141 13 the the DET cana-3423 141 14 mountain mountain NOUN cana-3423 141 15 pass pass NOUN cana-3423 141 16 theorem theorem NOUN cana-3423 141 17 for for ADP cana-3423 141 18 the the DET cana-3423 141 19 analysis analysis NOUN cana-3423 141 20 of of ADP cana-3423 141 21 the the DET cana-3423 141 22 critical critical ADJ cana-3423 141 23 points point NOUN cana-3423 141 24 of of ADP cana-3423 141 25 the the DET cana-3423 141 26 function function NOUN cana-3423 141 27 relating relate VERB cana-3423 141 28 to to ADP cana-3423 141 29 some some DET cana-3423 141 30 particular particular ADJ cana-3423 141 31 nonlinear nonlinear ADJ cana-3423 141 32 pdes pde NOUN cana-3423 141 33 to to PART cana-3423 141 34 describe describe VERB cana-3423 141 35 solutions solution NOUN cana-3423 141 36 ' ' PART cana-3423 141 37 stability stability NOUN cana-3423 141 38 and and CCONJ cana-3423 141 39 structure structure NOUN cana-3423 141 40 . . PUNCT cana-3423 142 1 the the DET cana-3423 142 2 critical critical ADJ cana-3423 142 3 points point NOUN cana-3423 142 4 corresponded correspond VERB cana-3423 142 5 with with ADP cana-3423 142 6 the the DET cana-3423 142 7 stable stable ADJ cana-3423 142 8 and and CCONJ cana-3423 142 9 unstable unstable ADJ cana-3423 142 10 equilibrium equilibrium NOUN cana-3423 142 11 solutions solution NOUN cana-3423 142 12 and and CCONJ cana-3423 142 13 the the DET cana-3423 142 14 variational variational ADJ cana-3423 142 15 equations equation NOUN cana-3423 142 16 identified identify VERB cana-3423 142 17 the the DET cana-3423 142 18 solution solution NOUN cana-3423 142 19 patterns pattern NOUN cana-3423 142 20 . . PUNCT cana-3423 143 1 the the DET cana-3423 143 2 critical critical ADJ cana-3423 143 3 points point NOUN cana-3423 143 4 obtained obtain VERB cana-3423 143 5 and and CCONJ cana-3423 143 6 the the DET cana-3423 143 7 stability stability NOUN cana-3423 143 8 characteristics characteristic NOUN cana-3423 143 9 of of ADP cana-3423 143 10 the the DET cana-3423 143 11 system system NOUN cana-3423 143 12 have have AUX cana-3423 143 13 been be AUX cana-3423 143 14 presented present VERB cana-3423 143 15 in in ADP cana-3423 143 16 the the DET cana-3423 143 17 following follow VERB cana-3423 143 18 table table NOUN cana-3423 143 19 2 2 NUM cana-3423 143 20 . . PUNCT cana-3423 143 21 table table NOUN cana-3423 143 22 2 2 NUM cana-3423 143 23 : : PUNCT cana-3423 143 24 critical critical ADJ cana-3423 143 25 points point NOUN cana-3423 143 26 and and CCONJ cana-3423 143 27 stability stability NOUN cana-3423 143 28 analysis analysis NOUN cana-3423 143 29 pde pde NOUN cana-3423 143 30 type type NOUN cana-3423 143 31 critical critical ADJ cana-3423 143 32 point point NOUN cana-3423 143 33 stability stability NOUN cana-3423 143 34 energy energy NOUN cana-3423 143 35 level level NOUN cana-3423 143 36 ( ( PUNCT cana-3423 143 37 j j NOUN cana-3423 143 38 ) ) PUNCT cana-3423 143 39 quasilinear quasilinear NOUN cana-3423 143 40 equation equation NOUN cana-3423 143 41 1 1 NUM cana-3423 143 42 0.5 0.5 NUM cana-3423 143 43 stable stable ADJ cana-3423 143 44 10 10 NUM cana-3423 143 45 semilinear semilinear NOUN cana-3423 143 46 equation equation NOUN cana-3423 143 47 2 2 NUM cana-3423 143 48 1.0 1.0 NUM cana-3423 143 49 unstable unstable ADJ cana-3423 143 50 12 12 NUM cana-3423 143 51 mixed mixed ADJ cana-3423 143 52 boundary boundary ADJ cana-3423 143 53 system system NOUN cana-3423 143 54 1.5 1.5 NUM cana-3423 143 55 stable stable ADJ cana-3423 143 56 8 8 NUM cana-3423 143 57 this this DET cana-3423 143 58 table table NOUN cana-3423 143 59 shows show VERB cana-3423 143 60 that that SCONJ cana-3423 143 61 variational variational ADJ cana-3423 143 62 methods method NOUN cana-3423 143 63 successfully successfully ADV cana-3423 143 64 identify identify VERB cana-3423 143 65 the the DET cana-3423 143 66 stability stability NOUN cana-3423 143 67 and and CCONJ cana-3423 143 68 energies energy NOUN cana-3423 143 69 of of ADP cana-3423 143 70 solutions solution NOUN cana-3423 143 71 as as ADP cana-3423 143 72 tools tool NOUN cana-3423 143 73 for for ADP cana-3423 143 74 measuring measure VERB cana-3423 143 75 the the DET cana-3423 143 76 nonlinear nonlinear ADJ cana-3423 143 77 dynamics dynamic NOUN cana-3423 143 78 of of ADP cana-3423 143 79 systems system NOUN cana-3423 143 80 . . PUNCT cana-3423 144 1 communications communication NOUN cana-3423 144 2 on on ADP cana-3423 144 3 applied apply VERB cana-3423 144 4 nonlinear nonlinear ADJ cana-3423 144 5 analysis analysis NOUN cana-3423 144 6 issn issn NOUN cana-3423 144 7 : : PUNCT cana-3423 144 8 1074 1074 NUM cana-3423 144 9 - - PUNCT cana-3423 144 10 133x 133x NUM cana-3423 144 11 vol vol NOUN cana-3423 144 12 32 32 NUM cana-3423 144 13 no no NOUN cana-3423 144 14 . . PUNCT cana-3423 145 1 7s 7 NOUN cana-3423 145 2 ( ( PUNCT cana-3423 145 3 2025 2025 NUM cana-3423 145 4 ) ) PUNCT cana-3423 146 1 https://internationalpubls.com https://internationalpubls.com X cana-3423 146 2 365 365 NUM cana-3423 146 3 5.4 5.4 NUM cana-3423 146 4 computational computational ADJ cana-3423 146 5 validation validation NOUN cana-3423 146 6 and and CCONJ cana-3423 146 7 error error NOUN cana-3423 146 8 analysis analysis NOUN cana-3423 146 9 to to PART cana-3423 146 10 evaluate evaluate VERB cana-3423 146 11 the the DET cana-3423 146 12 efficiency efficiency NOUN cana-3423 146 13 of of ADP cana-3423 146 14 the the DET cana-3423 146 15 utilized utilize VERB cana-3423 146 16 numerical numerical ADJ cana-3423 146 17 schemes scheme NOUN cana-3423 146 18 , , PUNCT cana-3423 146 19 the the DET cana-3423 146 20 error error NOUN cana-3423 146 21 analysis analysis NOUN cana-3423 146 22 and and CCONJ cana-3423 146 23 convergence convergence NOUN cana-3423 146 24 study study NOUN cana-3423 146 25 were be AUX cana-3423 146 26 conducted conduct VERB cana-3423 146 27 . . PUNCT cana-3423 147 1 we we PRON cana-3423 147 2 then then ADV cana-3423 147 3 compared compare VERB cana-3423 147 4 the the DET cana-3423 147 5 efficiency efficiency NOUN cana-3423 147 6 and and CCONJ cana-3423 147 7 error error NOUN cana-3423 147 8 of of ADP cana-3423 147 9 the the DET cana-3423 147 10 newton newton PROPN cana-3423 147 11 - - PUNCT cana-3423 147 12 raphson raphson NOUN cana-3423 147 13 as as ADV cana-3423 147 14 well well ADV cana-3423 147 15 as as ADP cana-3423 147 16 the the DET cana-3423 147 17 fixed fix VERB cana-3423 147 18 - - PUNCT cana-3423 147 19 point point NOUN cana-3423 147 20 iterative iterative NOUN cana-3423 147 21 schemes scheme NOUN cana-3423 147 22 . . PUNCT cana-3423 148 1 the the DET cana-3423 148 2 following follow VERB cana-3423 148 3 figure figure NOUN cana-3423 148 4 2 2 NUM cana-3423 148 5 shows show VERB cana-3423 148 6 the the DET cana-3423 148 7 error error NOUN cana-3423 148 8 convergence convergence NOUN cana-3423 148 9 for for ADP cana-3423 148 10 each each DET cana-3423 148 11 method method NOUN cana-3423 148 12 to to PART cana-3423 148 13 show show VERB cana-3423 148 14 how how SCONJ cana-3423 148 15 they they PRON cana-3423 148 16 scale scale VERB cana-3423 148 17 at at ADP cana-3423 148 18 different different ADJ cana-3423 148 19 mesh mesh NOUN cana-3423 148 20 densities density NOUN cana-3423 148 21 . . PUNCT cana-3423 149 1 figure figure NOUN cana-3423 149 2 2 2 NUM cana-3423 149 3 : : PUNCT cana-3423 149 4 error error NOUN cana-3423 149 5 convergence convergence NOUN cana-3423 149 6 of of ADP cana-3423 149 7 newton newton PROPN cana-3423 149 8 - - PUNCT cana-3423 149 9 raphson raphson NOUN cana-3423 149 10 and and CCONJ cana-3423 149 11 fixed fix VERB cana-3423 149 12 - - PUNCT cana-3423 149 13 point point NOUN cana-3423 149 14 schemes scheme NOUN cana-3423 149 15 a a DET cana-3423 149 16 line line NOUN cana-3423 149 17 plot plot NOUN cana-3423 149 18 comparing compare VERB cana-3423 149 19 error error NOUN cana-3423 149 20 convergence convergence NOUN cana-3423 149 21 rates rate NOUN cana-3423 149 22 for for ADP cana-3423 149 23 newton newton PROPN cana-3423 149 24 - - PUNCT cana-3423 149 25 raphson raphson NOUN cana-3423 149 26 and and CCONJ cana-3423 149 27 fixed fix VERB cana-3423 149 28 - - PUNCT cana-3423 149 29 point point NOUN cana-3423 149 30 schemes scheme NOUN cana-3423 149 31 across across ADP cana-3423 149 32 mesh mesh NOUN cana-3423 149 33 densities density NOUN cana-3423 149 34 ( ( PUNCT cana-3423 149 35 0.1 0.1 NUM cana-3423 149 36 , , PUNCT cana-3423 149 37 0.05 0.05 NUM cana-3423 149 38 , , PUNCT cana-3423 149 39 0.01 0.01 NUM cana-3423 149 40 ) ) PUNCT cana-3423 149 41 . . PUNCT cana-3423 150 1 newton newton PROPN cana-3423 150 2 - - PUNCT cana-3423 150 3 raphson raphson PROPN cana-3423 150 4 exhibits exhibit VERB cana-3423 150 5 a a DET cana-3423 150 6 faster fast ADJ cana-3423 150 7 convergence convergence NOUN cana-3423 150 8 rate rate NOUN cana-3423 150 9 with with ADP cana-3423 150 10 lower low ADJ cana-3423 150 11 overall overall ADJ cana-3423 150 12 error error NOUN cana-3423 150 13 . . PUNCT cana-3423 151 1 the the DET cana-3423 151 2 results result NOUN cana-3423 151 3 prove prove VERB cana-3423 151 4 that that SCONJ cana-3423 151 5 the the DET cana-3423 151 6 newton newton PROPN cana-3423 151 7 - - PUNCT cana-3423 151 8 raphson raphson NOUN cana-3423 151 9 scheme scheme NOUN cana-3423 151 10 is be AUX cana-3423 151 11 more more ADV cana-3423 151 12 efficient efficient ADJ cana-3423 151 13 than than ADP cana-3423 151 14 the the DET cana-3423 151 15 fixed fix VERB cana-3423 151 16 - - PUNCT cana-3423 151 17 point point NOUN cana-3423 151 18 scheme scheme NOUN cana-3423 151 19 in in ADP cana-3423 151 20 aspects aspect NOUN cana-3423 151 21 of of ADP cana-3423 151 22 speed speed NOUN cana-3423 151 23 and and CCONJ cana-3423 151 24 accuracy accuracy NOUN cana-3423 151 25 especially especially ADV cana-3423 151 26 in in ADP cana-3423 151 27 the the DET cana-3423 151 28 models model NOUN cana-3423 151 29 with with ADP cana-3423 151 30 nonlinear nonlinear ADJ cana-3423 151 31 dependencies dependency NOUN cana-3423 151 32 . . PUNCT cana-3423 152 1 5.5 5.5 NUM cana-3423 152 2 summary summary NOUN cana-3423 152 3 of of ADP cana-3423 152 4 key key ADJ cana-3423 152 5 findings finding NOUN cana-3423 152 6 the the DET cana-3423 152 7 following follow VERB cana-3423 152 8 points point NOUN cana-3423 152 9 summarize summarize VERB cana-3423 152 10 the the DET cana-3423 152 11 main main ADJ cana-3423 152 12 findings finding NOUN cana-3423 152 13 : : PUNCT cana-3423 152 14 1 1 X cana-3423 152 15 . . PUNCT cana-3423 152 16 fixed fix VERB cana-3423 152 17 - - PUNCT cana-3423 152 18 point point NOUN cana-3423 152 19 iterative iterative NOUN cana-3423 152 20 scheme scheme NOUN cana-3423 152 21 : : PUNCT cana-3423 152 22 obtained obtain VERB cana-3423 152 23 high high ADJ cana-3423 152 24 convergence convergence NOUN cana-3423 152 25 rates rate NOUN cana-3423 152 26 together together ADV cana-3423 152 27 with with ADP cana-3423 152 28 negligible negligible ADJ cana-3423 152 29 errors error NOUN cana-3423 152 30 for for ADP cana-3423 152 31 semilinear semilinear ADJ cana-3423 152 32 pdes pde NOUN cana-3423 152 33 with with ADP cana-3423 152 34 convergence convergence NOUN cana-3423 152 35 to to ADP cana-3423 152 36 nonlinearities nonlinearitie NOUN cana-3423 152 37 confirming confirm VERB cana-3423 152 38 solution solution NOUN cana-3423 152 39 uniqueness uniqueness NOUN cana-3423 152 40 in in ADP cana-3423 152 41 the the DET cana-3423 152 42 case case NOUN cana-3423 152 43 of of ADP cana-3423 152 44 proper proper ADJ cana-3423 152 45 boundary boundary ADJ cana-3423 152 46 conditions condition NOUN cana-3423 152 47 . . PUNCT cana-3423 153 1 2 2 X cana-3423 153 2 . . NOUN cana-3423 153 3 degree degree NOUN cana-3423 153 4 theory theory NOUN cana-3423 153 5 application application NOUN cana-3423 153 6 : : PUNCT cana-3423 153 7 a a DET cana-3423 153 8 capacity capacity NOUN cana-3423 153 9 to to PART cana-3423 153 10 demonstrate demonstrate VERB cana-3423 153 11 the the DET cana-3423 153 12 solution solution NOUN cana-3423 153 13 of of ADP cana-3423 153 14 multiple multiple ADJ cana-3423 153 15 solutions solution NOUN cana-3423 153 16 was be AUX cana-3423 153 17 also also ADV cana-3423 153 18 confirmed confirm VERB cana-3423 153 19 and and CCONJ cana-3423 153 20 their their PRON cana-3423 153 21 existence existence NOUN cana-3423 153 22 was be AUX cana-3423 153 23 also also ADV cana-3423 153 24 established establish VERB cana-3423 153 25 in in ADP cana-3423 153 26 high high ADJ cana-3423 153 27 - - PUNCT cana-3423 153 28 dimensional dimensional ADJ cana-3423 153 29 contexts context NOUN cana-3423 153 30 particularly particularly ADV cana-3423 153 31 where where SCONJ cana-3423 153 32 mixed mixed ADJ cana-3423 153 33 boundary boundary ADJ cana-3423 153 34 values value NOUN cana-3423 153 35 are be AUX cana-3423 153 36 involved involve VERB cana-3423 153 37 . . PUNCT cana-3423 154 1 3 3 X cana-3423 154 2 . . X cana-3423 154 3 critical critical ADJ cana-3423 154 4 point point NOUN cana-3423 154 5 analysis analysis NOUN cana-3423 154 6 : : PUNCT cana-3423 154 7 characterised characterise VERB cana-3423 154 8 steady steady ADJ cana-3423 154 9 and and CCONJ cana-3423 154 10 oscillatory oscillatory ADJ cana-3423 154 11 solutions solution NOUN cana-3423 154 12 according accord VERB cana-3423 154 13 to to ADP cana-3423 154 14 the the DET cana-3423 154 15 critical critical ADJ cana-3423 154 16 point point NOUN cana-3423 154 17 analysis analysis NOUN cana-3423 154 18 to to PART cana-3423 154 19 understand understand VERB cana-3423 154 20 the the DET cana-3423 154 21 stability stability NOUN cana-3423 154 22 of of ADP cana-3423 154 23 the the DET cana-3423 154 24 solutions solution NOUN cana-3423 154 25 . . PUNCT cana-3423 155 1 4 4 X cana-3423 155 2 . . X cana-3423 155 3 computational computational ADJ cana-3423 155 4 efficiency efficiency NOUN cana-3423 155 5 : : PUNCT cana-3423 155 6 newton newton PROPN cana-3423 155 7 - - PUNCT cana-3423 155 8 raphson raphson PROPN cana-3423 155 9 outperformed outperform VERB cana-3423 155 10 fixed fix VERB cana-3423 155 11 - - PUNCT cana-3423 155 12 point point NOUN cana-3423 155 13 methods method NOUN cana-3423 155 14 in in ADP cana-3423 155 15 terms term NOUN cana-3423 155 16 of of ADP cana-3423 155 17 convergence convergence NOUN cana-3423 155 18 rate rate NOUN cana-3423 155 19 and and CCONJ cana-3423 155 20 error error NOUN cana-3423 155 21 reduction reduction NOUN cana-3423 155 22 , , PUNCT cana-3423 155 23 validating validate VERB cana-3423 155 24 it it PRON cana-3423 155 25 as as ADP cana-3423 155 26 a a DET cana-3423 155 27 superior superior ADJ cana-3423 155 28 approach approach NOUN cana-3423 155 29 for for ADP cana-3423 155 30 complex complex ADJ cana-3423 155 31 nonlinearity nonlinearity NOUN cana-3423 155 32 . . PUNCT cana-3423 156 1 6 6 NUM cana-3423 156 2 . . X cana-3423 156 3 discussion discussion NOUN cana-3423 156 4 the the DET cana-3423 156 5 results result NOUN cana-3423 156 6 of of ADP cana-3423 156 7 this this DET cana-3423 156 8 study study NOUN cana-3423 156 9 are be AUX cana-3423 156 10 of of ADP cana-3423 156 11 great great ADJ cana-3423 156 12 importance importance NOUN cana-3423 156 13 to to AUX cana-3423 156 14 better well ADV cana-3423 156 15 understanding understand VERB cana-3423 156 16 how how SCONJ cana-3423 156 17 topological topological ADJ cana-3423 156 18 methods method NOUN cana-3423 156 19 can can AUX cana-3423 156 20 be be AUX cana-3423 156 21 applied apply VERB cana-3423 156 22 in in ADP cana-3423 156 23 solving solve VERB cana-3423 156 24 the the DET cana-3423 156 25 existence existence NOUN cana-3423 156 26 , , PUNCT cana-3423 156 27 uniqueness uniqueness NOUN cana-3423 156 28 , , PUNCT cana-3423 156 29 stability stability NOUN cana-3423 156 30 , , PUNCT cana-3423 156 31 and and CCONJ cana-3423 156 32 computation computation NOUN cana-3423 156 33 of of ADP cana-3423 156 34 the the DET cana-3423 156 35 solutions solution NOUN cana-3423 156 36 to to ADP cana-3423 156 37 nonlinear nonlinear ADJ cana-3423 156 38 pdes pde NOUN cana-3423 156 39 . . PUNCT cana-3423 157 1 combining combine VERB cana-3423 157 2 the the DET cana-3423 157 3 findings finding NOUN cana-3423 157 4 of of ADP cana-3423 157 5 fixed fix VERB cana-3423 157 6 - - PUNCT cana-3423 157 7 point point NOUN cana-3423 157 8 theory theory NOUN cana-3423 157 9 , , PUNCT cana-3423 157 10 the the DET cana-3423 157 11 degree degree NOUN cana-3423 157 12 theory theory NOUN cana-3423 157 13 together together ADV cana-3423 157 14 with with ADP cana-3423 157 15 variational variational ADJ cana-3423 157 16 methods method NOUN cana-3423 157 17 , , PUNCT cana-3423 157 18 this this DET cana-3423 157 19 thesis thesis NOUN cana-3423 157 20 articulates articulate VERB cana-3423 157 21 the the DET cana-3423 157 22 application application NOUN cana-3423 157 23 of of ADP cana-3423 157 24 multi multi ADJ cana-3423 157 25 - - ADJ cana-3423 157 26 faceted faceted ADJ cana-3423 157 27 solutions solution NOUN cana-3423 157 28 for for ADP cana-3423 157 29 the the DET cana-3423 157 30 solution solution NOUN cana-3423 157 31 of of ADP cana-3423 157 32 nonlinear nonlinear ADJ cana-3423 157 33 systems system NOUN cana-3423 157 34 especially especially ADV cana-3423 157 35 where where SCONJ cana-3423 157 36 classical classical ADJ cana-3423 157 37 techniques technique NOUN cana-3423 157 38 are be AUX cana-3423 157 39 unhelpful unhelpful ADJ cana-3423 157 40 . . PUNCT cana-3423 158 1 this this DET cana-3423 158 2 section section NOUN cana-3423 158 3 summarizes summarize VERB cana-3423 158 4 the the DET cana-3423 158 5 major major ADJ cana-3423 158 6 results result NOUN cana-3423 158 7 , , PUNCT cana-3423 158 8 compares compare VERB cana-3423 158 9 them they PRON cana-3423 158 10 with with ADP cana-3423 158 11 theory theory NOUN cana-3423 158 12 , , PUNCT cana-3423 158 13 states state VERB cana-3423 158 14 the the DET cana-3423 158 15 practicable practicable ADJ cana-3423 158 16 application application NOUN cana-3423 158 17 , , PUNCT cana-3423 158 18 notes note VERB cana-3423 158 19 limitations limitation NOUN cana-3423 158 20 , , PUNCT cana-3423 158 21 and and CCONJ cana-3423 158 22 defines define VERB cana-3423 158 23 further further ADJ cana-3423 158 24 research research NOUN cana-3423 158 25 prospects prospect NOUN cana-3423 158 26 . . PUNCT cana-3423 159 1 communications communication NOUN cana-3423 159 2 on on ADP cana-3423 159 3 applied apply VERB cana-3423 159 4 nonlinear nonlinear ADJ cana-3423 159 5 analysis analysis NOUN cana-3423 159 6 issn issn NOUN cana-3423 159 7 : : PUNCT cana-3423 159 8 1074 1074 NUM cana-3423 159 9 - - PUNCT cana-3423 159 10 133x 133x NUM cana-3423 159 11 vol vol NOUN cana-3423 159 12 32 32 NUM cana-3423 159 13 no no NOUN cana-3423 159 14 . . PUNCT cana-3423 160 1 7s 7 NOUN cana-3423 160 2 ( ( PUNCT cana-3423 160 3 2025 2025 NUM cana-3423 160 4 ) ) PUNCT cana-3423 161 1 https://internationalpubls.com https://internationalpubls.com X cana-3423 161 2 366 366 NUM cana-3423 161 3 6.1 6.1 NUM cana-3423 161 4 interpretation interpretation NOUN cana-3423 161 5 of of ADP cana-3423 161 6 key key ADJ cana-3423 161 7 findings finding NOUN cana-3423 161 8 the the DET cana-3423 161 9 results result NOUN cana-3423 161 10 reveal reveal VERB cana-3423 161 11 how how SCONJ cana-3423 161 12 the the DET cana-3423 161 13 topological topological ADJ cana-3423 161 14 methods method NOUN cana-3423 161 15 are be AUX cana-3423 161 16 useful useful ADJ cana-3423 161 17 in in ADP cana-3423 161 18 establishing establish VERB cana-3423 161 19 the the DET cana-3423 161 20 existence existence NOUN cana-3423 161 21 and and CCONJ cana-3423 161 22 stability stability NOUN cana-3423 161 23 of of ADP cana-3423 161 24 solutions solution NOUN cana-3423 161 25 in in ADP cana-3423 161 26 the the DET cana-3423 161 27 nonlinear nonlinear ADJ cana-3423 161 28 pdes pde NOUN cana-3423 161 29 . . PUNCT cana-3423 162 1 for for ADP cana-3423 162 2 instance instance NOUN cana-3423 162 3 , , PUNCT cana-3423 162 4 the the DET cana-3423 162 5 application application NOUN cana-3423 162 6 of of ADP cana-3423 162 7 the the DET cana-3423 162 8 banach banach ADV cana-3423 162 9 fixed fix VERB cana-3423 162 10 - - PUNCT cana-3423 162 11 point point NOUN cana-3423 162 12 theorem theorem NOUN cana-3423 162 13 in in ADP cana-3423 162 14 semi semi ADJ cana-3423 162 15 - - ADJ cana-3423 162 16 linear linear ADJ cana-3423 162 17 pdes pde NOUN cana-3423 162 18 provided provide VERB cana-3423 162 19 solution solution NOUN cana-3423 162 20 uniqueness uniqueness NOUN cana-3423 162 21 with with ADP cana-3423 162 22 convergence convergence NOUN cana-3423 162 23 rates rate NOUN cana-3423 162 24 purported purport VERB cana-3423 162 25 to to PART cana-3423 162 26 be be AUX cana-3423 162 27 highly highly ADV cana-3423 162 28 computational computational ADJ cana-3423 162 29 in in ADP cana-3423 162 30 some some DET cana-3423 162 31 boundary boundary ADJ cana-3423 162 32 conditions condition NOUN cana-3423 162 33 ( ( PUNCT cana-3423 162 34 medio medio NOUN cana-3423 162 35 & & CCONJ cana-3423 162 36 lines line NOUN cana-3423 162 37 , , PUNCT cana-3423 162 38 2001 2001 NUM cana-3423 162 39 ) ) PUNCT cana-3423 162 40 . . PUNCT cana-3423 163 1 this this PRON cana-3423 163 2 resonates resonate VERB cana-3423 163 3 with with ADP cana-3423 163 4 prior prior ADJ cana-3423 163 5 analyses analysis NOUN cana-3423 163 6 of of ADP cana-3423 163 7 the the DET cana-3423 163 8 fixed fix VERB cana-3423 163 9 point point NOUN cana-3423 163 10 approach approach NOUN cana-3423 163 11 in in ADP cana-3423 163 12 nonlinear nonlinear ADJ cana-3423 163 13 scenarios scenario NOUN cana-3423 163 14 as as ADP cana-3423 163 15 handy handy ADJ cana-3423 163 16 where where SCONJ cana-3423 163 17 contraction contraction NOUN cana-3423 163 18 mapping mapping NOUN cana-3423 163 19 is be AUX cana-3423 163 20 possible possible ADJ cana-3423 163 21 ( ( PUNCT cana-3423 163 22 evans evans PROPN cana-3423 163 23 , , PUNCT cana-3423 163 24 2010 2010 NUM cana-3423 163 25 ) ) PUNCT cana-3423 163 26 . . PUNCT cana-3423 164 1 this this DET cana-3423 164 2 work work NOUN cana-3423 164 3 further far ADV cana-3423 164 4 showed show VERB cana-3423 164 5 that that SCONJ cana-3423 164 6 the the DET cana-3423 164 7 presence presence NOUN cana-3423 164 8 of of ADP cana-3423 164 9 multiple multiple ADJ cana-3423 164 10 solutions solution NOUN cana-3423 164 11 for for ADP cana-3423 164 12 the the DET cana-3423 164 13 given give VERB cana-3423 164 14 controlled control VERB cana-3423 164 15 pde pde NOUN cana-3423 164 16 system system NOUN cana-3423 164 17 can can AUX cana-3423 164 18 be be AUX cana-3423 164 19 established establish VERB cana-3423 164 20 by by ADP cana-3423 164 21 degree degree NOUN cana-3423 164 22 theory theory NOUN cana-3423 164 23 . . PUNCT cana-3423 165 1 this this PRON cana-3423 165 2 is be AUX cana-3423 165 3 in in ADP cana-3423 165 4 concord concord PROPN cana-3423 165 5 with with ADP cana-3423 165 6 rabinowitz rabinowitz PROPN cana-3423 165 7 's 's PART cana-3423 165 8 ( ( PUNCT cana-3423 165 9 2015 2015 NUM cana-3423 165 10 ) ) PUNCT cana-3423 165 11 work work NOUN cana-3423 165 12 on on ADP cana-3423 165 13 degree degree NOUN cana-3423 165 14 theory theory NOUN cana-3423 165 15 in in ADP cana-3423 165 16 nonlinear nonlinear ADJ cana-3423 165 17 analysis analysis NOUN cana-3423 165 18 , , PUNCT cana-3423 165 19 where where SCONJ cana-3423 165 20 the the DET cana-3423 165 21 author author NOUN cana-3423 165 22 pointed point VERB cana-3423 165 23 out out ADP cana-3423 165 24 that that SCONJ cana-3423 165 25 the the DET cana-3423 165 26 topological topological ADJ cana-3423 165 27 degree degree NOUN cana-3423 165 28 methods method NOUN cana-3423 165 29 are be AUX cana-3423 165 30 most most ADV cana-3423 165 31 appropriate appropriate ADJ cana-3423 165 32 to to PART cana-3423 165 33 solve solve VERB cana-3423 165 34 problems problem NOUN cana-3423 165 35 of of ADP cana-3423 165 36 boundary boundary ADJ cana-3423 165 37 behaviors behavior NOUN cana-3423 165 38 when when SCONJ cana-3423 165 39 other other ADJ cana-3423 165 40 approaches approach NOUN cana-3423 165 41 may may AUX cana-3423 165 42 be be AUX cana-3423 165 43 inadequate inadequate ADJ cana-3423 165 44 . . PUNCT cana-3423 166 1 in in ADP cana-3423 166 2 the the DET cana-3423 166 3 same same ADJ cana-3423 166 4 way way NOUN cana-3423 166 5 , , PUNCT cana-3423 166 6 the the DET cana-3423 166 7 use use NOUN cana-3423 166 8 of of ADP cana-3423 166 9 variational variational ADJ cana-3423 166 10 methods method NOUN cana-3423 166 11 to to PART cana-3423 166 12 find find VERB cana-3423 166 13 critical critical ADJ cana-3423 166 14 points point NOUN cana-3423 166 15 established establish VERB cana-3423 166 16 the the DET cana-3423 166 17 stability stability NOUN cana-3423 166 18 of of ADP cana-3423 166 19 solutions solution NOUN cana-3423 166 20 and and CCONJ cana-3423 166 21 provided provide VERB cana-3423 166 22 analysis analysis NOUN cana-3423 166 23 into into ADP cana-3423 166 24 the the DET cana-3423 166 25 structure structure NOUN cana-3423 166 26 of of ADP cana-3423 166 27 the the DET cana-3423 166 28 solutions solution NOUN cana-3423 166 29 operationally operationally ADV cana-3423 166 30 confirming confirm VERB cana-3423 166 31 chang chang PROPN cana-3423 166 32 ’s ’s PART cana-3423 166 33 ( ( PUNCT cana-3423 166 34 2013 2013 NUM cana-3423 166 35 ) ) PUNCT cana-3423 166 36 proposal proposal NOUN cana-3423 166 37 of of ADP cana-3423 166 38 applying apply VERB cana-3423 166 39 critical critical ADJ cana-3423 166 40 point point NOUN cana-3423 166 41 theory theory NOUN cana-3423 166 42 in in ADP cana-3423 166 43 the the DET cana-3423 166 44 investigation investigation NOUN cana-3423 166 45 of of ADP cana-3423 166 46 stability stability NOUN cana-3423 166 47 of of ADP cana-3423 166 48 nonlinear nonlinear ADJ cana-3423 166 49 systems system NOUN cana-3423 166 50 . . PUNCT cana-3423 167 1 6.2 6.2 NUM cana-3423 167 2 relation relation NOUN cana-3423 167 3 to to ADP cana-3423 167 4 existing exist VERB cana-3423 167 5 literature literature NOUN cana-3423 167 6 this this DET cana-3423 167 7 work work NOUN cana-3423 167 8 is be AUX cana-3423 167 9 based base VERB cana-3423 167 10 on on ADP cana-3423 167 11 topological topological ADJ cana-3423 167 12 perspectives perspective NOUN cana-3423 167 13 that that PRON cana-3423 167 14 are be AUX cana-3423 167 15 developed develop VERB cana-3423 167 16 in in ADP cana-3423 167 17 the the DET cana-3423 167 18 framework framework NOUN cana-3423 167 19 of of ADP cana-3423 167 20 nonlinear nonlinear ADJ cana-3423 167 21 pdes pde NOUN cana-3423 167 22 furthering further VERB cana-3423 167 23 the the DET cana-3423 167 24 work work NOUN cana-3423 167 25 done do VERB cana-3423 167 26 in in ADP cana-3423 167 27 this this DET cana-3423 167 28 area area NOUN cana-3423 167 29 by by ADP cana-3423 167 30 expanding expand VERB cana-3423 167 31 topological topological ADJ cana-3423 167 32 methods method NOUN cana-3423 167 33 to to PART cana-3423 167 34 previously previously ADV cana-3423 167 35 explored explore VERB cana-3423 167 36 problems problem NOUN cana-3423 167 37 of of ADP cana-3423 167 38 higher high ADJ cana-3423 167 39 dimensions dimension NOUN cana-3423 167 40 and and CCONJ cana-3423 167 41 boundary boundary ADJ cana-3423 167 42 conditions condition NOUN cana-3423 167 43 . . PUNCT cana-3423 168 1 in in ADP cana-3423 168 2 this this DET cana-3423 168 3 work work NOUN cana-3423 168 4 , , PUNCT cana-3423 168 5 the the DET cana-3423 168 6 source source NOUN cana-3423 168 7 methods method NOUN cana-3423 168 8 are be AUX cana-3423 168 9 extended extend VERB cana-3423 168 10 to to ADP cana-3423 168 11 higher high ADJ cana-3423 168 12 dimensions dimension NOUN cana-3423 168 13 and and CCONJ cana-3423 168 14 applied apply VERB cana-3423 168 15 to to ADP cana-3423 168 16 a a DET cana-3423 168 17 variety variety NOUN cana-3423 168 18 of of ADP cana-3423 168 19 boundary boundary ADJ cana-3423 168 20 forms form NOUN cana-3423 168 21 , , PUNCT cana-3423 168 22 indicating indicate VERB cana-3423 168 23 that that SCONJ cana-3423 168 24 these these DET cana-3423 168 25 methods method NOUN cana-3423 168 26 may may AUX cana-3423 168 27 not not PART cana-3423 168 28 be be AUX cana-3423 168 29 limited limit VERB cana-3423 168 30 to to ADP cana-3423 168 31 the the DET cana-3423 168 32 lower low ADJ cana-3423 168 33 dimensions dimension NOUN cana-3423 168 34 and and CCONJ cana-3423 168 35 semi semi ADJ cana-3423 168 36 - - ADJ cana-3423 168 37 linear linear ADJ cana-3423 168 38 equations equation NOUN cana-3423 168 39 as as ADP cana-3423 168 40 berestycki berestycki NOUN cana-3423 168 41 and and CCONJ cana-3423 168 42 lions lion NOUN cana-3423 168 43 ( ( PUNCT cana-3423 168 44 2013 2013 NUM cana-3423 168 45 ) ) PUNCT cana-3423 168 46 came come VERB cana-3423 168 47 closer close ADV cana-3423 168 48 to to ADP cana-3423 168 49 pointing point VERB cana-3423 168 50 out out ADP cana-3423 168 51 . . PUNCT cana-3423 169 1 in in ADP cana-3423 169 2 addition addition NOUN cana-3423 169 3 , , PUNCT cana-3423 169 4 the the DET cana-3423 169 5 improvement improvement NOUN cana-3423 169 6 in in ADP cana-3423 169 7 computational computational ADJ cana-3423 169 8 speed speed NOUN cana-3423 169 9 with with ADP cana-3423 169 10 the the DET cana-3423 169 11 help help NOUN cana-3423 169 12 of of ADP cana-3423 169 13 newtonraphson newtonraphson PROPN cana-3423 169 14 ’s ’s PART cana-3423 169 15 scheme scheme NOUN cana-3423 169 16 over over ADP cana-3423 169 17 a a DET cana-3423 169 18 fixed fix VERB cana-3423 169 19 - - PUNCT cana-3423 169 20 point point NOUN cana-3423 169 21 method method NOUN cana-3423 169 22 supports support VERB cana-3423 169 23 the the DET cana-3423 169 24 statement statement NOUN cana-3423 169 25 of of ADP cana-3423 169 26 sΓΌli sΓΌli PROPN cana-3423 169 27 and and CCONJ cana-3423 169 28 mayers mayer NOUN cana-3423 169 29 ( ( PUNCT cana-3423 169 30 2003 2003 NUM cana-3423 169 31 ) ) PUNCT cana-3423 169 32 that that SCONJ cana-3423 169 33 the the DET cana-3423 169 34 newton newton PROPN cana-3423 169 35 - - PUNCT cana-3423 169 36 based base VERB cana-3423 169 37 method method NOUN cana-3423 169 38 is be AUX cana-3423 169 39 effective effective ADJ cana-3423 169 40 for for ADP cana-3423 169 41 convergence convergence NOUN cana-3423 169 42 in in ADP cana-3423 169 43 nonlinear nonlinear ADJ cana-3423 169 44 problems problem NOUN cana-3423 169 45 . . PUNCT cana-3423 170 1 6.3 6.3 NUM cana-3423 170 2 implications implication NOUN cana-3423 170 3 of of ADP cana-3423 170 4 the the DET cana-3423 170 5 findings finding NOUN cana-3423 170 6 the the DET cana-3423 170 7 findings finding NOUN cana-3423 170 8 of of ADP cana-3423 170 9 this this DET cana-3423 170 10 research research NOUN cana-3423 170 11 have have VERB cana-3423 170 12 several several ADJ cana-3423 170 13 practical practical ADJ cana-3423 170 14 and and CCONJ cana-3423 170 15 theoretical theoretical ADJ cana-3423 170 16 implications implication NOUN cana-3423 170 17 : : PUNCT cana-3423 170 18 1 1 X cana-3423 170 19 . . X cana-3423 170 20 practical practical ADJ cana-3423 170 21 applications application NOUN cana-3423 170 22 : : PUNCT cana-3423 170 23 the the DET cana-3423 170 24 ability ability NOUN cana-3423 170 25 to to PART cana-3423 170 26 model model VERB cana-3423 170 27 boundary boundary ADJ cana-3423 170 28 conditions condition NOUN cana-3423 170 29 using use VERB cana-3423 170 30 degree degree NOUN cana-3423 170 31 theory theory NOUN cana-3423 170 32 as as ADV cana-3423 170 33 well well ADV cana-3423 170 34 as as ADP cana-3423 170 35 leveraging leverage VERB cana-3423 170 36 variational variational ADJ cana-3423 170 37 methods method NOUN cana-3423 170 38 shows show VERB cana-3423 170 39 that that SCONJ cana-3423 170 40 topological topological ADJ cana-3423 170 41 techniques technique NOUN cana-3423 170 42 could could AUX cana-3423 170 43 be be AUX cana-3423 170 44 used use VERB cana-3423 170 45 in in ADP cana-3423 170 46 real real ADJ cana-3423 170 47 - - PUNCT cana-3423 170 48 world world NOUN cana-3423 170 49 problems problem NOUN cana-3423 170 50 in in ADP cana-3423 170 51 physics physics PROPN cana-3423 170 52 , , PUNCT cana-3423 170 53 engineering engineering NOUN cana-3423 170 54 , , PUNCT cana-3423 170 55 and and CCONJ cana-3423 170 56 environmental environmental ADJ cana-3423 170 57 modeling modeling NOUN cana-3423 170 58 where where SCONJ cana-3423 170 59 nonlinear nonlinear ADJ cana-3423 170 60 pdes pde NOUN cana-3423 170 61 are be AUX cana-3423 170 62 found find VERB cana-3423 170 63 . . PUNCT cana-3423 171 1 2 2 X cana-3423 171 2 . . X cana-3423 171 3 enhanced enhance VERB cana-3423 171 4 computational computational ADJ cana-3423 171 5 techniques technique NOUN cana-3423 171 6 : : PUNCT cana-3423 171 7 the the DET cana-3423 171 8 comparison comparison NOUN cana-3423 171 9 of of ADP cana-3423 171 10 newton newton PROPN cana-3423 171 11 - - PUNCT cana-3423 171 12 raphson raphson NOUN cana-3423 171 13 and and CCONJ cana-3423 171 14 fixed fix VERB cana-3423 171 15 - - PUNCT cana-3423 171 16 point point NOUN cana-3423 171 17 methods method NOUN cana-3423 171 18 is be AUX cana-3423 171 19 useful useful ADJ cana-3423 171 20 when when SCONJ cana-3423 171 21 choosing choose VERB cana-3423 171 22 casing casing NOUN cana-3423 171 23 strategies strategy NOUN cana-3423 171 24 depending depend VERB cana-3423 171 25 on on ADP cana-3423 171 26 cost cost NOUN cana-3423 171 27 efficiency efficiency NOUN cana-3423 171 28 and and CCONJ cana-3423 171 29 the the DET cana-3423 171 30 quantity quantity NOUN cana-3423 171 31 of of ADP cana-3423 171 32 error error NOUN cana-3423 171 33 . . PUNCT cana-3423 172 1 this this PRON cana-3423 172 2 has have VERB cana-3423 172 3 implications implication NOUN cana-3423 172 4 for for ADP cana-3423 172 5 constructing construct VERB cana-3423 172 6 faster fast ADV cana-3423 172 7 and and CCONJ cana-3423 172 8 more more ADV cana-3423 172 9 accurate accurate ADJ cana-3423 172 10 solvers solver NOUN cana-3423 172 11 of of ADP cana-3423 172 12 nonlinear nonlinear ADJ cana-3423 172 13 pdes pde NOUN cana-3423 172 14 in in ADP cana-3423 172 15 applied apply VERB cana-3423 172 16 fields field NOUN cana-3423 172 17 . . PUNCT cana-3423 173 1 3 3 X cana-3423 173 2 . . X cana-3423 173 3 theoretical theoretical ADJ cana-3423 173 4 insights insight NOUN cana-3423 173 5 : : PUNCT cana-3423 173 6 this this DET cana-3423 173 7 work work NOUN cana-3423 173 8 is be AUX cana-3423 173 9 valuable valuable ADJ cana-3423 173 10 because because SCONJ cana-3423 173 11 expanding expand VERB cana-3423 173 12 the the DET cana-3423 173 13 domain domain NOUN cana-3423 173 14 of of ADP cana-3423 173 15 the the DET cana-3423 173 16 topological topological ADJ cana-3423 173 17 approach approach NOUN cana-3423 173 18 to to ADP cana-3423 173 19 higher high ADJ cana-3423 173 20 dimensions dimension NOUN cana-3423 173 21 leads lead VERB cana-3423 173 22 to to ADP cana-3423 173 23 a a DET cana-3423 173 24 better well ADJ cana-3423 173 25 understanding understanding NOUN cana-3423 173 26 of of ADP cana-3423 173 27 nonlinear nonlinear ADJ cana-3423 173 28 systems system NOUN cana-3423 173 29 , , PUNCT cana-3423 173 30 perhaps perhaps ADV cana-3423 173 31 creating create VERB cana-3423 173 32 new new ADJ cana-3423 173 33 directions direction NOUN cana-3423 173 34 for for ADP cana-3423 173 35 future future ADJ cana-3423 173 36 study study NOUN cana-3423 173 37 in in ADP cana-3423 173 38 dynamical dynamical ADJ cana-3423 173 39 systems system NOUN cana-3423 173 40 and and CCONJ cana-3423 173 41 chaos chaos NOUN cana-3423 173 42 . . PUNCT cana-3423 174 1 6.4 6.4 NUM cana-3423 174 2 limitations limitation NOUN cana-3423 174 3 of of ADP cana-3423 174 4 the the DET cana-3423 174 5 study study NOUN cana-3423 174 6 despite despite SCONJ cana-3423 174 7 its its PRON cana-3423 174 8 contributions contribution NOUN cana-3423 174 9 , , PUNCT cana-3423 174 10 this this DET cana-3423 174 11 study study NOUN cana-3423 174 12 has have VERB cana-3423 174 13 limitations limitation NOUN cana-3423 174 14 that that PRON cana-3423 174 15 must must AUX cana-3423 174 16 be be AUX cana-3423 174 17 acknowledged acknowledge VERB cana-3423 174 18 : : PUNCT cana-3423 174 19 β€’ β€’ NUM cana-3423 174 20 boundary boundary ADJ cana-3423 174 21 condition condition NOUN cana-3423 174 22 constraints constraint NOUN cana-3423 174 23 : : PUNCT cana-3423 174 24 despite despite SCONJ cana-3423 174 25 the the DET cana-3423 174 26 broad broad ADJ cana-3423 174 27 range range NOUN cana-3423 174 28 of of ADP cana-3423 174 29 cases case NOUN cana-3423 174 30 with with ADP cana-3423 174 31 boundary boundary ADJ cana-3423 174 32 conditions condition NOUN cana-3423 174 33 examined examine VERB cana-3423 174 34 in in ADP cana-3423 174 35 the the DET cana-3423 174 36 study study NOUN cana-3423 174 37 , , PUNCT cana-3423 174 38 some some PRON cana-3423 174 39 of of ADP cana-3423 174 40 the the DET cana-3423 174 41 boundary boundary ADJ cana-3423 174 42 cases case NOUN cana-3423 174 43 were be AUX cana-3423 174 44 mixed mixed ADJ cana-3423 174 45 , , PUNCT cana-3423 174 46 and and CCONJ cana-3423 174 47 due due ADP cana-3423 174 48 to to ADP cana-3423 174 49 the the DET cana-3423 174 50 interconnections interconnection NOUN cana-3423 174 51 therein therein ADV cana-3423 174 52 , , PUNCT cana-3423 174 53 they they PRON cana-3423 174 54 at at ADP cana-3423 174 55 times time NOUN cana-3423 174 56 constrained constrain VERB cana-3423 174 57 the the DET cana-3423 174 58 degree degree NOUN cana-3423 174 59 theory theory NOUN cana-3423 174 60 applications application NOUN cana-3423 174 61 that that PRON cana-3423 174 62 were be AUX cana-3423 174 63 promising promise VERB cana-3423 174 64 to to PART cana-3423 174 65 solve solve VERB cana-3423 174 66 the the DET cana-3423 174 67 problem problem NOUN cana-3423 174 68 more more ADV cana-3423 174 69 accurately accurately ADV cana-3423 174 70 given give VERB cana-3423 174 71 imprecise imprecise ADJ cana-3423 174 72 boundary boundary ADJ cana-3423 174 73 conditions condition NOUN cana-3423 174 74 , , PUNCT cana-3423 174 75 particularly particularly ADV cana-3423 174 76 in in ADP cana-3423 174 77 irregular irregular ADJ cana-3423 174 78 systems system NOUN cana-3423 174 79 . . PUNCT cana-3423 175 1 it it PRON cana-3423 175 2 is be AUX cana-3423 175 3 suggested suggest VERB cana-3423 175 4 that that SCONJ cana-3423 175 5 further further ADJ cana-3423 175 6 studies study NOUN cana-3423 175 7 should should AUX cana-3423 175 8 employ employ VERB cana-3423 175 9 more more ADV cana-3423 175 10 sophisticated sophisticated ADJ cana-3423 175 11 degree degree NOUN cana-3423 175 12 - - PUNCT cana-3423 175 13 theoretic theoretic NOUN cana-3423 175 14 techniques technique NOUN cana-3423 175 15 which which PRON cana-3423 175 16 can can AUX cana-3423 175 17 deal deal VERB cana-3423 175 18 with with ADP cana-3423 175 19 these these DET cana-3423 175 20 issues issue NOUN cana-3423 175 21 . . PUNCT cana-3423 176 1 communications communication NOUN cana-3423 176 2 on on ADP cana-3423 176 3 applied apply VERB cana-3423 176 4 nonlinear nonlinear ADJ cana-3423 176 5 analysis analysis NOUN cana-3423 176 6 issn issn NOUN cana-3423 176 7 : : PUNCT cana-3423 176 8 1074 1074 NUM cana-3423 176 9 - - PUNCT cana-3423 176 10 133x 133x NUM cana-3423 176 11 vol vol NOUN cana-3423 176 12 32 32 NUM cana-3423 176 13 no no NOUN cana-3423 176 14 . . PUNCT cana-3423 177 1 7s 7 NOUN cana-3423 177 2 ( ( PUNCT cana-3423 177 3 2025 2025 NUM cana-3423 177 4 ) ) PUNCT cana-3423 178 1 https://internationalpubls.com https://internationalpubls.com X cana-3423 178 2 367 367 NUM cana-3423 178 3 β€’ β€’ NUM cana-3423 178 4 computational computational ADJ cana-3423 178 5 scope scope NOUN cana-3423 178 6 : : PUNCT cana-3423 178 7 the the DET cana-3423 178 8 present present ADJ cana-3423 178 9 computations computation NOUN cana-3423 178 10 were be AUX cana-3423 178 11 strictly strictly ADV cana-3423 178 12 confined confine VERB cana-3423 178 13 to to ADP cana-3423 178 14 the the DET cana-3423 178 15 finite finite ADJ cana-3423 178 16 difference difference NOUN cana-3423 178 17 and and CCONJ cana-3423 178 18 finite finite ADJ cana-3423 178 19 element element NOUN cana-3423 178 20 techniques technique NOUN cana-3423 178 21 . . PUNCT cana-3423 179 1 it it PRON cana-3423 179 2 is be AUX cana-3423 179 3 clear clear ADJ cana-3423 179 4 that that SCONJ cana-3423 179 5 other other ADJ cana-3423 179 6 , , PUNCT cana-3423 179 7 more more ADV cana-3423 179 8 efficient efficient ADJ cana-3423 179 9 techniques technique NOUN cana-3423 179 10 , , PUNCT cana-3423 179 11 like like ADP cana-3423 179 12 spectral spectral ADJ cana-3423 179 13 methods method NOUN cana-3423 179 14 , , PUNCT cana-3423 179 15 might might AUX cana-3423 179 16 help help VERB cana-3423 179 17 increase increase VERB cana-3423 179 18 the the DET cana-3423 179 19 precision precision NOUN cana-3423 179 20 of of ADP cana-3423 179 21 solutions solution NOUN cana-3423 179 22 for for ADP cana-3423 179 23 high high ADJ cana-3423 179 24 - - PUNCT cana-3423 179 25 dimensional dimensional ADJ cana-3423 179 26 pdes pde NOUN cana-3423 179 27 in in ADP cana-3423 179 28 fine fine ADV cana-3423 179 29 - - PUNCT cana-3423 179 30 grained grain VERB cana-3423 179 31 cases case NOUN cana-3423 179 32 . . PUNCT cana-3423 180 1 β€’ β€’ NUM cana-3423 180 2 assumptions assumption NOUN cana-3423 180 3 of of ADP cana-3423 180 4 contraction contraction NOUN cana-3423 180 5 mapping mapping NOUN cana-3423 180 6 : : PUNCT cana-3423 180 7 by by ADP cana-3423 180 8 depending depend VERB cana-3423 180 9 on on ADP cana-3423 180 10 contraction contraction NOUN cana-3423 180 11 mapping mapping NOUN cana-3423 180 12 conditions condition NOUN cana-3423 180 13 fixedpoint fixedpoint NOUN cana-3423 180 14 results result NOUN cana-3423 180 15 could could AUX cana-3423 180 16 prove prove VERB cana-3423 180 17 inadequate inadequate ADJ cana-3423 180 18 in in ADP cana-3423 180 19 a a DET cana-3423 180 20 global global ADJ cana-3423 180 21 solution solution NOUN cana-3423 180 22 of of ADP cana-3423 180 23 classes class NOUN cana-3423 180 24 of of ADP cana-3423 180 25 nonlinear nonlinear ADJ cana-3423 180 26 pdes pde NOUN cana-3423 180 27 . . PUNCT cana-3423 181 1 this this PRON cana-3423 181 2 restricts restrict VERB cana-3423 181 3 the the DET cana-3423 181 4 applicability applicability NOUN cana-3423 181 5 of of ADP cana-3423 181 6 the the DET cana-3423 181 7 fixed fix VERB cana-3423 181 8 point point NOUN cana-3423 181 9 to to ADP cana-3423 181 10 a a DET cana-3423 181 11 specific specific ADJ cana-3423 181 12 class class NOUN cana-3423 181 13 of of ADP cana-3423 181 14 nonlinear nonlinear ADJ cana-3423 181 15 situations situation NOUN cana-3423 181 16 and and CCONJ cana-3423 181 17 there there PRON cana-3423 181 18 is be VERB cana-3423 181 19 a a DET cana-3423 181 20 need need NOUN cana-3423 181 21 to to PART cana-3423 181 22 assume assume VERB cana-3423 181 23 the the DET cana-3423 181 24 like like NOUN cana-3423 181 25 to to PART cana-3423 181 26 expand expand VERB cana-3423 181 27 the the DET cana-3423 181 28 scope scope NOUN cana-3423 181 29 of of ADP cana-3423 181 30 the the DET cana-3423 181 31 fixed fix VERB cana-3423 181 32 point point NOUN cana-3423 181 33 approach approach NOUN cana-3423 181 34 to to PART cana-3423 181 35 cover cover VERB cana-3423 181 36 a a DET cana-3423 181 37 wider wide ADJ cana-3423 181 38 area area NOUN cana-3423 181 39 . . PUNCT cana-3423 182 1 6.5 6.5 NUM cana-3423 182 2 suggestions suggestion NOUN cana-3423 182 3 for for ADP cana-3423 182 4 future future ADJ cana-3423 182 5 research research NOUN cana-3423 182 6 building building NOUN cana-3423 182 7 on on ADP cana-3423 182 8 the the DET cana-3423 182 9 insights insight NOUN cana-3423 182 10 and and CCONJ cana-3423 182 11 limitations limitation NOUN cana-3423 182 12 of of ADP cana-3423 182 13 this this DET cana-3423 182 14 study study NOUN cana-3423 182 15 , , PUNCT cana-3423 182 16 several several ADJ cana-3423 182 17 directions direction NOUN cana-3423 182 18 for for ADP cana-3423 182 19 future future ADJ cana-3423 182 20 research research NOUN cana-3423 182 21 are be AUX cana-3423 182 22 proposed propose VERB cana-3423 182 23 : : PUNCT cana-3423 182 24 1 1 X cana-3423 182 25 . . NUM cana-3423 182 26 extended extend VERB cana-3423 182 27 boundary boundary ADJ cana-3423 182 28 conditions condition NOUN cana-3423 182 29 : : PUNCT cana-3423 182 30 possible possible ADJ cana-3423 182 31 areas area NOUN cana-3423 182 32 for for ADP cana-3423 182 33 future future ADJ cana-3423 182 34 research research NOUN cana-3423 182 35 include include VERB cana-3423 182 36 studies study NOUN cana-3423 182 37 of of ADP cana-3423 182 38 degree degree NOUN cana-3423 182 39 theoretical theoretical ADJ cana-3423 182 40 methods method NOUN cana-3423 182 41 at at ADP cana-3423 182 42 object object NOUN cana-3423 182 43 ’s ’s PART cana-3423 182 44 faces face NOUN cana-3423 182 45 with with ADP cana-3423 182 46 strongly strongly ADV cana-3423 182 47 irregular irregular ADJ cana-3423 182 48 or or CCONJ cana-3423 182 49 a a DET cana-3423 182 50 mix mix NOUN cana-3423 182 51 of of ADP cana-3423 182 52 dirichlet dirichlet PROPN cana-3423 182 53 and and CCONJ cana-3423 182 54 neumann neumann PROPN cana-3423 182 55 conditions condition NOUN cana-3423 182 56 . . PUNCT cana-3423 183 1 application application NOUN cana-3423 183 2 of of ADP cana-3423 183 3 augmentation augmentation NOUN cana-3423 183 4 research research NOUN cana-3423 183 5 in in ADP cana-3423 183 6 envisioning envision VERB cana-3423 183 7 future future ADJ cana-3423 183 8 developments development NOUN cana-3423 183 9 relevant relevant ADJ cana-3423 183 10 to to ADP cana-3423 183 11 topological topological ADJ cana-3423 183 12 degree degree NOUN cana-3423 183 13 theory theory NOUN cana-3423 183 14 might might AUX cana-3423 183 15 contribute contribute VERB cana-3423 183 16 to to ADP cana-3423 183 17 refining refine VERB cana-3423 183 18 modeling modeling NOUN cana-3423 183 19 in in ADP cana-3423 183 20 the the DET cana-3423 183 21 mentioned mention VERB cana-3423 183 22 domains domain NOUN cana-3423 183 23 . . PUNCT cana-3423 184 1 2 2 X cana-3423 184 2 . . X cana-3423 184 3 application application NOUN cana-3423 184 4 of of ADP cana-3423 184 5 spectral spectral ADJ cana-3423 184 6 methods method NOUN cana-3423 184 7 : : PUNCT cana-3423 184 8 combining combine VERB cana-3423 184 9 spectral spectral ADJ cana-3423 184 10 type type NOUN cana-3423 184 11 with with ADP cana-3423 184 12 such such ADJ cana-3423 184 13 topological topological ADJ cana-3423 184 14 methods method NOUN cana-3423 184 15 could could AUX cana-3423 184 16 enhance enhance VERB cana-3423 184 17 accuracy accuracy NOUN cana-3423 184 18 in in ADP cana-3423 184 19 cases case NOUN cana-3423 184 20 where where SCONJ cana-3423 184 21 basic basic ADJ cana-3423 184 22 order order NOUN cana-3423 184 23 methods method NOUN cana-3423 184 24 in in ADP cana-3423 184 25 high high ADJ cana-3423 184 26 - - PUNCT cana-3423 184 27 dimensions dimension NOUN cana-3423 184 28 fail fail NOUN cana-3423 184 29 . . PUNCT cana-3423 185 1 this this PRON cana-3423 185 2 would would AUX cana-3423 185 3 also also ADV cana-3423 185 4 solve solve VERB cana-3423 185 5 their their PRON cana-3423 185 6 convergence convergence NOUN cana-3423 185 7 issues issue NOUN cana-3423 185 8 as as SCONJ cana-3423 185 9 seen see VERB cana-3423 185 10 with with ADP cana-3423 185 11 computationally computationally ADV cana-3423 185 12 complex complex ADJ cana-3423 185 13 situations situation NOUN cana-3423 185 14 . . PUNCT cana-3423 186 1 3 3 X cana-3423 186 2 . . X cana-3423 186 3 hybrid hybrid ADJ cana-3423 186 4 analytical analytical ADJ cana-3423 186 5 - - PUNCT cana-3423 186 6 computational computational ADJ cana-3423 186 7 models model NOUN cana-3423 186 8 : : PUNCT cana-3423 186 9 it it PRON cana-3423 186 10 is be AUX cana-3423 186 11 also also ADV cana-3423 186 12 possible possible ADJ cana-3423 186 13 that that SCONJ cana-3423 186 14 creating create VERB cana-3423 186 15 new new ADJ cana-3423 186 16 models model NOUN cana-3423 186 17 that that PRON cana-3423 186 18 would would AUX cana-3423 186 19 combine combine VERB cana-3423 186 20 topological topological ADJ cana-3423 186 21 approaches approach NOUN cana-3423 186 22 with with ADP cana-3423 186 23 machine machine NOUN cana-3423 186 24 learning learn VERB cana-3423 186 25 techniques technique NOUN cana-3423 186 26 will will AUX cana-3423 186 27 provide provide VERB cana-3423 186 28 faster fast ADJ cana-3423 186 29 and and CCONJ cana-3423 186 30 more more ADV cana-3423 186 31 flexible flexible ADJ cana-3423 186 32 solutions solution NOUN cana-3423 186 33 . . PUNCT cana-3423 187 1 for for ADP cana-3423 187 2 instance instance NOUN cana-3423 187 3 , , PUNCT cana-3423 187 4 learning learn VERB cana-3423 187 5 the the DET cana-3423 187 6 initial initial ADJ cana-3423 187 7 guess guess NOUN cana-3423 187 8 in in ADP cana-3423 187 9 iterative iterative ADJ cana-3423 187 10 schemes scheme NOUN cana-3423 187 11 could could AUX cana-3423 187 12 , , PUNCT cana-3423 187 13 in in ADP cana-3423 187 14 turn turn NOUN cana-3423 187 15 , , PUNCT cana-3423 187 16 dramatically dramatically ADV cana-3423 187 17 boost boost VERB cana-3423 187 18 the the DET cana-3423 187 19 enhanced enhanced ADJ cana-3423 187 20 rate rate NOUN cana-3423 187 21 of of ADP cana-3423 187 22 system system NOUN cana-3423 187 23 convergence convergence NOUN cana-3423 187 24 precisely precisely ADV cana-3423 187 25 within within ADP cana-3423 187 26 dynamic dynamic ADJ cana-3423 187 27 nonlinear nonlinear ADJ cana-3423 187 28 environments environment NOUN cana-3423 187 29 . . PUNCT cana-3423 188 1 4 4 X cana-3423 188 2 . . X cana-3423 188 3 cross cross ADJ cana-3423 188 4 - - ADJ cana-3423 188 5 disciplinary disciplinary ADJ cana-3423 188 6 applications application NOUN cana-3423 188 7 : : PUNCT cana-3423 188 8 since since SCONJ cana-3423 188 9 the the DET cana-3423 188 10 use use NOUN cana-3423 188 11 of of ADP cana-3423 188 12 neither neither CCONJ cana-3423 188 13 linear linear ADJ cana-3423 188 14 pde pde NOUN cana-3423 188 15 is be AUX cana-3423 188 16 premises premise NOUN cana-3423 188 17 - - PUNCT cana-3423 188 18 bound bind VERB cana-3423 188 19 , , PUNCT cana-3423 188 20 future future ADJ cana-3423 188 21 work work NOUN cana-3423 188 22 can can AUX cana-3423 188 23 incline incline VERB cana-3423 188 24 to to PART cana-3423 188 25 employ employ VERB cana-3423 188 26 these these DET cana-3423 188 27 topological topological ADJ cana-3423 188 28 techniques technique NOUN cana-3423 188 29 in in ADP cana-3423 188 30 diverse diverse ADJ cana-3423 188 31 domains domain NOUN cana-3423 188 32 comprising comprise VERB cana-3423 188 33 finance finance NOUN cana-3423 188 34 , , PUNCT cana-3423 188 35 climatology climatology NOUN cana-3423 188 36 , , PUNCT cana-3423 188 37 and and CCONJ cana-3423 188 38 bioengineering bioengineering NOUN cana-3423 188 39 . . PUNCT cana-3423 189 1 conversely conversely ADV cana-3423 189 2 , , PUNCT cana-3423 189 3 each each DET cana-3423 189 4 field field NOUN cana-3423 189 5 might might AUX cana-3423 189 6 pose pose VERB cana-3423 189 7 problems problem NOUN cana-3423 189 8 that that PRON cana-3423 189 9 could could AUX cana-3423 189 10 enhance enhance VERB cana-3423 189 11 the the DET cana-3423 189 12 methodologies methodology NOUN cana-3423 189 13 discussed discuss VERB cana-3423 189 14 here here ADV cana-3423 189 15 further far ADV cana-3423 189 16 projecting project VERB cana-3423 189 17 a a DET cana-3423 189 18 basis basis NOUN cana-3423 189 19 for for ADP cana-3423 189 20 facelift facelift NOUN cana-3423 189 21 . . PUNCT cana-3423 190 1 conclusion conclusion NOUN cana-3423 190 2 the the DET cana-3423 190 3 study study NOUN cana-3423 190 4 contributes contribute VERB cana-3423 190 5 to to ADP cana-3423 190 6 the the DET cana-3423 190 7 problem problem NOUN cana-3423 190 8 of of ADP cana-3423 190 9 applying apply VERB cana-3423 190 10 topological topological ADJ cana-3423 190 11 methods method NOUN cana-3423 190 12 for for ADP cana-3423 190 13 the the DET cana-3423 190 14 analysis analysis NOUN cana-3423 190 15 of of ADP cana-3423 190 16 nonlinear nonlinear ADJ cana-3423 190 17 pdes pde NOUN cana-3423 190 18 and and CCONJ cana-3423 190 19 shows show VERB cana-3423 190 20 that that SCONJ cana-3423 190 21 these these DET cana-3423 190 22 methods method NOUN cana-3423 190 23 allow allow VERB cana-3423 190 24 to to PART cana-3423 190 25 establish establish VERB cana-3423 190 26 the the DET cana-3423 190 27 existence existence NOUN cana-3423 190 28 and and CCONJ cana-3423 190 29 uniqueness uniqueness NOUN cana-3423 190 30 of of ADP cana-3423 190 31 solutions solution NOUN cana-3423 190 32 and and CCONJ cana-3423 190 33 their their PRON cana-3423 190 34 stability stability NOUN cana-3423 190 35 and and CCONJ cana-3423 190 36 efficiency efficiency NOUN cana-3423 190 37 of of ADP cana-3423 190 38 calculation calculation NOUN cana-3423 190 39 in in ADP cana-3423 190 40 practice practice NOUN cana-3423 190 41 . . PUNCT cana-3423 191 1 based base VERB cana-3423 191 2 on on ADP cana-3423 191 3 the the DET cana-3423 191 4 findings finding NOUN cana-3423 191 5 , , PUNCT cana-3423 191 6 it it PRON cana-3423 191 7 becomes become VERB cana-3423 191 8 clear clear ADJ cana-3423 191 9 banach banach NOUN cana-3423 191 10 fixed fix VERB cana-3423 191 11 - - PUNCT cana-3423 191 12 point point NOUN cana-3423 191 13 theorem theorem NOUN cana-3423 191 14 delivers deliver VERB cana-3423 191 15 an an DET cana-3423 191 16 accurate accurate ADJ cana-3423 191 17 convergence convergence NOUN cana-3423 191 18 of of ADP cana-3423 191 19 solution solution NOUN cana-3423 191 20 for for ADP cana-3423 191 21 semilinear semilinear ADJ cana-3423 191 22 pdes pde NOUN cana-3423 191 23 if if SCONJ cana-3423 191 24 some some DET cana-3423 191 25 boundary boundary ADJ cana-3423 191 26 conditions condition NOUN cana-3423 191 27 are be AUX cana-3423 191 28 met meet VERB cana-3423 191 29 while while SCONJ cana-3423 191 30 degree degree NOUN cana-3423 191 31 theory theory NOUN cana-3423 191 32 offers offer VERB cana-3423 191 33 a a DET cana-3423 191 34 systematic systematic ADJ cana-3423 191 35 approach approach NOUN cana-3423 191 36 to to ADP cana-3423 191 37 handling handle VERB cana-3423 191 38 multiplicity multiplicity NOUN cana-3423 191 39 of of ADP cana-3423 191 40 solutions solution NOUN cana-3423 191 41 in in ADP cana-3423 191 42 higher high ADJ cana-3423 191 43 dimensional dimensional ADJ cana-3423 191 44 space space NOUN cana-3423 191 45 systems system NOUN cana-3423 191 46 . . PUNCT cana-3423 192 1 further far ADV cana-3423 192 2 , , PUNCT cana-3423 192 3 the the DET cana-3423 192 4 variational variational ADJ cana-3423 192 5 techniques technique NOUN cana-3423 192 6 identify identify VERB cana-3423 192 7 stable stable ADJ cana-3423 192 8 critical critical ADJ cana-3423 192 9 points point NOUN cana-3423 192 10 and and CCONJ cana-3423 192 11 therefore therefore ADV cana-3423 192 12 provide provide VERB cana-3423 192 13 insight insight NOUN cana-3423 192 14 into into ADP cana-3423 192 15 solution solution NOUN cana-3423 192 16 stability stability NOUN cana-3423 192 17 for for ADP cana-3423 192 18 various various ADJ cana-3423 192 19 nonlinear nonlinear ADJ cana-3423 192 20 systems system NOUN cana-3423 192 21 . . PUNCT cana-3423 193 1 more more ADV cana-3423 193 2 specifically specifically ADV cana-3423 193 3 , , PUNCT cana-3423 193 4 these these DET cana-3423 193 5 results result NOUN cana-3423 193 6 greatly greatly ADV cana-3423 193 7 advance advance VERB cana-3423 193 8 the the DET cana-3423 193 9 existing exist VERB cana-3423 193 10 body body NOUN cana-3423 193 11 of of ADP cana-3423 193 12 knowledge knowledge NOUN cana-3423 193 13 by by ADP cana-3423 193 14 demonstrating demonstrate VERB cana-3423 193 15 that that SCONJ cana-3423 193 16 topological topological ADJ cana-3423 193 17 methods method NOUN cana-3423 193 18 can can AUX cana-3423 193 19 be be AUX cana-3423 193 20 fruitfully fruitfully ADV cana-3423 193 21 applied apply VERB cana-3423 193 22 in in ADP cana-3423 193 23 more more ADJ cana-3423 193 24 complex complex ADJ cana-3423 193 25 settings setting NOUN cana-3423 193 26 and and CCONJ cana-3423 193 27 boundary boundary ADJ cana-3423 193 28 conditions condition NOUN cana-3423 193 29 than than SCONJ cana-3423 193 30 previously previously ADV cana-3423 193 31 analyzed analyze VERB cana-3423 193 32 . . PUNCT cana-3423 194 1 this this PRON cana-3423 194 2 is be AUX cana-3423 194 3 important important ADJ cana-3423 194 4 to to PART cana-3423 194 5 stress stress VERB cana-3423 194 6 when when SCONJ cana-3423 194 7 arguing argue VERB cana-3423 194 8 for for ADP cana-3423 194 9 the the DET cana-3423 194 10 value value NOUN cana-3423 194 11 of of ADP cana-3423 194 12 topological topological ADJ cana-3423 194 13 approaches approach NOUN cana-3423 194 14 in in ADP cana-3423 194 15 contemporary contemporary ADJ cana-3423 194 16 mathematics mathematic NOUN cana-3423 194 17 . . PUNCT cana-3423 195 1 a a DET cana-3423 195 2 comparison comparison NOUN cana-3423 195 3 of of ADP cana-3423 195 4 the the DET cana-3423 195 5 gains gain NOUN cana-3423 195 6 found find VERB cana-3423 195 7 in in ADP cana-3423 195 8 newton newton PROPN cana-3423 195 9 - - PUNCT cana-3423 195 10 raphson raphson PROPN cana-3423 195 11 and and CCONJ cana-3423 195 12 fixed fix VERB cana-3423 195 13 - - PUNCT cana-3423 195 14 point point NOUN cana-3423 195 15 schemes scheme NOUN cana-3423 195 16 is be AUX cana-3423 195 17 valuable valuable ADJ cana-3423 195 18 to to PART cana-3423 195 19 observe observe VERB cana-3423 195 20 optimal optimal ADJ cana-3423 195 21 computational computational ADJ cana-3423 195 22 techniques technique NOUN cana-3423 195 23 , , PUNCT cana-3423 195 24 which which PRON cana-3423 195 25 further far ADV cana-3423 195 26 improve improve VERB cana-3423 195 27 the the DET cana-3423 195 28 nonlinear nonlinear ADJ cana-3423 195 29 pde pde NOUN cana-3423 195 30 solvers solver NOUN cana-3423 195 31 . . PUNCT cana-3423 196 1 these these DET cana-3423 196 2 methods method NOUN cana-3423 196 3 are be AUX cana-3423 196 4 applicable applicable ADJ cana-3423 196 5 in in ADP cana-3423 196 6 functional functional ADJ cana-3423 196 7 areas area NOUN cana-3423 196 8 of of ADP cana-3423 196 9 modeling model VERB cana-3423 196 10 real real ADJ cana-3423 196 11 - - PUNCT cana-3423 196 12 world world NOUN cana-3423 196 13 systems system NOUN cana-3423 196 14 which which PRON cana-3423 196 15 include include VERB cana-3423 196 16 engineering engineering NOUN cana-3423 196 17 , , PUNCT cana-3423 196 18 physics physics NOUN cana-3423 196 19 , , PUNCT cana-3423 196 20 physical physical ADJ cana-3423 196 21 system system NOUN cana-3423 196 22 modeling modeling NOUN cana-3423 196 23 , , PUNCT cana-3423 196 24 and and CCONJ cana-3423 196 25 environmental environmental ADJ cana-3423 196 26 systems system NOUN cana-3423 196 27 modeling model VERB cana-3423 196 28 where where SCONJ cana-3423 196 29 non non ADJ cana-3423 196 30 - - ADJ cana-3423 196 31 linear linear ADJ cana-3423 196 32 dynamics dynamic NOUN cana-3423 196 33 are be AUX cana-3423 196 34 key key ADJ cana-3423 196 35 determinants determinant NOUN cana-3423 196 36 . . PUNCT cana-3423 197 1 in in ADP cana-3423 197 2 sum sum NOUN cana-3423 197 3 , , PUNCT cana-3423 197 4 this this DET cana-3423 197 5 study study NOUN cana-3423 197 6 highlights highlight VERB cana-3423 197 7 the the DET cana-3423 197 8 theoretical theoretical ADJ cana-3423 197 9 and and CCONJ cana-3423 197 10 potential potential ADJ cana-3423 197 11 application application NOUN cana-3423 197 12 of of ADP cana-3423 197 13 topological topological ADJ cana-3423 197 14 methods method NOUN cana-3423 197 15 in in ADP cana-3423 197 16 nonlinear nonlinear ADJ cana-3423 197 17 analysis analysis NOUN cana-3423 197 18 for for ADP cana-3423 197 19 future future ADJ cana-3423 197 20 research research NOUN cana-3423 197 21 in in ADP cana-3423 197 22 this this DET cana-3423 197 23 area area NOUN cana-3423 197 24 , , PUNCT cana-3423 197 25 not not PART cana-3423 197 26 only only ADV cana-3423 197 27 the the DET cana-3423 197 28 communications communication NOUN cana-3423 197 29 on on ADP cana-3423 197 30 applied apply VERB cana-3423 197 31 nonlinear nonlinear ADJ cana-3423 197 32 analysis analysis NOUN cana-3423 197 33 issn issn NOUN cana-3423 197 34 : : PUNCT cana-3423 197 35 1074 1074 NUM cana-3423 197 36 - - PUNCT cana-3423 197 37 133x 133x NUM cana-3423 197 38 vol vol NOUN cana-3423 197 39 32 32 NUM cana-3423 197 40 no no NOUN cana-3423 197 41 . . PUNCT cana-3423 198 1 7s 7 NOUN cana-3423 198 2 ( ( PUNCT cana-3423 198 3 2025 2025 NUM cana-3423 198 4 ) ) PUNCT cana-3423 199 1 https://internationalpubls.com https://internationalpubls.com X cana-3423 199 2 368 368 NUM cana-3423 199 3 ability ability NOUN cana-3423 199 4 to to PART cana-3423 199 5 deal deal VERB cana-3423 199 6 with with ADP cana-3423 199 7 different different ADJ cana-3423 199 8 boundary boundary ADJ cana-3423 199 9 conditions condition NOUN cana-3423 199 10 but but CCONJ cana-3423 199 11 also also ADV cana-3423 199 12 the the DET cana-3423 199 13 optimization optimization NOUN cana-3423 199 14 of of ADP cana-3423 199 15 existing exist VERB cana-3423 199 16 computational computational ADJ cana-3423 199 17 techniques technique NOUN cana-3423 199 18 for for ADP cana-3423 199 19 multidimensional multidimensional ADJ cana-3423 199 20 nonlinear nonlinear ADJ cana-3423 199 21 pdes pde NOUN cana-3423 199 22 . . PUNCT cana-3423 200 1 references reference NOUN cana-3423 200 2 [ [ X cana-3423 200 3 1 1 X cana-3423 200 4 ] ] PUNCT cana-3423 200 5 ambrosetti ambrosetti PROPN cana-3423 200 6 , , PUNCT cana-3423 200 7 a. a. PROPN cana-3423 200 8 , , PUNCT cana-3423 200 9 & & CCONJ cana-3423 200 10 prodi prodi PROPN cana-3423 200 11 , , PUNCT cana-3423 200 12 g. g. PROPN cana-3423 200 13 ( ( PUNCT cana-3423 200 14 1995 1995 NUM cana-3423 200 15 ) ) PUNCT cana-3423 200 16 . . PUNCT cana-3423 201 1 a a DET cana-3423 201 2 primer primer NOUN cana-3423 201 3 of of ADP cana-3423 201 4 nonlinear nonlinear ADJ cana-3423 201 5 analysis analysis NOUN cana-3423 201 6 . . PUNCT cana-3423 202 1 cambridge cambridge PROPN cana-3423 202 2 university university PROPN cana-3423 202 3 press press NOUN cana-3423 202 4 . . PUNCT cana-3423 203 1 [ [ X cana-3423 203 2 2 2 NUM cana-3423 203 3 ] ] PUNCT cana-3423 203 4 aubin aubin PROPN cana-3423 203 5 , , PUNCT cana-3423 203 6 j.-p j.-p PROPN cana-3423 203 7 . . PROPN cana-3423 203 8 , , PUNCT cana-3423 203 9 & & CCONJ cana-3423 203 10 frankowska frankowska PROPN cana-3423 203 11 , , PUNCT cana-3423 203 12 h. h. PROPN cana-3423 203 13 ( ( PUNCT cana-3423 203 14 2009 2009 NUM cana-3423 203 15 ) ) PUNCT cana-3423 203 16 . . PUNCT cana-3423 204 1 set set NOUN cana-3423 204 2 - - PUNCT cana-3423 204 3 valued value VERB cana-3423 204 4 analysis analysis NOUN cana-3423 204 5 . . PUNCT cana-3423 205 1 birkhΓ€user birkhΓ€user NOUN cana-3423 205 2 . . PUNCT cana-3423 206 1 [ [ X cana-3423 206 2 3 3 NUM cana-3423 206 3 ] ] SYM cana-3423 206 4 berestycki berestycki NOUN cana-3423 206 5 , , PUNCT cana-3423 206 6 h. h. PROPN cana-3423 206 7 , , PUNCT cana-3423 206 8 & & CCONJ cana-3423 206 9 lions lion NOUN cana-3423 206 10 , , PUNCT cana-3423 206 11 p. p. NOUN cana-3423 206 12 l. l. PROPN cana-3423 206 13 ( ( PUNCT cana-3423 206 14 2013 2013 NUM cana-3423 206 15 ) ) PUNCT cana-3423 206 16 . . PUNCT cana-3423 207 1 " " PUNCT cana-3423 207 2 existence existence NOUN cana-3423 207 3 of of ADP cana-3423 207 4 positive positive ADJ cana-3423 207 5 solutions solution NOUN cana-3423 207 6 in in ADP cana-3423 207 7 semilinear semilinear NOUN cana-3423 207 8 and and CCONJ cana-3423 207 9 quasilinear quasilinear NOUN cana-3423 207 10 pdes pde NOUN cana-3423 207 11 . . PUNCT cana-3423 207 12 " " PUNCT cana-3423 208 1 annals annals NOUN cana-3423 208 2 of of ADP cana-3423 208 3 mathematics mathematic NOUN cana-3423 208 4 , , PUNCT cana-3423 208 5 118(2 118(2 NUM cana-3423 208 6 ) ) PUNCT cana-3423 208 7 , , PUNCT cana-3423 208 8 731 731 NUM cana-3423 208 9 - - SYM cana-3423 208 10 742 742 NUM cana-3423 208 11 . . PUNCT cana-3423 209 1 [ [ X cana-3423 209 2 4 4 NUM cana-3423 209 3 ] ] SYM cana-3423 209 4 brezis brezis NOUN cana-3423 209 5 , , PUNCT cana-3423 209 6 h. h. PROPN cana-3423 209 7 ( ( PUNCT cana-3423 209 8 2010 2010 NUM cana-3423 209 9 ) ) PUNCT cana-3423 209 10 . . PUNCT cana-3423 210 1 functional functional ADJ cana-3423 210 2 analysis analysis NOUN cana-3423 210 3 , , PUNCT cana-3423 210 4 sobolev sobolev NOUN cana-3423 210 5 spaces space NOUN cana-3423 210 6 , , PUNCT cana-3423 210 7 and and CCONJ cana-3423 210 8 partial partial ADJ cana-3423 210 9 differential differential ADJ cana-3423 210 10 equations equation NOUN cana-3423 210 11 . . PUNCT cana-3423 211 1 springer springer NOUN cana-3423 211 2 . . PUNCT cana-3423 212 1 [ [ X cana-3423 212 2 5 5 NUM cana-3423 212 3 ] ] X cana-3423 212 4 chang chang PROPN cana-3423 212 5 , , PUNCT cana-3423 212 6 k. k. PROPN cana-3423 212 7 c. c. PROPN cana-3423 212 8 ( ( PUNCT cana-3423 212 9 2013 2013 NUM cana-3423 212 10 ) ) PUNCT cana-3423 212 11 . . PUNCT cana-3423 213 1 methods method NOUN cana-3423 213 2 in in ADP cana-3423 213 3 nonlinear nonlinear ADJ cana-3423 213 4 analysis analysis NOUN cana-3423 213 5 . . PUNCT cana-3423 214 1 springer springer NOUN cana-3423 214 2 . . PUNCT cana-3423 215 1 [ [ X cana-3423 215 2 6 6 NUM cana-3423 215 3 ] ] PUNCT cana-3423 215 4 evans evans PROPN cana-3423 215 5 , , PUNCT cana-3423 215 6 l. l. PROPN cana-3423 215 7 c. c. PROPN cana-3423 215 8 ( ( PUNCT cana-3423 215 9 2010 2010 NUM cana-3423 215 10 ) ) PUNCT cana-3423 215 11 . . PUNCT cana-3423 216 1 partial partial ADJ cana-3423 216 2 differential differential ADJ cana-3423 216 3 equations equation NOUN cana-3423 216 4 . . PUNCT cana-3423 217 1 american american PROPN cana-3423 217 2 mathematical mathematical PROPN cana-3423 217 3 society society NOUN cana-3423 217 4 . . PUNCT cana-3423 218 1 [ [ X cana-3423 218 2 7 7 NUM cana-3423 218 3 ] ] X cana-3423 218 4 mawhin mawhin NOUN cana-3423 218 5 , , PUNCT cana-3423 218 6 j. j. PROPN cana-3423 218 7 ( ( PUNCT cana-3423 218 8 2017 2017 NUM cana-3423 218 9 ) ) PUNCT cana-3423 218 10 . . PUNCT cana-3423 219 1 " " PUNCT cana-3423 219 2 topological topological ADJ cana-3423 219 3 degree degree NOUN cana-3423 219 4 methods method NOUN cana-3423 219 5 for for ADP cana-3423 219 6 nonlinear nonlinear ADJ cana-3423 219 7 boundary boundary ADJ cana-3423 219 8 value value NOUN cana-3423 219 9 problems problem NOUN cana-3423 219 10 . . PUNCT cana-3423 219 11 " " PUNCT cana-3423 220 1 journal journal NOUN cana-3423 220 2 of of ADP cana-3423 220 3 differential differential ADJ cana-3423 220 4 equations equation NOUN cana-3423 220 5 , , PUNCT cana-3423 220 6 54(4 54(4 NUM cana-3423 220 7 ) ) PUNCT cana-3423 220 8 , , PUNCT cana-3423 220 9 845 845 NUM cana-3423 220 10 - - SYM cana-3423 220 11 878 878 NUM cana-3423 220 12 . . PUNCT cana-3423 221 1 [ [ X cana-3423 221 2 8 8 NUM cana-3423 221 3 ] ] X cana-3423 221 4 rabinowitz rabinowitz NOUN cana-3423 221 5 , , PUNCT cana-3423 221 6 p. p. NOUN cana-3423 221 7 h. h. PROPN cana-3423 221 8 ( ( PUNCT cana-3423 221 9 1986 1986 NUM cana-3423 221 10 ) ) PUNCT cana-3423 221 11 . . PUNCT cana-3423 222 1 minimax minimax NOUN cana-3423 222 2 methods method NOUN cana-3423 222 3 in in ADP cana-3423 222 4 critical critical ADJ cana-3423 222 5 point point NOUN cana-3423 222 6 theory theory NOUN cana-3423 222 7 with with ADP cana-3423 222 8 applications application NOUN cana-3423 222 9 to to PART cana-3423 222 10 differential differential VERB cana-3423 222 11 equations equation NOUN cana-3423 222 12 . . PUNCT cana-3423 223 1 american american PROPN cana-3423 223 2 mathematical mathematical PROPN cana-3423 223 3 society society NOUN cana-3423 223 4 . . PUNCT cana-3423 224 1 [ [ X cana-3423 224 2 9 9 NUM cana-3423 224 3 ] ] X cana-3423 224 4 sΓΌli sΓΌli NOUN cana-3423 224 5 , , PUNCT cana-3423 224 6 e. e. PROPN cana-3423 224 7 , , PUNCT cana-3423 224 8 & & CCONJ cana-3423 224 9 mayers mayers PROPN cana-3423 224 10 , , PUNCT cana-3423 224 11 d. d. PROPN cana-3423 224 12 f. f. PROPN cana-3423 224 13 ( ( PUNCT cana-3423 224 14 2003 2003 NUM cana-3423 224 15 ) ) PUNCT cana-3423 224 16 . . PUNCT cana-3423 225 1 an an DET cana-3423 225 2 introduction introduction NOUN cana-3423 225 3 to to ADP cana-3423 225 4 numerical numerical ADJ cana-3423 225 5 analysis analysis NOUN cana-3423 225 6 . . PUNCT cana-3423 226 1 cambridge cambridge PROPN cana-3423 226 2 university university PROPN cana-3423 226 3 press press NOUN cana-3423 226 4 . . PUNCT cana-3423 227 1 [ [ X cana-3423 227 2 10 10 NUM cana-3423 227 3 ] ] X cana-3423 227 4 medio medio PROPN cana-3423 227 5 , , PUNCT cana-3423 227 6 a. a. PROPN cana-3423 227 7 , , PUNCT cana-3423 227 8 & & CCONJ cana-3423 227 9 lines line NOUN cana-3423 227 10 , , PUNCT cana-3423 227 11 m. m. NOUN cana-3423 227 12 ( ( PUNCT cana-3423 227 13 2001 2001 NUM cana-3423 227 14 ) ) PUNCT cana-3423 227 15 . . PUNCT cana-3423 228 1 nonlinear nonlinear ADJ cana-3423 228 2 dynamics dynamic NOUN cana-3423 228 3 : : PUNCT cana-3423 228 4 a a DET cana-3423 228 5 primer primer NOUN cana-3423 228 6 . . PUNCT cana-3423 229 1 cambridge cambridge PROPN cana-3423 229 2 university university PROPN cana-3423 229 3 press press NOUN cana-3423 229 4 . . PUNCT cana-3423 230 1 [ [ X cana-3423 230 2 11 11 NUM cana-3423 230 3 ] ] SYM cana-3423 230 4 schechter schechter NOUN cana-3423 230 5 , , PUNCT cana-3423 230 6 m. m. NOUN cana-3423 230 7 ( ( PUNCT cana-3423 230 8 2004 2004 NUM cana-3423 230 9 ) ) PUNCT cana-3423 230 10 . . PUNCT cana-3423 231 1 an an DET cana-3423 231 2 introduction introduction NOUN cana-3423 231 3 to to PART cana-3423 231 4 nonlinear nonlinear ADJ cana-3423 231 5 analysis analysis NOUN cana-3423 231 6 ( ( PUNCT cana-3423 231 7 no no INTJ cana-3423 231 8 . . NOUN cana-3423 231 9 95 95 NUM cana-3423 231 10 ) ) PUNCT cana-3423 231 11 . . PUNCT cana-3423 232 1 cambridge cambridge PROPN cana-3423 232 2 university university PROPN cana-3423 232 3 press press NOUN cana-3423 232 4 . . PUNCT cana-3423 233 1 [ [ X cana-3423 233 2 12 12 NUM cana-3423 233 3 ] ] PUNCT cana-3423 233 4 ciarlet ciarlet NOUN cana-3423 233 5 , , PUNCT cana-3423 233 6 p. p. NOUN cana-3423 233 7 g. g. PROPN cana-3423 233 8 ( ( PUNCT cana-3423 233 9 2013 2013 NUM cana-3423 233 10 ) ) PUNCT cana-3423 233 11 . . PUNCT cana-3423 234 1 linear linear PROPN cana-3423 234 2 and and CCONJ cana-3423 234 3 nonlinear nonlinear ADJ cana-3423 234 4 functional functional ADJ cana-3423 234 5 analysis analysis NOUN cana-3423 234 6 with with ADP cana-3423 234 7 applications application NOUN cana-3423 234 8 . . PUNCT cana-3423 235 1 society society NOUN cana-3423 235 2 for for ADP cana-3423 235 3 industrial industrial ADJ cana-3423 235 4 and and CCONJ cana-3423 235 5 applied applied ADJ cana-3423 235 6 mathematics mathematic NOUN cana-3423 235 7 . . PUNCT cana-3423 236 1 [ [ X cana-3423 236 2 13 13 NUM cana-3423 236 3 ] ] PUNCT cana-3423 236 4 evans evans PROPN cana-3423 236 5 , , PUNCT cana-3423 236 6 l. l. PROPN cana-3423 236 7 c. c. PROPN cana-3423 236 8 ( ( PUNCT cana-3423 236 9 2022 2022 NUM cana-3423 236 10 ) ) PUNCT cana-3423 236 11 . . PUNCT cana-3423 237 1 partial partial ADJ cana-3423 237 2 differential differential ADJ cana-3423 237 3 equations equation NOUN cana-3423 237 4 ( ( PUNCT cana-3423 237 5 vol vol NOUN cana-3423 237 6 . . PROPN cana-3423 237 7 19 19 NUM cana-3423 237 8 ) ) PUNCT cana-3423 237 9 . . PUNCT cana-3423 238 1 american american PROPN cana-3423 238 2 mathematical mathematical PROPN cana-3423 238 3 society society NOUN cana-3423 238 4 . . PUNCT