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PUNCT iajs-3665 24 1 the the DET iajs-3665 24 2 riemann riemann PROPN iajs-3665 24 3 - - PUNCT iajs-3665 24 4 liouville liouville VERB iajs-3665 24 5 derivatives derivative NOUN iajs-3665 24 6 used use VERB iajs-3665 24 7 numerous numerous ADJ iajs-3665 24 8 analytical analytical ADJ iajs-3665 24 9 and and CCONJ iajs-3665 24 10 numerical numerical ADJ iajs-3665 24 11 techniques technique NOUN iajs-3665 24 12 to to PART iajs-3665 24 13 solve solve VERB iajs-3665 24 14 distinguished distinguish VERB iajs-3665 24 15 foc foc ADJ iajs-3665 24 16 problem problem NOUN iajs-3665 24 17 types type NOUN iajs-3665 24 18 ( ( PUNCT iajs-3665 24 19 3 3 X iajs-3665 24 20 ) ) PUNCT iajs-3665 24 21 presented present VERB iajs-3665 24 22 a a DET iajs-3665 24 23 broad broad ADJ iajs-3665 24 24 characterization characterization NOUN iajs-3665 24 25 of of ADP iajs-3665 24 26 foc foc ADJ iajs-3665 24 27 issues issue NOUN iajs-3665 24 28 as as ADV iajs-3665 24 29 well well ADV iajs-3665 24 30 as as ADP iajs-3665 24 31 a a DET iajs-3665 24 32 way way NOUN iajs-3665 24 33 for for ADP iajs-3665 24 34 solving solve VERB iajs-3665 24 35 these these DET iajs-3665 24 36 problems problem NOUN iajs-3665 24 37 using use VERB iajs-3665 24 38 the the DET iajs-3665 24 39 variational variational ADJ iajs-3665 24 40 virtual virtual ADJ iajs-3665 24 41 method method NOUN iajs-3665 24 42 and and CCONJ iajs-3665 24 43 the the DET iajs-3665 24 44 lagrange lagrange ADJ iajs-3665 24 45 multiplier multipli ADJ iajs-3665 24 46 technique technique NOUN iajs-3665 24 47 . . PUNCT iajs-3665 25 1 it it PRON iajs-3665 25 2 is be AUX iajs-3665 25 3 commonly commonly ADV iajs-3665 25 4 used use VERB iajs-3665 25 5 in in ADP iajs-3665 25 6 the the DET iajs-3665 25 7 domains domain NOUN iajs-3665 25 8 of of ADP iajs-3665 25 9 operations operation NOUN iajs-3665 25 10 research research NOUN iajs-3665 25 11 , , PUNCT iajs-3665 25 12 engineering engineering NOUN iajs-3665 25 13 , , PUNCT iajs-3665 25 14 and and CCONJ iajs-3665 25 15 science science NOUN iajs-3665 25 16 . . PUNCT iajs-3665 26 1 a a DET iajs-3665 26 2 dynamic dynamic ADJ iajs-3665 26 3 system system NOUN iajs-3665 26 4 might might AUX iajs-3665 26 5 be be AUX iajs-3665 26 6 a a DET iajs-3665 26 7 spacecraft spacecraft NOUN iajs-3665 26 8 getting get VERB iajs-3665 26 9 controls control NOUN iajs-3665 26 10 that that PRON iajs-3665 26 11 correspond correspond VERB iajs-3665 26 12 to to ADP iajs-3665 26 13 rocket rocket NOUN iajs-3665 26 14 engines engine NOUN iajs-3665 26 15 , , PUNCT iajs-3665 26 16 with with ADP iajs-3665 26 17 the the DET iajs-3665 26 18 purpose purpose NOUN iajs-3665 26 19 of of ADP iajs-3665 26 20 reaching reach VERB iajs-3665 26 21 the the DET iajs-3665 26 22 moon moon NOUN iajs-3665 26 23 's 's PART iajs-3665 26 24 surface surface NOUN iajs-3665 26 25 with with ADP iajs-3665 26 26 the the DET iajs-3665 26 27 least least ADJ iajs-3665 26 28 amount amount NOUN iajs-3665 26 29 of of ADP iajs-3665 26 30 energy energy NOUN iajs-3665 26 31 . . PUNCT iajs-3665 27 1 additionally additionally ADV iajs-3665 27 2 , , PUNCT iajs-3665 27 3 fractional fractional ADJ iajs-3665 27 4 dynamics dynamic NOUN iajs-3665 27 5 , , PUNCT iajs-3665 27 6 a a DET iajs-3665 27 7 subfield subfield NOUN iajs-3665 27 8 of of ADP iajs-3665 27 9 mathematics mathematic NOUN iajs-3665 27 10 and and CCONJ iajs-3665 27 11 physics physics NOUN iajs-3665 27 12 , , PUNCT iajs-3665 27 13 studies study VERB iajs-3665 27 14 the the DET iajs-3665 27 15 behavior behavior NOUN iajs-3665 27 16 of of ADP iajs-3665 27 17 systems system NOUN iajs-3665 27 18 and and CCONJ iajs-3665 27 19 objects object NOUN iajs-3665 27 20 by by ADP iajs-3665 27 21 differentiating differentiate VERB iajs-3665 27 22 fractional fractional ADJ iajs-3665 27 23 orders order NOUN iajs-3665 27 24 . . PUNCT iajs-3665 28 1 numerous numerous ADJ iajs-3665 28 2 experts expert NOUN iajs-3665 28 3 , , PUNCT iajs-3665 28 4 including include VERB iajs-3665 28 5 mathematicians mathematician NOUN iajs-3665 28 6 , , PUNCT iajs-3665 28 7 physicists physicist NOUN iajs-3665 28 8 , , PUNCT iajs-3665 28 9 applied applied ADJ iajs-3665 28 10 researchers researcher NOUN iajs-3665 28 11 , , PUNCT iajs-3665 28 12 and and CCONJ iajs-3665 28 13 practitioners practitioner NOUN iajs-3665 28 14 , , PUNCT iajs-3665 28 15 showed show VERB iajs-3665 28 16 interest interest NOUN iajs-3665 28 17 in in ADP iajs-3665 28 18 the the DET iajs-3665 28 19 fresh fresh ADJ iajs-3665 28 20 knowledge knowledge NOUN iajs-3665 28 21 gained gain VERB iajs-3665 28 22 from from ADP iajs-3665 28 23 investigations investigation NOUN iajs-3665 28 24 on on ADP iajs-3665 28 25 fractional fractional ADJ iajs-3665 28 26 dynamical dynamical ADJ iajs-3665 28 27 systems system NOUN iajs-3665 28 28 . . PUNCT iajs-3665 29 1 a a DET iajs-3665 29 2 literature literature NOUN iajs-3665 29 3 evaluation evaluation NOUN iajs-3665 29 4 on on ADP iajs-3665 29 5 numerical numerical ADJ iajs-3665 29 6 analysis analysis NOUN iajs-3665 29 7 of of ADP iajs-3665 29 8 fractional fractional ADJ iajs-3665 29 9 - - PUNCT iajs-3665 29 10 order order NOUN iajs-3665 29 11 optimal optimal ADJ iajs-3665 29 12 controls control NOUN iajs-3665 29 13 systems system NOUN iajs-3665 29 14 ( ( PUNCT iajs-3665 29 15 focps focps PROPN iajs-3665 29 16 ) ) PUNCT iajs-3665 29 17 is be AUX iajs-3665 29 18 also also ADV iajs-3665 29 19 required require VERB iajs-3665 29 20 . . PUNCT iajs-3665 30 1 it it PRON iajs-3665 30 2 is be AUX iajs-3665 30 3 additionally additionally ADV iajs-3665 30 4 essential essential ADJ iajs-3665 30 5 for for ADP iajs-3665 30 6 conducting conduct VERB iajs-3665 30 7 a a DET iajs-3665 30 8 literature literature NOUN iajs-3665 30 9 review review NOUN iajs-3665 30 10 on on ADP iajs-3665 30 11 numerical numerical ADJ iajs-3665 30 12 analysis analysis NOUN iajs-3665 30 13 of of ADP iajs-3665 30 14 ( ( PUNCT iajs-3665 30 15 focps focps PROPN iajs-3665 30 16 ) ) PUNCT iajs-3665 30 17 . . PUNCT iajs-3665 31 1 for for ADP iajs-3665 31 2 focps focp NOUN iajs-3665 31 3 , , PUNCT iajs-3665 31 4 agrawa agrawa PROPN iajs-3665 31 5 ( ( PUNCT iajs-3665 31 6 3 3 X iajs-3665 31 7 ) ) PUNCT iajs-3665 31 8 performed perform VERB iajs-3665 31 9 a a DET iajs-3665 31 10 formulation formulation NOUN iajs-3665 31 11 and and CCONJ iajs-3665 31 12 numerical numerical ADJ iajs-3665 31 13 methodology methodology NOUN iajs-3665 31 14 for for ADP iajs-3665 31 15 this this DET iajs-3665 31 16 problem problem NOUN iajs-3665 31 17 . . PUNCT iajs-3665 32 1 some some DET iajs-3665 32 2 forms form NOUN iajs-3665 32 3 of of ADP iajs-3665 32 4 focps focp NOUN iajs-3665 32 5 were be AUX iajs-3665 32 6 numerically numerically ADV iajs-3665 32 7 solved solve VERB iajs-3665 32 8 by by ADP iajs-3665 32 9 ( ( PUNCT iajs-3665 32 10 4,5 4,5 NUM iajs-3665 32 11 ) ) PUNCT iajs-3665 32 12 . . PUNCT iajs-3665 33 1 furthermore furthermore ADV iajs-3665 33 2 , , PUNCT iajs-3665 33 3 akbarian akbarian VERB iajs-3665 33 4 and and CCONJ iajs-3665 33 5 keyanpour keyanpour NOUN iajs-3665 33 6 ( ( PUNCT iajs-3665 33 7 6 6 NUM iajs-3665 33 8 ) ) PUNCT iajs-3665 33 9 proposed propose VERB iajs-3665 33 10 a a DET iajs-3665 33 11 new new ADJ iajs-3665 33 12 approach approach NOUN iajs-3665 33 13 for for ADP iajs-3665 33 14 numerically numerically ADV iajs-3665 33 15 solving solve VERB iajs-3665 33 16 focps focp NOUN iajs-3665 33 17 . . PUNCT iajs-3665 34 1 bhrawy bhrawy PROPN iajs-3665 34 2 et et PROPN iajs-3665 34 3 al al PROPN iajs-3665 34 4 . . PROPN iajs-3665 35 1 ( ( PUNCT iajs-3665 35 2 7 7 X iajs-3665 35 3 ) ) PUNCT iajs-3665 35 4 established establish VERB iajs-3665 35 5 a a DET iajs-3665 35 6 practical practical ADJ iajs-3665 35 7 numerical numerical ADJ iajs-3665 35 8 technique technique NOUN iajs-3665 35 9 for for ADP iajs-3665 35 10 the the DET iajs-3665 35 11 quadratic quadratic ADJ iajs-3665 35 12 performance performance NOUN iajs-3665 35 13 index index NOUN iajs-3665 35 14 solution solution NOUN iajs-3665 35 15 for for ADP iajs-3665 35 16 multidimensional multidimensional ADJ iajs-3665 35 17 focps focp NOUN iajs-3665 35 18 . . PUNCT iajs-3665 36 1 parallel parallel ADJ iajs-3665 36 2 to to ADP iajs-3665 36 3 these these DET iajs-3665 36 4 researchers researcher NOUN iajs-3665 36 5 , , PUNCT iajs-3665 36 6 sweilam sweilam PROPN iajs-3665 36 7 et et PROPN iajs-3665 36 8 al al PROPN iajs-3665 36 9 . . PROPN iajs-3665 36 10 used use VERB iajs-3665 36 11 the the DET iajs-3665 36 12 shifted shift VERB iajs-3665 36 13 chebyshev chebyshev NOUN iajs-3665 36 14 polynomials polynomial NOUN iajs-3665 36 15 of of ADP iajs-3665 36 16 second second ADJ iajs-3665 36 17 type type NOUN iajs-3665 36 18 ( ( PUNCT iajs-3665 36 19 8) 8) NUM iajs-3665 36 20 but but CCONJ iajs-3665 36 21 a a DET iajs-3665 36 22 discrete discrete NOUN iajs-3665 36 23 - - PUNCT iajs-3665 36 24 technique technique NOUN iajs-3665 36 25 for for ADP iajs-3665 36 26 solving solve VERB iajs-3665 36 27 focps focp NOUN iajs-3665 36 28 introduced introduce VERB iajs-3665 36 29 by by ADP iajs-3665 36 30 almeida almeida PROPN iajs-3665 36 31 and and CCONJ iajs-3665 36 32 torres torre VERB iajs-3665 36 33 ( ( PUNCT iajs-3665 36 34 9 9 NUM iajs-3665 36 35 ) ) PUNCT iajs-3665 36 36 . . PUNCT iajs-3665 37 1 following follow VERB iajs-3665 37 2 that that PRON iajs-3665 37 3 , , PUNCT iajs-3665 37 4 they they PRON iajs-3665 37 5 solved solve VERB iajs-3665 37 6 certain certain ADJ iajs-3665 37 7 types type NOUN iajs-3665 37 8 of of ADP iajs-3665 37 9 focps focp NOUN iajs-3665 37 10 using use VERB iajs-3665 37 11 collocation collocation NOUN iajs-3665 37 12 method method NOUN iajs-3665 37 13 with with ADP iajs-3665 37 14 legendre legendre PROPN iajs-3665 37 15 spectral spectral PROPN iajs-3665 37 16 , , PUNCT iajs-3665 37 17 and and CCONJ iajs-3665 37 18 a a DET iajs-3665 37 19 number number NOUN iajs-3665 37 20 of of ADP iajs-3665 37 21 authors author NOUN iajs-3665 37 22 examined examine VERB iajs-3665 37 23 a a DET iajs-3665 37 24 variety variety NOUN iajs-3665 37 25 of of ADP iajs-3665 37 26 fdes fde NOUN iajs-3665 37 27 . . PUNCT iajs-3665 38 1 moreover moreover ADV iajs-3665 38 2 , , PUNCT iajs-3665 38 3 ahmad ahmad PROPN iajs-3665 38 4 and and CCONJ iajs-3665 38 5 el el PROPN iajs-3665 38 6 - - PUNCT iajs-3665 38 7 khazali khazali PROPN iajs-3665 38 8 ( ( PUNCT iajs-3665 38 9 10 10 NUM iajs-3665 38 10 ) ) PUNCT iajs-3665 38 11 introduced introduce VERB iajs-3665 38 12 a a DET iajs-3665 38 13 dynamical dynamical ADJ iajs-3665 38 14 model model NOUN iajs-3665 38 15 of of ADP iajs-3665 38 16 fractional fractional ADJ iajs-3665 38 17 - - PUNCT iajs-3665 38 18 order order NOUN iajs-3665 38 19 , , PUNCT iajs-3665 38 20 and and CCONJ iajs-3665 38 21 david david PROPN iajs-3665 38 22 et et PROPN iajs-3665 38 23 al al PROPN iajs-3665 38 24 . . PUNCT iajs-3665 39 1 while while SCONJ iajs-3665 39 2 ( ( PUNCT iajs-3665 39 3 11 11 NUM iajs-3665 39 4 ) ) PUNCT iajs-3665 39 5 investigated investigate VERB iajs-3665 39 6 fractional fractional ADJ iajs-3665 39 7 - - PUNCT iajs-3665 39 8 order order NOUN iajs-3665 39 9 calculus calculus NOUN iajs-3665 39 10 . . PUNCT iajs-3665 40 1 lastly lastly ADV iajs-3665 40 2 , , PUNCT iajs-3665 40 3 some some DET iajs-3665 40 4 authors author NOUN iajs-3665 40 5 studied study VERB iajs-3665 40 6 the the DET iajs-3665 40 7 solutions solution NOUN iajs-3665 40 8 of of ADP iajs-3665 40 9 optimal optimal ADJ iajs-3665 40 10 control control NOUN iajs-3665 40 11 systems system NOUN iajs-3665 40 12 using use VERB iajs-3665 40 13 different different ADJ iajs-3665 40 14 approaches approach NOUN iajs-3665 40 15 ( ( PUNCT iajs-3665 40 16 12 12 NUM iajs-3665 40 17 - - SYM iajs-3665 40 18 19 19 NUM iajs-3665 40 19 ) ) PUNCT iajs-3665 40 20 . . PUNCT iajs-3665 41 1 the the DET iajs-3665 41 2 direct direct ADJ iajs-3665 41 3 searching searching NOUN iajs-3665 41 4 method method NOUN iajs-3665 41 5 is be AUX iajs-3665 41 6 being be AUX iajs-3665 41 7 investigated investigate VERB iajs-3665 41 8 in in ADP iajs-3665 41 9 order order NOUN iajs-3665 41 10 to to PART iajs-3665 41 11 overcome overcome VERB iajs-3665 41 12 the the DET iajs-3665 41 13 unconstrained unconstrained ADJ iajs-3665 41 14 optimization optimization NOUN iajs-3665 41 15 difficulties difficulty NOUN iajs-3665 41 16 . . PUNCT iajs-3665 42 1 however however ADV iajs-3665 42 2 , , PUNCT iajs-3665 42 3 the the DET iajs-3665 42 4 collocation collocation NOUN iajs-3665 42 5 method method NOUN iajs-3665 42 6 with with ADP iajs-3665 42 7 the the DET iajs-3665 42 8 base base NOUN iajs-3665 42 9 of of ADP iajs-3665 42 10 a a DET iajs-3665 42 11 b b NOUN iajs-3665 42 12 - - PUNCT iajs-3665 42 13 cubic cubic ADJ iajs-3665 42 14 spline spline NOUN iajs-3665 42 15 was be AUX iajs-3665 42 16 implemented implement VERB iajs-3665 42 17 to to PART iajs-3665 42 18 solve solve VERB iajs-3665 42 19 the the DET iajs-3665 42 20 problem problem NOUN iajs-3665 42 21 focps focp NOUN iajs-3665 42 22 in in ADP iajs-3665 42 23 both both DET iajs-3665 42 24 cases case NOUN iajs-3665 42 25 of of ADP iajs-3665 42 26 ( ( PUNCT iajs-3665 42 27 tt tt PROPN iajs-3665 42 28 ) ) PUNCT iajs-3665 42 29 . . PUNCT iajs-3665 43 1 furthermore furthermore ADV iajs-3665 43 2 , , PUNCT iajs-3665 43 3 the the DET iajs-3665 43 4 direct direct ADJ iajs-3665 43 5 searching searching NOUN iajs-3665 43 6 method method NOUN iajs-3665 43 7 is be AUX iajs-3665 43 8 being be AUX iajs-3665 43 9 investigated investigate VERB iajs-3665 43 10 in in ADP iajs-3665 43 11 order order NOUN iajs-3665 43 12 to to PART iajs-3665 43 13 overcome overcome VERB iajs-3665 43 14 the the DET iajs-3665 43 15 unconstrained unconstrained ADJ iajs-3665 43 16 optimization optimization NOUN iajs-3665 43 17 difficulties difficulty NOUN iajs-3665 43 18 . . PUNCT iajs-3665 44 1 in in ADP iajs-3665 44 2 this this DET iajs-3665 44 3 paper paper NOUN iajs-3665 44 4 , , PUNCT iajs-3665 44 5 we we PRON iajs-3665 44 6 have have AUX iajs-3665 44 7 approximated approximate VERB iajs-3665 44 8 the the DET iajs-3665 44 9 numerical numerical ADJ iajs-3665 44 10 solutions solution NOUN iajs-3665 44 11 for for ADP iajs-3665 44 12 a a DET iajs-3665 44 13 fractional fractional ADJ iajs-3665 44 14 - - PUNCT iajs-3665 44 15 order order NOUN iajs-3665 44 16 optimal optimal ADJ iajs-3665 44 17 control control NOUN iajs-3665 44 18 systems system NOUN iajs-3665 44 19 in in ADP iajs-3665 44 20 both both CCONJ iajs-3665 44 21 free free ADJ iajs-3665 44 22 and and CCONJ iajs-3665 44 23 non non ADJ iajs-3665 44 24 - - ADJ iajs-3665 44 25 free free ADJ iajs-3665 44 26 terminal terminal ADJ iajs-3665 44 27 time time NOUN iajs-3665 44 28 ( ( PUNCT iajs-3665 44 29 tt tt PROPN iajs-3665 44 30 ) ) PUNCT iajs-3665 44 31 . . PUNCT iajs-3665 45 1 after after ADV iajs-3665 45 2 all all ADV iajs-3665 45 3 , , PUNCT iajs-3665 45 4 the the DET iajs-3665 45 5 collocation collocation NOUN iajs-3665 45 6 approach approach NOUN iajs-3665 45 7 considers consider VERB iajs-3665 45 8 a a DET iajs-3665 45 9 numerical numerical ADJ iajs-3665 45 10 solution solution NOUN iajs-3665 45 11 to to ADP iajs-3665 45 12 this this DET iajs-3665 45 13 problem problem NOUN iajs-3665 45 14 by by ADP iajs-3665 45 15 using use VERB iajs-3665 45 16 a a DET iajs-3665 45 17 b b NOUN iajs-3665 45 18 - - PUNCT iajs-3665 45 19 cubic cubic ADJ iajs-3665 45 20 spline spline NOUN iajs-3665 45 21 base base NOUN iajs-3665 45 22 . . PUNCT iajs-3665 46 1 accordingly accordingly ADV iajs-3665 46 2 , , PUNCT iajs-3665 46 3 the the DET iajs-3665 46 4 computations computation NOUN iajs-3665 46 5 in in ADP iajs-3665 46 6 the the DET iajs-3665 46 7 proposed propose VERB iajs-3665 46 8 method method NOUN iajs-3665 46 9 are be AUX iajs-3665 46 10 calculated calculate VERB iajs-3665 46 11 the the DET iajs-3665 46 12 methodology methodology NOUN iajs-3665 46 13 for for ADP iajs-3665 46 14 the the DET iajs-3665 46 15 well well ADV iajs-3665 46 16 - - PUNCT iajs-3665 46 17 known know VERB iajs-3665 46 18 hooke hooke NOUN iajs-3665 46 19 and and CCONJ iajs-3665 46 20 jeeves jeeve NOUN iajs-3665 46 21 approach approach NOUN iajs-3665 46 22 . . PUNCT iajs-3665 47 1 2 2 X iajs-3665 47 2 . . X iajs-3665 47 3 preliminary preliminary ADJ iajs-3665 47 4 we we PRON iajs-3665 47 5 have have AUX iajs-3665 47 6 discussed discuss VERB iajs-3665 47 7 some some DET iajs-3665 47 8 background background NOUN iajs-3665 47 9 knowledge knowledge NOUN iajs-3665 47 10 and and CCONJ iajs-3665 47 11 concepts concept NOUN iajs-3665 47 12 concerning concern VERB iajs-3665 47 13 the the DET iajs-3665 47 14 research research NOUN iajs-3665 47 15 problem problem NOUN iajs-3665 47 16 in in ADP iajs-3665 47 17 this this DET iajs-3665 47 18 section section NOUN iajs-3665 47 19 . . PUNCT iajs-3665 48 1 ihjpas ihjpas PROPN iajs-3665 48 2 . . PUNCT iajs-3665 49 1 2025 2025 NUM iajs-3665 49 2 , , PUNCT iajs-3665 49 3 38(2 38(2 NUM iajs-3665 49 4 ) ) PUNCT iajs-3665 49 5 340 340 NUM iajs-3665 49 6 2.1 2.1 NUM iajs-3665 49 7 fractional fractional ADJ iajs-3665 49 8 derivatives derivative NOUN iajs-3665 49 9 background background NOUN iajs-3665 49 10 information information NOUN iajs-3665 49 11 and and CCONJ iajs-3665 49 12 basic basic ADJ iajs-3665 49 13 definitions definition NOUN iajs-3665 49 14 with with ADP iajs-3665 49 15 regard regard NOUN iajs-3665 49 16 to to ADP iajs-3665 49 17 the the DET iajs-3665 49 18 study study NOUN iajs-3665 49 19 's 's PART iajs-3665 49 20 problem problem NOUN iajs-3665 49 21 have have AUX iajs-3665 49 22 been be AUX iajs-3665 49 23 provided provide VERB iajs-3665 49 24 . . PUNCT iajs-3665 50 1 this this DET iajs-3665 50 2 section section NOUN iajs-3665 50 3 provides provide VERB iajs-3665 50 4 the the DET iajs-3665 50 5 essential essential ADJ iajs-3665 50 6 definitions definition NOUN iajs-3665 50 7 of of ADP iajs-3665 50 8 fractional fractional ADJ iajs-3665 50 9 derivatives derivative NOUN iajs-3665 50 10 including include VERB iajs-3665 50 11 the the DET iajs-3665 50 12 terminal terminal ADJ iajs-3665 50 13 times time NOUN iajs-3665 50 14 ( ( PUNCT iajs-3665 50 15 tt tt PROPN iajs-3665 50 16 ) ) PUNCT iajs-3665 50 17 of of ADP iajs-3665 50 18 focps focp NOUN iajs-3665 50 19 . . PUNCT iajs-3665 51 1 using use VERB iajs-3665 51 2 the the DET iajs-3665 51 3 riemann riemann PROPN iajs-3665 51 4 - - PUNCT iajs-3665 51 5 liouville liouville VERB iajs-3665 51 6 derivative derivative ADJ iajs-3665 51 7 formulation formulation NOUN iajs-3665 51 8 , , PUNCT iajs-3665 51 9 the the DET iajs-3665 51 10 fractional fractional ADJ iajs-3665 51 11 derivative derivative NOUN iajs-3665 51 12 of of ADP iajs-3665 51 13 the the DET iajs-3665 51 14 function function NOUN iajs-3665 51 15 f f PROPN iajs-3665 51 16 is be AUX iajs-3665 51 17 defined define VERB iajs-3665 51 18 as as SCONJ iajs-3665 51 19 follows follow VERB iajs-3665 51 20 for for ADP iajs-3665 51 21 𝜎 𝜎 PROPN iajs-3665 51 22 ∈ ∈ PROPN iajs-3665 52 1 [ [ X iajs-3665 52 2 𝑛 𝑛 NOUN iajs-3665 52 3 βˆ’ βˆ’ PROPN iajs-3665 52 4 1 1 NUM iajs-3665 52 5 , , PUNCT iajs-3665 52 6 𝑛 𝑛 NOUN iajs-3665 52 7 ) ) PUNCT iajs-3665 52 8 , , PUNCT iajs-3665 52 9 and and CCONJ iajs-3665 52 10 𝑛 𝑛 DET iajs-3665 52 11 ∈ ∈ PROPN iajs-3665 52 12 𝑁 𝑁 PROPN iajs-3665 52 13 , , PUNCT iajs-3665 52 14 definition definition NOUN iajs-3665 52 15 2.1 2.1 NUM iajs-3665 52 16 . . PUNCT iajs-3665 53 1 the the DET iajs-3665 53 2 following follow VERB iajs-3665 53 3 are be AUX iajs-3665 53 4 left leave VERB iajs-3665 53 5 and and CCONJ iajs-3665 53 6 right right ADJ iajs-3665 53 7 sides side NOUN iajs-3665 53 8 of of ADP iajs-3665 53 9 the the DET iajs-3665 53 10 fractional fractional ADJ iajs-3665 53 11 - - PUNCT iajs-3665 53 12 integral integral ADJ iajs-3665 53 13 riemann riemann NOUN iajs-3665 53 14 - - PUNCT iajs-3665 53 15 liouville liouville NOUN iajs-3665 53 16 ( ( PUNCT iajs-3665 53 17 rl rl NOUN iajs-3665 53 18 ) ) PUNCT iajs-3665 53 19 operator operator NOUN iajs-3665 53 20 of of ADP iajs-3665 53 21 order order NOUN iajs-3665 53 22 𝜎 𝜎 PROPN iajs-3665 53 23 > > X iajs-3665 53 24 0 0 NUM iajs-3665 53 25 : : PUNCT iajs-3665 53 26 adΟ„ adΟ„ NOUN iajs-3665 53 27 Οƒy(Ο„ Οƒy(Ο„ PRON iajs-3665 53 28 ) ) PUNCT iajs-3665 53 29 = = SYM iajs-3665 53 30 1 1 NUM iajs-3665 53 31 Ξ³(nβˆ’Οƒ Ξ³(nβˆ’Οƒ PROPN iajs-3665 53 32 ) ) PUNCT iajs-3665 53 33 dn dn PROPN iajs-3665 53 34 dΟ„n dΟ„n PROPN iajs-3665 53 35 ∫ ∫ PROPN iajs-3665 53 36 y(Ο„ y(Ο„ PROPN iajs-3665 53 37 ) ) PUNCT iajs-3665 53 38 ( ( PUNCT iajs-3665 53 39 Ο„βˆ’π‘₯)nβˆ’Οƒβˆ’1 Ο„βˆ’π‘₯)nβˆ’Οƒβˆ’1 X iajs-3665 53 40 Ο„ Ο„ PROPN iajs-3665 53 41 a a DET iajs-3665 53 42 dπ‘₯ dπ‘₯ PROPN iajs-3665 53 43 , , PUNCT iajs-3665 53 44 ( ( PUNCT iajs-3665 53 45 1 1 X iajs-3665 53 46 ) ) PUNCT iajs-3665 53 47 and and CCONJ iajs-3665 53 48 adΟ„ adΟ„ VERB iajs-3665 53 49 Οƒy(Ο„ Οƒy(Ο„ PRON iajs-3665 53 50 ) ) PUNCT iajs-3665 53 51 = = SYM iajs-3665 53 52 1 1 NUM iajs-3665 53 53 Ξ³(nβˆ’Οƒ Ξ³(nβˆ’Οƒ PROPN iajs-3665 53 54 ) ) PUNCT iajs-3665 53 55 dn dn PROPN iajs-3665 53 56 dΟ„n dΟ„n PROPN iajs-3665 53 57 ∫ ∫ PROPN iajs-3665 54 1 ( ( PUNCT iajs-3665 54 2 Ο„ Ο„ X iajs-3665 54 3 βˆ’ βˆ’ PROPN iajs-3665 54 4 s)nβˆ’Οƒβˆ’1 s)nβˆ’Οƒβˆ’1 PROPN iajs-3665 54 5 Ο„ Ο„ PROPN iajs-3665 54 6 a a DET iajs-3665 54 7 y(s)ds y(s)ds NOUN iajs-3665 54 8 , , PUNCT iajs-3665 54 9 ( ( PUNCT iajs-3665 54 10 2 2 X iajs-3665 54 11 ) ) PUNCT iajs-3665 54 12 caputo caputo NOUN iajs-3665 54 13 derivative derivative NOUN iajs-3665 54 14 of of ADP iajs-3665 54 15 the the DET iajs-3665 54 16 function function NOUN iajs-3665 54 17 𝑓 𝑓 X iajs-3665 54 18 ; ; PUNCT iajs-3665 54 19 for for ADP iajs-3665 54 20 𝜎 𝜎 PROPN iajs-3665 54 21 ∈ ∈ PROPN iajs-3665 55 1 [ [ X iajs-3665 55 2 𝑛 𝑛 NOUN iajs-3665 55 3 βˆ’ βˆ’ PROPN iajs-3665 55 4 1 1 NUM iajs-3665 55 5 , , PUNCT iajs-3665 55 6 𝑛 𝑛 NOUN iajs-3665 55 7 ) ) PUNCT iajs-3665 55 8 , , PUNCT iajs-3665 55 9 is be AUX iajs-3665 55 10 defined define VERB iajs-3665 55 11 as as SCONJ iajs-3665 55 12 follows follow VERB iajs-3665 55 13 : : PUNCT iajs-3665 55 14 definition definition NOUN iajs-3665 55 15 2.2.the 2.2.the DET iajs-3665 55 16 fractional fractional PROPN iajs-3665 55 17 - - PUNCT iajs-3665 55 18 caputo caputo NOUN iajs-3665 55 19 - - PUNCT iajs-3665 55 20 derivative derivative NOUN iajs-3665 55 21 has have VERB iajs-3665 55 22 the the DET iajs-3665 55 23 form form NOUN iajs-3665 55 24 as as SCONJ iajs-3665 55 25 follows follow VERB iajs-3665 55 26 adΟ„ adΟ„ NOUN iajs-3665 55 27 σ𝑓 σ𝑓 PROPN iajs-3665 55 28 ( ( PUNCT iajs-3665 55 29 Ο„ Ο„ X iajs-3665 55 30 ) ) PUNCT iajs-3665 55 31 = = SYM iajs-3665 55 32 1 1 NUM iajs-3665 55 33 Ξ³(nβˆ’Οƒ Ξ³(nβˆ’Οƒ PROPN iajs-3665 55 34 ) ) PUNCT iajs-3665 55 35 ∫ ∫ PROPN iajs-3665 55 36 𝑓(𝑛 𝑓(𝑛 X iajs-3665 55 37 ) ) PUNCT iajs-3665 55 38 ( ( PUNCT iajs-3665 55 39 x x X iajs-3665 55 40 ) ) PUNCT iajs-3665 55 41 ( ( PUNCT iajs-3665 56 1 Ο„βˆ’π‘₯)Οƒβˆ’nβˆ’1 Ο„βˆ’π‘₯)Οƒβˆ’nβˆ’1 PROPN iajs-3665 56 2 Ο„ Ο„ PROPN iajs-3665 56 3 a a DET iajs-3665 56 4 dx dx PROPN iajs-3665 56 5 , , PUNCT iajs-3665 56 6 ( ( PUNCT iajs-3665 56 7 3 3 X iajs-3665 56 8 ) ) PUNCT iajs-3665 56 9 the the DET iajs-3665 56 10 fractional fractional ADJ iajs-3665 56 11 - - PUNCT iajs-3665 56 12 type type NOUN iajs-3665 56 13 derivatives derivative NOUN iajs-3665 56 14 and and CCONJ iajs-3665 56 15 integrals integral NOUN iajs-3665 56 16 for for ADP iajs-3665 56 17 Οƒ Οƒ PROPN iajs-3665 56 18 > > X iajs-3665 56 19 0 0 NUM iajs-3665 56 20 ; ; PUNCT iajs-3665 56 21 n n CCONJ iajs-3665 56 22 βˆ’ βˆ’ PROPN iajs-3665 56 23 1 1 NUM iajs-3665 56 24 ≀ ≀ NUM iajs-3665 56 25 Οƒ Οƒ NOUN iajs-3665 56 26 < < X iajs-3665 56 27 n n X iajs-3665 56 28 have have VERB iajs-3665 56 29 the the DET iajs-3665 56 30 property property NOUN iajs-3665 56 31 of of ADP iajs-3665 56 32 linearity linearity NOUN iajs-3665 56 33 where where SCONJ iajs-3665 56 34 𝑓 𝑓 DET iajs-3665 56 35 ∢ ∢ NOUN iajs-3665 56 36 [ [ X iajs-3665 56 37 π‘Ž;∞ π‘Ž;∞ NOUN iajs-3665 56 38 ) ) PUNCT iajs-3665 56 39 β†’ β†’ SYM iajs-3665 56 40 β„œ β„œ PROPN iajs-3665 56 41 and and CCONJ iajs-3665 56 42 π‘Ž π‘Ž ADP iajs-3665 56 43 > > X iajs-3665 56 44 0 0 NUM iajs-3665 56 45 . . PUNCT iajs-3665 57 1 2.2 2.2 NUM iajs-3665 57 2 a a DET iajs-3665 57 3 quasi quasi ADJ iajs-3665 57 4 - - ADJ iajs-3665 57 5 linear linear ADJ iajs-3665 57 6 fodes fode NOUN iajs-3665 57 7 of of ADP iajs-3665 57 8 first first ADJ iajs-3665 57 9 - - PUNCT iajs-3665 57 10 order order NOUN iajs-3665 57 11 the the DET iajs-3665 57 12 quasi quasi NOUN iajs-3665 57 13 linear linear PROPN iajs-3665 57 14 first first ADJ iajs-3665 57 15 - - PUNCT iajs-3665 57 16 order order NOUN iajs-3665 57 17 fractional fractional ADJ iajs-3665 57 18 ordinary ordinary ADJ iajs-3665 57 19 differential differential ADJ iajs-3665 57 20 equation equation NOUN iajs-3665 57 21 is be AUX iajs-3665 57 22 written write VERB iajs-3665 57 23 as as ADP iajs-3665 57 24 follow follow NOUN iajs-3665 57 25 : : PUNCT iajs-3665 57 26 π·βˆπ‘’(𝜏 π·βˆπ‘’(𝜏 NOUN iajs-3665 57 27 ) ) PUNCT iajs-3665 57 28 = = SYM iajs-3665 57 29 𝛷(𝜏 𝛷(𝜏 PROPN iajs-3665 57 30 , , PUNCT iajs-3665 57 31 𝑒(𝜏 𝑒(𝜏 PROPN iajs-3665 57 32 ) ) PUNCT iajs-3665 57 33 , , PUNCT iajs-3665 57 34 π‘’κžŒ(𝜏 π‘’κžŒ(𝜏 NUM iajs-3665 57 35 ) ) PUNCT iajs-3665 57 36 ) ) PUNCT iajs-3665 57 37 ; ; PUNCT iajs-3665 57 38 0 0 NUM iajs-3665 57 39 < < X iajs-3665 57 40 𝜏 𝜏 X iajs-3665 57 41 < < X iajs-3665 57 42 𝑏 𝑏 NOUN iajs-3665 57 43 , , PUNCT iajs-3665 57 44 0 0 PUNCT iajs-3665 57 45 < < X iajs-3665 57 46 ∝ ∝ PROPN iajs-3665 57 47 < < X iajs-3665 57 48 1 1 NUM iajs-3665 57 49 ( ( PUNCT iajs-3665 57 50 4 4 NUM iajs-3665 57 51 ) ) PUNCT iajs-3665 57 52 with with ADP iajs-3665 57 53 the the DET iajs-3665 57 54 initial initial ADJ iajs-3665 57 55 condition condition NOUN iajs-3665 57 56 ( ( PUNCT iajs-3665 57 57 ic ic X iajs-3665 57 58 ) ) PUNCT iajs-3665 57 59 𝑒(0 𝑒(0 PROPN iajs-3665 57 60 ) ) PUNCT iajs-3665 57 61 = = SYM iajs-3665 57 62 πœ‰0 πœ‰0 NOUN iajs-3665 57 63 . . PUNCT iajs-3665 58 1 ( ( PUNCT iajs-3665 58 2 5 5 X iajs-3665 58 3 ) ) SYM iajs-3665 58 4 2.2.1 2.2.1 NUM iajs-3665 58 5 boundary boundary ADJ iajs-3665 58 6 value value NOUN iajs-3665 58 7 problem problem NOUN iajs-3665 58 8 of of ADP iajs-3665 58 9 second second ADJ iajs-3665 58 10 - - PUNCT iajs-3665 58 11 order order NOUN iajs-3665 58 12 fodes fode NOUN iajs-3665 58 13 . . PUNCT iajs-3665 59 1 consider consider VERB iajs-3665 59 2 the the DET iajs-3665 59 3 following follow VERB iajs-3665 59 4 quasi quasi ADJ iajs-3665 59 5 - - ADJ iajs-3665 59 6 linear linear ADJ iajs-3665 59 7 second second ADJ iajs-3665 59 8 - - PUNCT iajs-3665 59 9 order order NOUN iajs-3665 59 10 fractional fractional ADJ iajs-3665 59 11 differential differential NOUN iajs-3665 59 12 equation equation NOUN iajs-3665 59 13 : : PUNCT iajs-3665 59 14 π·βˆπ›·(Ο› π·βˆπ›·(Ο› X iajs-3665 59 15 ) ) PUNCT iajs-3665 60 1 = = SYM iajs-3665 60 2 πœ“(𝜏 πœ“(𝜏 NOUN iajs-3665 60 3 , , PUNCT iajs-3665 60 4 𝛷(Ο›),π›·κžŒ(Ο›),π›·κžŒκžŒ(Ο› 𝛷(Ο›),π›·κžŒ(Ο›),π›·κžŒκžŒ(Ο› NOUN iajs-3665 60 5 ) ) PUNCT iajs-3665 60 6 ) ) PUNCT iajs-3665 60 7 ; ; PUNCT iajs-3665 60 8 0 0 PUNCT iajs-3665 60 9 < < X iajs-3665 60 10 Ο› Ο› X iajs-3665 60 11 < < X iajs-3665 60 12 1 1 NUM iajs-3665 60 13 , , PUNCT iajs-3665 60 14 0 0 NUM iajs-3665 61 1 < < X iajs-3665 61 2 ∝ ∝ X iajs-3665 61 3 < < X iajs-3665 61 4 2 2 NUM iajs-3665 61 5 ( ( PUNCT iajs-3665 61 6 6 6 NUM iajs-3665 61 7 ) ) PUNCT iajs-3665 61 8 with with ADP iajs-3665 61 9 the the DET iajs-3665 61 10 boundary boundary ADJ iajs-3665 61 11 conditions condition NOUN iajs-3665 61 12 ( ( PUNCT iajs-3665 61 13 bcs bcs NOUN iajs-3665 61 14 ) ) PUNCT iajs-3665 61 15 𝛷(0 𝛷(0 NUM iajs-3665 61 16 ) ) PUNCT iajs-3665 61 17 = = NOUN iajs-3665 61 18 πœ‰0 πœ‰0 NOUN iajs-3665 61 19 ; ; PUNCT iajs-3665 61 20 𝛷(1 𝛷(1 PUNCT iajs-3665 61 21 ) ) PUNCT iajs-3665 61 22 = = SYM iajs-3665 61 23 πœ‰1 πœ‰1 PROPN iajs-3665 61 24 . . PUNCT iajs-3665 62 1 ( ( PUNCT iajs-3665 62 2 7 7 NUM iajs-3665 62 3 ) ) PUNCT iajs-3665 62 4 2.3 2.3 NUM iajs-3665 62 5 . . PUNCT iajs-3665 63 1 spline spline NOUN iajs-3665 63 2 functions function NOUN iajs-3665 63 3 we we PRON iajs-3665 63 4 reviewed review VERB iajs-3665 63 5 the the DET iajs-3665 63 6 cubic cubic ADJ iajs-3665 63 7 b b NOUN iajs-3665 63 8 - - PUNCT iajs-3665 63 9 spline spline NOUN iajs-3665 63 10 basis basis NOUN iajs-3665 63 11 and and CCONJ iajs-3665 63 12 how how SCONJ iajs-3665 63 13 it it PRON iajs-3665 63 14 may may AUX iajs-3665 63 15 be be AUX iajs-3665 63 16 utilized utilize VERB iajs-3665 63 17 to to PART iajs-3665 63 18 solve solve VERB iajs-3665 63 19 odes ode NOUN iajs-3665 63 20 in in ADP iajs-3665 63 21 the the DET iajs-3665 63 22 next next ADJ iajs-3665 63 23 section section NOUN iajs-3665 63 24 . . PUNCT iajs-3665 64 1 spline spline PROPN iajs-3665 64 2 refers refer VERB iajs-3665 64 3 to to ADP iajs-3665 64 4 a a DET iajs-3665 64 5 comprehensive comprehensive ADJ iajs-3665 64 6 class class NOUN iajs-3665 64 7 of of ADP iajs-3665 64 8 smooth smooth ADJ iajs-3665 64 9 functions function NOUN iajs-3665 64 10 used use VERB iajs-3665 64 11 in in ADP iajs-3665 64 12 data data NOUN iajs-3665 64 13 interpolation interpolation NOUN iajs-3665 64 14 applications application NOUN iajs-3665 64 15 ( ( PUNCT iajs-3665 64 16 20 20 NUM iajs-3665 64 17 ) ) PUNCT iajs-3665 64 18 . . PUNCT iajs-3665 65 1 in in ADP iajs-3665 65 2 general general ADJ iajs-3665 65 3 , , PUNCT iajs-3665 65 4 the the DET iajs-3665 65 5 spline spline NOUN iajs-3665 65 6 function function NOUN iajs-3665 65 7 for for ADP iajs-3665 65 8 interpolation interpolation NOUN iajs-3665 65 9 is be AUX iajs-3665 65 10 derived derive VERB iajs-3665 65 11 by by ADP iajs-3665 65 12 reducing reduce VERB iajs-3665 65 13 the the DET iajs-3665 65 14 appropriate appropriate ADJ iajs-3665 65 15 roughness roughness NOUN iajs-3665 65 16 measures measure NOUN iajs-3665 65 17 while while SCONJ iajs-3665 65 18 maintaining maintain VERB iajs-3665 65 19 the the DET iajs-3665 65 20 interpolation interpolation NOUN iajs-3665 65 21 criteria criterion NOUN iajs-3665 65 22 in in ADP iajs-3665 65 23 mind mind NOUN iajs-3665 65 24 . . PUNCT iajs-3665 66 1 smoothing smooth VERB iajs-3665 66 2 splines spline NOUN iajs-3665 66 3 are be AUX iajs-3665 66 4 the the DET iajs-3665 66 5 best good ADJ iajs-3665 66 6 base base NOUN iajs-3665 66 7 for for ADP iajs-3665 66 8 interpolation interpolation NOUN iajs-3665 66 9 the the DET iajs-3665 66 10 solution solution NOUN iajs-3665 66 11 of of ADP iajs-3665 66 12 differential differential ADJ iajs-3665 66 13 equations equation NOUN iajs-3665 66 14 . . PUNCT iajs-3665 67 1 the the DET iajs-3665 67 2 base base NOUN iajs-3665 67 3 𝛷(Ο› 𝛷(Ο› NOUN iajs-3665 67 4 ) ) PUNCT iajs-3665 67 5 = = NOUN iajs-3665 67 6 { { PUNCT iajs-3665 67 7 𝛷1(Ο› 𝛷1(Ο› NUM iajs-3665 67 8 ) ) PUNCT iajs-3665 67 9 , , PUNCT iajs-3665 67 10 𝛷2(Ο› 𝛷2(Ο› NUM iajs-3665 67 11 ) ) PUNCT iajs-3665 67 12 , , PUNCT iajs-3665 67 13 … … PUNCT iajs-3665 67 14 , , PUNCT iajs-3665 67 15 𝛷𝑛(Ο› 𝛷𝑛(Ο› NOUN iajs-3665 67 16 ) ) PUNCT iajs-3665 67 17 } } PUNCT iajs-3665 67 18 has have AUX iajs-3665 67 19 been be AUX iajs-3665 67 20 called call VERB iajs-3665 67 21 as as ADP iajs-3665 67 22 spline spline NOUN iajs-3665 67 23 base base NOUN iajs-3665 67 24 of of ADP iajs-3665 67 25 π‘›π‘‘β„Žorder π‘›π‘‘β„Žorder NOUN iajs-3665 67 26 if if SCONJ iajs-3665 67 27 all all DET iajs-3665 67 28 basis basis NOUN iajs-3665 67 29 functions function NOUN iajs-3665 67 30 satisfy satisfy VERB iajs-3665 67 31 the the DET iajs-3665 67 32 conditions condition NOUN iajs-3665 67 33 𝛷𝑗(Ο› 𝛷𝑗(Ο› NOUN iajs-3665 67 34 ) ) PUNCT iajs-3665 67 35 ∈ ∈ PROPN iajs-3665 67 36 𝐢 𝐢 PROPN iajs-3665 67 37 π‘›βˆ’1(βˆ’βˆž,∞ π‘›βˆ’1(βˆ’βˆž,∞ PROPN iajs-3665 67 38 ) ) PUNCT iajs-3665 67 39 for for ADP iajs-3665 67 40 𝑗 𝑗 NOUN iajs-3665 67 41 = = SYM iajs-3665 67 42 1,2 1,2 NUM iajs-3665 67 43 , , PUNCT iajs-3665 67 44 … … PUNCT iajs-3665 67 45 , , PUNCT iajs-3665 67 46 𝑛. 𝑛. NOUN iajs-3665 67 47 firstly firstly ADV iajs-3665 67 48 , , PUNCT iajs-3665 67 49 we we PRON iajs-3665 67 50 partition partition VERB iajs-3665 67 51 the the DET iajs-3665 67 52 unit unit NOUN iajs-3665 67 53 interval interval NOUN iajs-3665 67 54 [ [ X iajs-3665 67 55 0,1 0,1 NUM iajs-3665 67 56 ] ] PUNCT iajs-3665 67 57 by by ADP iajs-3665 67 58 selecting select VERB iajs-3665 67 59 a a DET iajs-3665 67 60 positive positive ADJ iajs-3665 67 61 integer integer NOUN iajs-3665 67 62 π‘š π‘š PROPN iajs-3665 67 63 with with ADP iajs-3665 67 64 the the DET iajs-3665 67 65 norm norm NOUN iajs-3665 67 66 of of ADP iajs-3665 67 67 the the DET iajs-3665 67 68 partition partition NOUN iajs-3665 67 69 Δ§ Δ§ NOUN iajs-3665 67 70 = = NOUN iajs-3665 67 71 1 1 NUM iajs-3665 67 72 π‘š+1 π‘š+1 NUM iajs-3665 67 73 . . PUNCT iajs-3665 68 1 the the DET iajs-3665 68 2 mesh mesh NOUN iajs-3665 68 3 nodes nod VERB iajs-3665 68 4 π‘₯π‘˜ π‘₯π‘˜ NOUN iajs-3665 69 1 = = NOUN iajs-3665 69 2 π‘˜β„Ž π‘˜β„Ž PROPN iajs-3665 69 3 , , PUNCT iajs-3665 69 4 for for ADP iajs-3665 69 5 π‘˜ π‘˜ PRON iajs-3665 69 6 = = SYM iajs-3665 69 7 0,1 0,1 NUM iajs-3665 69 8 , , PUNCT iajs-3665 69 9 … … PUNCT iajs-3665 69 10 , , PUNCT iajs-3665 69 11 𝑛 𝑛 PRON iajs-3665 69 12 + + NOUN iajs-3665 69 13 1 1 X iajs-3665 69 14 . . PUNCT iajs-3665 69 15 then then ADV iajs-3665 69 16 , , PUNCT iajs-3665 69 17 the the DET iajs-3665 69 18 basic basic ADJ iajs-3665 69 19 functions function NOUN iajs-3665 69 20 { { PUNCT iajs-3665 69 21 π›·π‘˜(Ο›)}π‘˜=0 π›·π‘˜(Ο›)}π‘˜=0 X iajs-3665 69 22 𝑛+1 𝑛+1 NOUN iajs-3665 69 23 on on ADP iajs-3665 69 24 the the DET iajs-3665 69 25 interval interval NOUN iajs-3665 69 26 𝐼 𝐼 ADP iajs-3665 69 27 = = PUNCT iajs-3665 70 1 [ [ X iajs-3665 70 2 0,1 0,1 NUM iajs-3665 70 3 ] ] PUNCT iajs-3665 70 4 are be AUX iajs-3665 70 5 given give VERB iajs-3665 70 6 : : PUNCT iajs-3665 70 7 2.3.1 2.3.1 NUM iajs-3665 70 8 cubic cubic ADJ iajs-3665 70 9 b b PROPN iajs-3665 70 10 - - PUNCT iajs-3665 70 11 spline spline NOUN iajs-3665 70 12 base base NOUN iajs-3665 70 13 ( ( PUNCT iajs-3665 70 14 20 20 NUM iajs-3665 70 15 ) ) PUNCT iajs-3665 70 16 some some DET iajs-3665 70 17 researchers researcher NOUN iajs-3665 70 18 used use VERB iajs-3665 70 19 the the DET iajs-3665 70 20 b b NOUN iajs-3665 70 21 - - PUNCT iajs-3665 70 22 cubic cubic ADJ iajs-3665 70 23 spline spline NOUN iajs-3665 70 24 base base NOUN iajs-3665 70 25 , , PUNCT iajs-3665 70 26 which which PRON iajs-3665 70 27 is be AUX iajs-3665 70 28 defined define VERB iajs-3665 70 29 as as ADP iajs-3665 70 30 follows follow VERB iajs-3665 70 31 : : PUNCT iajs-3665 70 32 ihjpas ihjpas PROPN iajs-3665 70 33 . . PUNCT iajs-3665 71 1 2025 2025 NUM iajs-3665 71 2 , , PUNCT iajs-3665 71 3 38(2 38(2 NUM iajs-3665 71 4 ) ) PUNCT iajs-3665 71 5 341 341 NUM iajs-3665 71 6 𝑠(Ο› 𝑠(Ο› NUM iajs-3665 71 7 ) ) PUNCT iajs-3665 71 8 = = SYM iajs-3665 71 9 1 1 NUM iajs-3665 71 10 4 4 NUM iajs-3665 71 11 { { PUNCT iajs-3665 71 12 0 0 NUM iajs-3665 71 13 , , PUNCT iajs-3665 71 14 Ο› Ο› X iajs-3665 71 15 < < X iajs-3665 71 16 βˆ’2 βˆ’2 X iajs-3665 71 17 ( ( PUNCT iajs-3665 71 18 2 2 NUM iajs-3665 71 19 + + CCONJ iajs-3665 71 20 Ο›)3 Ο›)3 PROPN iajs-3665 71 21 , , PUNCT iajs-3665 71 22 βˆ’ βˆ’ PROPN iajs-3665 71 23 2 2 NUM iajs-3665 71 24 ≀ ≀ NOUN iajs-3665 71 25 Ο› Ο› PROPN iajs-3665 71 26 ≀ ≀ NUM iajs-3665 71 27 βˆ’1 βˆ’1 NOUN iajs-3665 71 28 ( ( PUNCT iajs-3665 71 29 2 2 NUM iajs-3665 72 1 + + CCONJ iajs-3665 72 2 Ο›)3 Ο›)3 PROPN iajs-3665 72 3 βˆ’ βˆ’ PROPN iajs-3665 72 4 4(1 4(1 NUM iajs-3665 73 1 + + CCONJ iajs-3665 73 2 Ο›)3 Ο›)3 PROPN iajs-3665 73 3 , , PUNCT iajs-3665 73 4 βˆ’ βˆ’ PROPN iajs-3665 73 5 1 1 NUM iajs-3665 73 6 < < X iajs-3665 73 7 Ο› Ο› PROPN iajs-3665 73 8 ≀ ≀ ADV iajs-3665 73 9 0 0 NUM iajs-3665 74 1 ( ( PUNCT iajs-3665 74 2 2 2 NUM iajs-3665 74 3 βˆ’ βˆ’ NOUN iajs-3665 74 4 Ο›)3 Ο›)3 PROPN iajs-3665 74 5 βˆ’ βˆ’ PROPN iajs-3665 74 6 4(1 4(1 NOUN iajs-3665 74 7 βˆ’ βˆ’ ADP iajs-3665 75 1 Ο›)3 Ο›)3 PROPN iajs-3665 75 2 , , PUNCT iajs-3665 75 3 0 0 PUNCT iajs-3665 75 4 < < X iajs-3665 75 5 Ο› Ο› PROPN iajs-3665 75 6 ≀ ≀ ADV iajs-3665 75 7 1 1 NUM iajs-3665 75 8 ( ( PUNCT iajs-3665 75 9 2 2 NUM iajs-3665 75 10 βˆ’ βˆ’ NOUN iajs-3665 75 11 Ο›)3 Ο›)3 NOUN iajs-3665 75 12 , , PUNCT iajs-3665 75 13 1 1 NUM iajs-3665 75 14 < < X iajs-3665 75 15 Ο› Ο› PROPN iajs-3665 75 16 ≀ ≀ ADV iajs-3665 75 17 2 2 NUM iajs-3665 75 18 0 0 NUM iajs-3665 75 19 , , PUNCT iajs-3665 75 20 π‘₯ π‘₯ PROPN iajs-3665 75 21 > > X iajs-3665 75 22 2 2 NUM iajs-3665 75 23 ( ( PUNCT iajs-3665 75 24 8) 8) NUM iajs-3665 75 25 where where SCONJ iajs-3665 75 26 𝑠(Ο› 𝑠(Ο› NUM iajs-3665 75 27 ) ) PUNCT iajs-3665 75 28 ∈ ∈ PROPN iajs-3665 75 29 𝐢0 𝐢0 NOUN iajs-3665 75 30 2 2 NUM iajs-3665 75 31 ( ( PUNCT iajs-3665 75 32 βˆ’βˆž,∞ βˆ’βˆž,∞ NOUN iajs-3665 75 33 ) ) PUNCT iajs-3665 75 34 . . PUNCT iajs-3665 76 1 to to PART iajs-3665 76 2 construct construct VERB iajs-3665 76 3 a a DET iajs-3665 76 4 cubic cubic ADJ iajs-3665 76 5 spline spline NOUN iajs-3665 76 6 , , PUNCT iajs-3665 76 7 start start VERB iajs-3665 76 8 with with ADP iajs-3665 76 9 a a DET iajs-3665 76 10 base base NOUN iajs-3665 76 11 that that PRON iajs-3665 76 12 satisfies satisfy VERB iajs-3665 76 13 the the DET iajs-3665 76 14 bcs bcs NOUN iajs-3665 76 15 π›·π‘˜(0 π›·π‘˜(0 NOUN iajs-3665 76 16 ) ) PUNCT iajs-3665 76 17 = = SYM iajs-3665 76 18 π›·π‘˜(1 π›·π‘˜(1 NOUN iajs-3665 76 19 ) ) PUNCT iajs-3665 76 20 for for ADP iajs-3665 76 21 π‘˜ π‘˜ PROPN iajs-3665 76 22 = = SYM iajs-3665 76 23 𝑛 𝑛 PROPN iajs-3665 76 24 , , PUNCT iajs-3665 76 25 𝑛 𝑛 PRON iajs-3665 76 26 βˆ’ βˆ’ PROPN iajs-3665 76 27 1 1 NUM iajs-3665 76 28 , , PUNCT iajs-3665 76 29 . . PUNCT iajs-3665 77 1 . . PUNCT iajs-3665 78 1 , , PUNCT iajs-3665 78 2 1 1 NUM iajs-3665 78 3 however however ADV iajs-3665 78 4 , , PUNCT iajs-3665 78 5 the the DET iajs-3665 78 6 component component NOUN iajs-3665 78 7 cubic cubic PROPN iajs-3665 78 8 spline spline NOUN iajs-3665 78 9 functions function NOUN iajs-3665 78 10 illustrated illustrate VERB iajs-3665 78 11 following follow VERB iajs-3665 78 12 have have AUX iajs-3665 78 13 been be AUX iajs-3665 78 14 created create VERB iajs-3665 78 15 : : PUNCT iajs-3665 78 16 𝛷𝑖(Ο› 𝛷𝑖(Ο› X iajs-3665 78 17 ) ) PUNCT iajs-3665 78 18 = = PRON iajs-3665 78 19 { { PUNCT iajs-3665 78 20 ( ( PUNCT iajs-3665 78 21 Ο› Ο› X iajs-3665 78 22 Δ§ Δ§ NOUN iajs-3665 78 23 ) ) PUNCT iajs-3665 79 1 βˆ’ βˆ’ PROPN iajs-3665 79 2 4𝑆 4𝑆 NOUN iajs-3665 79 3 ( ( PUNCT iajs-3665 79 4 Ο› Ο› PROPN iajs-3665 79 5 + + CCONJ iajs-3665 79 6 Δ§ Δ§ PROPN iajs-3665 79 7 Δ§ Δ§ NOUN iajs-3665 79 8 ) ) PUNCT iajs-3665 79 9 , , PUNCT iajs-3665 79 10 𝑖 𝑖 PUNCT iajs-3665 79 11 = = SYM iajs-3665 79 12 0 0 PUNCT iajs-3665 80 1 ( ( PUNCT iajs-3665 80 2 Ο› Ο› NOUN iajs-3665 80 3 βˆ’ βˆ’ PROPN iajs-3665 80 4 Δ§ Δ§ PROPN iajs-3665 80 5 Δ§ Δ§ NOUN iajs-3665 80 6 ) ) PUNCT iajs-3665 80 7 βˆ’ βˆ’ PROPN iajs-3665 80 8 𝑆 𝑆 PROPN iajs-3665 80 9 ( ( PUNCT iajs-3665 80 10 Ο› Ο› PROPN iajs-3665 80 11 + + CCONJ iajs-3665 80 12 Δ§ Δ§ PROPN iajs-3665 80 13 Δ§ Δ§ NOUN iajs-3665 80 14 ) ) PUNCT iajs-3665 80 15 , , PUNCT iajs-3665 80 16 𝑖 𝑖 SYM iajs-3665 80 17 = = SYM iajs-3665 80 18 1 1 NUM iajs-3665 80 19 ( ( PUNCT iajs-3665 80 20 Ο› Ο› NOUN iajs-3665 80 21 βˆ’ βˆ’ NOUN iajs-3665 81 1 𝑖ħ 𝑖ħ INTJ iajs-3665 82 1 Δ§ Δ§ NOUN iajs-3665 82 2 ) ) PUNCT iajs-3665 82 3 , , PUNCT iajs-3665 82 4 2 2 NUM iajs-3665 82 5 ≀ ≀ NOUN iajs-3665 82 6 𝑖 𝑖 SYM iajs-3665 82 7 ≀ ≀ NUM iajs-3665 82 8 π‘š π‘š X iajs-3665 82 9 ( ( PUNCT iajs-3665 82 10 Ο› Ο› NOUN iajs-3665 82 11 βˆ’ βˆ’ PROPN iajs-3665 82 12 π‘šΔ§ π‘šΔ§ NOUN iajs-3665 82 13 Δ§ Δ§ NOUN iajs-3665 82 14 ) ) PUNCT iajs-3665 82 15 βˆ’ βˆ’ PROPN iajs-3665 82 16 𝑆 𝑆 PROPN iajs-3665 82 17 ( ( PUNCT iajs-3665 82 18 Ο›(π‘š Ο›(π‘š PROPN iajs-3665 82 19 + + CCONJ iajs-3665 82 20 2)Δ§ 2)Δ§ NUM iajs-3665 82 21 Δ§ Δ§ NOUN iajs-3665 82 22 ) ) PUNCT iajs-3665 82 23 , , PUNCT iajs-3665 82 24 𝑖 𝑖 X iajs-3665 82 25 = = SYM iajs-3665 82 26 π‘š π‘š X iajs-3665 82 27 ( ( PUNCT iajs-3665 82 28 Ο› Ο› X iajs-3665 82 29 βˆ’ βˆ’ PROPN iajs-3665 82 30 ( ( PUNCT iajs-3665 82 31 1 1 NUM iajs-3665 82 32 + + NUM iajs-3665 82 33 π‘š)Δ§ π‘š)Δ§ X iajs-3665 82 34 Δ§ Δ§ NOUN iajs-3665 82 35 ) ) PUNCT iajs-3665 82 36 βˆ’ βˆ’ PROPN iajs-3665 82 37 4𝑆 4𝑆 NOUN iajs-3665 82 38 ( ( PUNCT iajs-3665 82 39 Ο› Ο› PROPN iajs-3665 82 40 βˆ’ βˆ’ PROPN iajs-3665 82 41 ( ( PUNCT iajs-3665 82 42 2 2 NUM iajs-3665 82 43 + + NUM iajs-3665 82 44 π‘š)Δ§ π‘š)Δ§ NOUN iajs-3665 82 45 Δ§ Δ§ NOUN iajs-3665 82 46 ) ) PUNCT iajs-3665 82 47 , , PUNCT iajs-3665 82 48 𝑖 𝑖 SYM iajs-3665 82 49 = = SYM iajs-3665 82 50 1 1 NUM iajs-3665 82 51 + + NUM iajs-3665 82 52 π‘š π‘š X iajs-3665 82 53 ( ( PUNCT iajs-3665 82 54 9 9 NUM iajs-3665 82 55 ) ) PUNCT iajs-3665 82 56 in in ADP iajs-3665 82 57 the the DET iajs-3665 82 58 following follow VERB iajs-3665 82 59 figure figure NOUN iajs-3665 82 60 1 1 NUM iajs-3665 82 61 represents represent VERB iajs-3665 82 62 the the DET iajs-3665 82 63 cubic cubic ADJ iajs-3665 82 64 spline spline NOUN iajs-3665 82 65 functions function NOUN iajs-3665 82 66 . . PUNCT iajs-3665 83 1 table table NOUN iajs-3665 83 2 1 1 NUM iajs-3665 83 3 . . PUNCT iajs-3665 84 1 values value NOUN iajs-3665 84 2 at at ADP iajs-3665 84 3 node node NOUN iajs-3665 84 4 points point NOUN iajs-3665 84 5 cubic cubic ADJ iajs-3665 84 6 b b PROPN iajs-3665 84 7 - - PUNCT iajs-3665 84 8 spline spline NOUN iajs-3665 84 9 π‘₯π‘˜ π‘₯π‘˜ NOUN iajs-3665 84 10 π›·π‘˜(π‘₯π‘˜ π›·π‘˜(π‘₯π‘˜ ADV iajs-3665 84 11 ) ) PUNCT iajs-3665 85 1 π›·π‘˜ π›·π‘˜ ADP iajs-3665 85 2 ꞌ ꞌ PRON iajs-3665 85 3 ( ( PUNCT iajs-3665 85 4 π‘₯π‘˜ π‘₯π‘˜ NOUN iajs-3665 85 5 ) ) PUNCT iajs-3665 85 6 π›·π‘˜ π›·π‘˜ NOUN iajs-3665 85 7 ꞌꞌ ꞌꞌ ADP iajs-3665 85 8 ( ( PUNCT iajs-3665 85 9 π‘₯π‘˜ π‘₯π‘˜ NOUN iajs-3665 85 10 ) ) PUNCT iajs-3665 85 11 π‘₯π‘˜βˆ’2 π‘₯π‘˜βˆ’2 NOUN iajs-3665 85 12 0 0 NUM iajs-3665 85 13 0 0 NUM iajs-3665 85 14 0 0 NUM iajs-3665 85 15 π‘₯π‘˜βˆ’1 π‘₯π‘˜βˆ’1 PROPN iajs-3665 85 16 1 1 NUM iajs-3665 85 17 4 4 NUM iajs-3665 85 18 3 3 NUM iajs-3665 85 19 4 4 NUM iajs-3665 85 20 βˆ’ βˆ’ NOUN iajs-3665 85 21 3 3 NUM iajs-3665 85 22 2 2 NUM iajs-3665 85 23 π‘₯π‘˜ π‘₯π‘˜ NOUN iajs-3665 85 24 1 1 NUM iajs-3665 85 25 0 0 NUM iajs-3665 85 26 βˆ’ βˆ’ NOUN iajs-3665 85 27 3 3 NUM iajs-3665 85 28 4 4 NUM iajs-3665 85 29 π‘₯π‘˜+1 π‘₯π‘˜+1 NUM iajs-3665 85 30 1 1 NUM iajs-3665 85 31 4 4 NUM iajs-3665 85 32 βˆ’ βˆ’ NOUN iajs-3665 85 33 3 3 NUM iajs-3665 85 34 4 4 NUM iajs-3665 85 35 βˆ’ βˆ’ NOUN iajs-3665 85 36 3 3 NUM iajs-3665 85 37 2 2 NUM iajs-3665 85 38 π‘₯π‘˜+2 π‘₯π‘˜+2 NUM iajs-3665 85 39 0 0 NUM iajs-3665 85 40 0 0 NUM iajs-3665 85 41 0 0 NUM iajs-3665 85 42 2.3.2 2.3.2 NUM iajs-3665 85 43 fractional fractional ADJ iajs-3665 85 44 b b X iajs-3665 85 45 - - PUNCT iajs-3665 85 46 cubic cubic ADJ iajs-3665 85 47 spline spline NOUN iajs-3665 85 48 in in ADP iajs-3665 85 49 this this DET iajs-3665 85 50 subsection subsection NOUN iajs-3665 85 51 , , PUNCT iajs-3665 85 52 the the DET iajs-3665 85 53 derivative derivative NOUN iajs-3665 85 54 of of ADP iajs-3665 85 55 𝛼order 𝛼order NOUN iajs-3665 85 56 fractional fractional NOUN iajs-3665 85 57 of of ADP iajs-3665 85 58 a a DET iajs-3665 85 59 cubic cubic ADJ iajs-3665 85 60 b b NOUN iajs-3665 85 61 - - PUNCT iajs-3665 85 62 spline spline NOUN iajs-3665 85 63 is be AUX iajs-3665 85 64 calculated calculate VERB iajs-3665 85 65 as as SCONJ iajs-3665 85 66 follows follow VERB iajs-3665 85 67 : : PUNCT iajs-3665 85 68 𝐽π‘₯ 𝐽π‘₯ PROPN iajs-3665 85 69 ∝(π‘₯ ∝(π‘₯ ADJ iajs-3665 85 70 ) ) PUNCT iajs-3665 86 1 = = SYM iajs-3665 86 2 1 1 NUM iajs-3665 86 3 Ξ³ Ξ³ X iajs-3665 86 4 ( ( PUNCT iajs-3665 86 5 ∝ ∝ PROPN iajs-3665 86 6 +4 +4 PROPN iajs-3665 86 7 ) ) PUNCT iajs-3665 86 8 { { PUNCT iajs-3665 86 9 0 0 NUM iajs-3665 86 10 , , PUNCT iajs-3665 86 11 Ο› Ο› PRON iajs-3665 86 12 < < X iajs-3665 86 13 βˆ’2 βˆ’2 PROPN iajs-3665 86 14 3 3 NUM iajs-3665 86 15 2 2 NUM iajs-3665 86 16 π‘₯∝+3 π‘₯∝+3 NOUN iajs-3665 86 17 , , PUNCT iajs-3665 86 18 βˆ’ βˆ’ PROPN iajs-3665 86 19 2 2 NUM iajs-3665 86 20 = = NOUN iajs-3665 86 21 < < X iajs-3665 86 22 π‘₯ π‘₯ X iajs-3665 86 23 < < X iajs-3665 86 24 = = X iajs-3665 86 25 βˆ’1 βˆ’1 NOUN iajs-3665 86 26 3 3 NUM iajs-3665 86 27 2 2 NUM iajs-3665 86 28 π‘₯∝+3 π‘₯∝+3 NOUN iajs-3665 86 29 βˆ’ βˆ’ PROPN iajs-3665 86 30 6π‘₯1 6π‘₯1 NUM iajs-3665 86 31 ∝+3 ∝+3 NOUN iajs-3665 86 32 , , PUNCT iajs-3665 86 33 βˆ’ βˆ’ PROPN iajs-3665 86 34 1 1 NUM iajs-3665 86 35 < < X iajs-3665 86 36 π‘₯ π‘₯ X iajs-3665 86 37 < < X iajs-3665 86 38 = = SYM iajs-3665 86 39 0 0 NUM iajs-3665 86 40 3 3 NUM iajs-3665 86 41 2 2 NUM iajs-3665 86 42 π‘₯∝+3 π‘₯∝+3 NOUN iajs-3665 86 43 βˆ’ βˆ’ PROPN iajs-3665 86 44 6π‘₯1 6π‘₯1 NUM iajs-3665 86 45 ∝+3 ∝+3 NOUN iajs-3665 86 46 + + CCONJ iajs-3665 86 47 9π‘₯2 9π‘₯2 NUM iajs-3665 86 48 ∝+3 ∝+3 NOUN iajs-3665 86 49 ; ; PUNCT iajs-3665 86 50 0 0 NUM iajs-3665 86 51 < < X iajs-3665 86 52 π‘₯ π‘₯ X iajs-3665 86 53 < < X iajs-3665 86 54 1 1 NUM iajs-3665 86 55 3 3 NUM iajs-3665 86 56 2 2 NUM iajs-3665 86 57 π‘₯∝+3 π‘₯∝+3 NOUN iajs-3665 86 58 βˆ’ βˆ’ PROPN iajs-3665 86 59 6π‘₯1 6π‘₯1 NUM iajs-3665 86 60 ∝+3 ∝+3 NOUN iajs-3665 86 61 + + CCONJ iajs-3665 86 62 9π‘₯2 9π‘₯2 NUM iajs-3665 86 63 ∝+3 ∝+3 NOUN iajs-3665 86 64 βˆ’ βˆ’ NOUN iajs-3665 86 65 6π‘₯3 6π‘₯3 NUM iajs-3665 86 66 ∝+3 ∝+3 NOUN iajs-3665 86 67 ; ; PUNCT iajs-3665 86 68 1 1 NUM iajs-3665 86 69 < < X iajs-3665 86 70 π‘₯ π‘₯ X iajs-3665 86 71 < < X iajs-3665 86 72 = = X iajs-3665 86 73 2 2 NUM iajs-3665 86 74 0 0 NUM iajs-3665 86 75 , , PUNCT iajs-3665 86 76 π‘₯ π‘₯ PROPN iajs-3665 86 77 > > X iajs-3665 86 78 2 2 NUM iajs-3665 86 79 . . PUNCT iajs-3665 87 1 ( ( PUNCT iajs-3665 87 2 10 10 NUM iajs-3665 87 3 ) ) PUNCT iajs-3665 87 4 ihjpas ihjpa NOUN iajs-3665 87 5 . . PUNCT iajs-3665 88 1 2025 2025 NUM iajs-3665 88 2 , , PUNCT iajs-3665 88 3 38(2 38(2 NUM iajs-3665 88 4 ) ) PUNCT iajs-3665 88 5 342 342 NUM iajs-3665 88 6 where where SCONJ iajs-3665 88 7 π‘₯1 π‘₯1 NOUN iajs-3665 88 8 = = PROPN iajs-3665 88 9 βˆ’1 βˆ’1 NOUN iajs-3665 88 10 + + CCONJ iajs-3665 88 11 π‘₯ π‘₯ NOUN iajs-3665 88 12 , , PUNCT iajs-3665 88 13 π‘₯2 π‘₯2 NOUN iajs-3665 88 14 = = SYM iajs-3665 88 15 βˆ’2 βˆ’2 NOUN iajs-3665 88 16 + + CCONJ iajs-3665 88 17 π‘₯ π‘₯ NOUN iajs-3665 88 18 and and CCONJ iajs-3665 88 19 π‘₯3 π‘₯3 NOUN iajs-3665 88 20 = = PUNCT iajs-3665 88 21 βˆ’3 βˆ’3 PROPN iajs-3665 89 1 + + CCONJ iajs-3665 89 2 π‘₯ π‘₯ PROPN iajs-3665 89 3 i i PRON iajs-3665 89 4 ii ii PROPN iajs-3665 89 5 figure figure NOUN iajs-3665 89 6 1 1 NUM iajs-3665 89 7 . . PUNCT iajs-3665 90 1 ( ( PUNCT iajs-3665 90 2 i i NOUN iajs-3665 90 3 ) ) PUNCT iajs-3665 90 4 b b X iajs-3665 90 5 - - PUNCT iajs-3665 90 6 cubic cubic ADJ iajs-3665 90 7 spline spline NOUN iajs-3665 90 8 and and CCONJ iajs-3665 90 9 ( ( PUNCT iajs-3665 90 10 ii ii NOUN iajs-3665 90 11 ) ) PUNCT iajs-3665 90 12 compound compound NOUN iajs-3665 90 13 b b X iajs-3665 90 14 - - PUNCT iajs-3665 90 15 cubic cubic ADJ iajs-3665 90 16 functions function NOUN iajs-3665 90 17 3 3 NUM iajs-3665 90 18 . . PUNCT iajs-3665 90 19 materials material NOUN iajs-3665 90 20 and and CCONJ iajs-3665 90 21 methods method NOUN iajs-3665 90 22 3.1 3.1 NUM iajs-3665 90 23 . . PUNCT iajs-3665 91 1 analysis analysis NOUN iajs-3665 91 2 of of ADP iajs-3665 91 3 the the DET iajs-3665 91 4 numerical numerical ADJ iajs-3665 91 5 collocation collocation NOUN iajs-3665 91 6 method method NOUN iajs-3665 91 7 for for ADP iajs-3665 91 8 solving solve VERB iajs-3665 91 9 second second ADJ iajs-3665 91 10 - - PUNCT iajs-3665 91 11 order order NOUN iajs-3665 91 12 quasi quasi ADJ iajs-3665 91 13 - - ADJ iajs-3665 91 14 linear linear ADJ iajs-3665 91 15 fodes fode NOUN iajs-3665 91 16 using use VERB iajs-3665 91 17 b b NOUN iajs-3665 91 18 - - PUNCT iajs-3665 91 19 cubic cubic ADJ iajs-3665 91 20 spline spline NOUN iajs-3665 91 21 base base NOUN iajs-3665 91 22 collocation collocation NOUN iajs-3665 91 23 points point NOUN iajs-3665 91 24 should should AUX iajs-3665 91 25 be be AUX iajs-3665 91 26 defined define VERB iajs-3665 91 27 π‘₯𝑗 π‘₯𝑗 NOUN iajs-3665 91 28 = = PROPN iajs-3665 91 29 π‘Ž π‘Ž PROPN iajs-3665 91 30 + + NUM iajs-3665 91 31 π‘—β„Ž π‘—β„Ž NOUN iajs-3665 91 32 for for ADP iajs-3665 91 33 𝑗 𝑗 NOUN iajs-3665 91 34 = = SYM iajs-3665 91 35 0,1 0,1 NUM iajs-3665 91 36 , , PUNCT iajs-3665 91 37 … … PUNCT iajs-3665 91 38 , , PUNCT iajs-3665 91 39 π‘š π‘š PART iajs-3665 91 40 discretize discretize VERB iajs-3665 91 41 the the DET iajs-3665 91 42 functions function NOUN iajs-3665 91 43 𝛷(𝜁 𝛷(𝜁 NUM iajs-3665 91 44 ) ) PUNCT iajs-3665 91 45 = = PRON iajs-3665 91 46 { { PUNCT iajs-3665 91 47 𝛷0(𝜁),𝛷1(𝜁 𝛷0(𝜁),𝛷1(𝜁 NOUN iajs-3665 91 48 ) ) PUNCT iajs-3665 91 49 , , PUNCT iajs-3665 91 50 , , PUNCT iajs-3665 91 51 . . PUNCT iajs-3665 91 52 . . PUNCT iajs-3665 92 1 , , PUNCT iajs-3665 92 2 π›·π‘š(𝜁 π›·π‘š(𝜁 PROPN iajs-3665 92 3 ) ) PUNCT iajs-3665 92 4 } } PUNCT iajs-3665 92 5 . . PUNCT iajs-3665 93 1 suppose suppose VERB iajs-3665 93 2 𝑦(𝜁 𝑦(𝜁 ADJ iajs-3665 93 3 ) ) PUNCT iajs-3665 93 4 = = SYM iajs-3665 93 5 βˆ‘π‘π‘–π›·π‘–(𝜁 βˆ‘π‘π‘–π›·π‘–(𝜁 NOUN iajs-3665 93 6 ) ) PUNCT iajs-3665 93 7 π‘š π‘š ADP iajs-3665 93 8 𝑖=0 𝑖=0 PROPN iajs-3665 93 9 . . PUNCT iajs-3665 94 1 ( ( PUNCT iajs-3665 94 2 11 11 NUM iajs-3665 94 3 ) ) PUNCT iajs-3665 94 4 when when SCONJ iajs-3665 94 5 the the DET iajs-3665 94 6 approximation approximation NOUN iajs-3665 94 7 of of ADP iajs-3665 94 8 the the DET iajs-3665 94 9 function function NOUN iajs-3665 94 10 𝑦(π‘₯ 𝑦(π‘₯ NOUN iajs-3665 94 11 ) ) PUNCT iajs-3665 94 12 at at ADP iajs-3665 94 13 the the DET iajs-3665 94 14 point point NOUN iajs-3665 94 15 π‘₯𝑗 π‘₯𝑗 NOUN iajs-3665 94 16 in in ADP iajs-3665 94 17 the the DET iajs-3665 94 18 domain domain NOUN iajs-3665 94 19 of of ADP iajs-3665 94 20 the the DET iajs-3665 94 21 ode ode PROPN iajs-3665 94 22 , , PUNCT iajs-3665 94 23 then then ADV iajs-3665 94 24 , , PUNCT iajs-3665 94 25 we we PRON iajs-3665 94 26 get get VERB iajs-3665 94 27 the the DET iajs-3665 94 28 matrix matrix NOUN iajs-3665 94 29 of of ADP iajs-3665 94 30 coefficient coefficient NOUN iajs-3665 94 31 as as ADP iajs-3665 94 32 Ο†i Ο†i ADV iajs-3665 94 33 , , PUNCT iajs-3665 94 34 j j PROPN iajs-3665 94 35 = = PROPN iajs-3665 94 36 Ο†i(xj Ο†i(xj PROPN iajs-3665 94 37 ) ) PUNCT iajs-3665 94 38 and and CCONJ iajs-3665 94 39 Ο†i Ο†i ADP iajs-3665 94 40 , , PUNCT iajs-3665 94 41 j j PROPN iajs-3665 94 42 ꞌ ꞌ PROPN iajs-3665 94 43 = = PUNCT iajs-3665 94 44 Ο†i Ο†i ADP iajs-3665 94 45 ꞌ(xj ꞌ(xj PROPN iajs-3665 94 46 ) ) PUNCT iajs-3665 94 47 with with ADP iajs-3665 94 48 dimensions dimension NOUN iajs-3665 94 49 π‘š π‘š ADP iajs-3665 94 50 Γ—π‘š. Γ—π‘š. VERB iajs-3665 94 51 the the DET iajs-3665 94 52 function function NOUN iajs-3665 94 53 y(x y(x NOUN iajs-3665 94 54 ) ) PUNCT iajs-3665 94 55 suppose suppose VERB iajs-3665 94 56 to to PART iajs-3665 94 57 be be AUX iajs-3665 94 58 integrable integrable ADJ iajs-3665 94 59 in in ADP iajs-3665 94 60 the the DET iajs-3665 94 61 domain domain NOUN iajs-3665 94 62 of of ADP iajs-3665 94 63 interval interval NOUN iajs-3665 94 64 ( ( PUNCT iajs-3665 94 65 0,1 0,1 NOUN iajs-3665 94 66 ) ) PUNCT iajs-3665 94 67 which which PRON iajs-3665 94 68 it it PRON iajs-3665 94 69 be be AUX iajs-3665 94 70 represented represent VERB iajs-3665 94 71 as as ADP iajs-3665 94 72 a a DET iajs-3665 94 73 finite finite ADJ iajs-3665 94 74 sum sum NOUN iajs-3665 94 75 of of ADP iajs-3665 94 76 the the DET iajs-3665 94 77 b b PROPN iajs-3665 94 78 - - PUNCT iajs-3665 94 79 spline spline NOUN iajs-3665 94 80 base base NOUN iajs-3665 94 81 . . PUNCT iajs-3665 95 1 the the DET iajs-3665 95 2 previous previous ADJ iajs-3665 95 3 series series NOUN iajs-3665 95 4 terminates terminate VERB iajs-3665 95 5 in in ADP iajs-3665 95 6 finite finite ADJ iajs-3665 95 7 terms term NOUN iajs-3665 95 8 if if SCONJ iajs-3665 95 9 y(x y(x NOUN iajs-3665 95 10 ) ) PUNCT iajs-3665 95 11 is be AUX iajs-3665 95 12 a a DET iajs-3665 95 13 piecewise piecewise NOUN iajs-3665 95 14 constant constant ADJ iajs-3665 95 15 or or CCONJ iajs-3665 95 16 can can AUX iajs-3665 95 17 be be AUX iajs-3665 95 18 approximated approximate VERB iajs-3665 95 19 as as ADP iajs-3665 95 20 a a DET iajs-3665 95 21 piecewise piecewise NOUN iajs-3665 95 22 constant constant ADJ iajs-3665 95 23 across across ADP iajs-3665 95 24 each each DET iajs-3665 95 25 subinterval subinterval NOUN iajs-3665 95 26 . . PUNCT iajs-3665 96 1 consider consider VERB iajs-3665 96 2 the the DET iajs-3665 96 3 general general ADJ iajs-3665 96 4 quasi quasi ADJ iajs-3665 96 5 - - ADJ iajs-3665 96 6 linear linear ADJ iajs-3665 96 7 second second ADJ iajs-3665 96 8 - - PUNCT iajs-3665 96 9 order order NOUN iajs-3665 96 10 fode fode NOUN iajs-3665 96 11 in in ADP iajs-3665 96 12 equation equation NOUN iajs-3665 96 13 ( ( PUNCT iajs-3665 96 14 4 4 NUM iajs-3665 96 15 ) ) PUNCT iajs-3665 96 16 with with ADP iajs-3665 96 17 bcs bc NOUN iajs-3665 96 18 in in ADP iajs-3665 96 19 equation equation NOUN iajs-3665 96 20 ( ( PUNCT iajs-3665 96 21 5 5 NUM iajs-3665 96 22 ) ) PUNCT iajs-3665 96 23 . . PUNCT iajs-3665 97 1 substitute substitute VERB iajs-3665 97 2 the the DET iajs-3665 97 3 approximation approximation NOUN iajs-3665 97 4 in in ADP iajs-3665 97 5 equation equation NOUN iajs-3665 97 6 ( ( PUNCT iajs-3665 97 7 11 11 NUM iajs-3665 97 8 ) ) PUNCT iajs-3665 97 9 which which PRON iajs-3665 97 10 satisfy satisfy VERB iajs-3665 97 11 the the DET iajs-3665 97 12 bcs bcs NOUN iajs-3665 97 13 in in ADP iajs-3665 97 14 equation equation NOUN iajs-3665 97 15 ( ( PUNCT iajs-3665 97 16 4 4 NUM iajs-3665 97 17 ) ) PUNCT iajs-3665 97 18 at at ADP iajs-3665 97 19 the the DET iajs-3665 97 20 point point NOUN iajs-3665 97 21 π‘₯ π‘₯ NOUN iajs-3665 97 22 = = PUNCT iajs-3665 97 23 𝑑𝑗 𝑑𝑗 PROPN iajs-3665 97 24 for for ADP iajs-3665 97 25 𝑗 𝑗 NOUN iajs-3665 97 26 = = SYM iajs-3665 97 27 0,1 0,1 NUM iajs-3665 97 28 , , PUNCT iajs-3665 97 29 … … PUNCT iajs-3665 97 30 , , PUNCT iajs-3665 97 31 𝑛. 𝑛. NOUN iajs-3665 97 32 to to PART iajs-3665 97 33 obtain obtain VERB iajs-3665 97 34 βˆ‘π›Όπ‘–π· βˆ‘π›Όπ‘–π· ADJ iajs-3665 97 35 𝛼 𝛼 NOUN iajs-3665 97 36 ( ( PUNCT iajs-3665 97 37 𝛷𝑖(πœπ‘— 𝛷𝑖(πœπ‘— NOUN iajs-3665 97 38 ) ) PUNCT iajs-3665 97 39 ) ) PUNCT iajs-3665 98 1 = = PUNCT iajs-3665 98 2 𝑛 𝑛 PRON iajs-3665 98 3 𝑖=1 𝑖=1 PROPN iajs-3665 98 4 𝛷(πœπ‘— 𝛷(πœπ‘— NOUN iajs-3665 98 5 , , PUNCT iajs-3665 98 6 βˆ‘π›Όπ‘–π›·π‘–(πœπ‘— βˆ‘π›Όπ‘–π›·π‘–(πœπ‘— NOUN iajs-3665 98 7 ) ) PUNCT iajs-3665 98 8 , , PUNCT iajs-3665 98 9 βˆ‘π›Όπ‘–π›·π‘– βˆ‘π›Όπ‘–π›·π‘– PROPN iajs-3665 98 10 ꞌ(πœπ‘— ꞌ(πœπ‘— NOUN iajs-3665 98 11 ) ) PUNCT iajs-3665 98 12 𝑛 𝑛 PRON iajs-3665 98 13 𝑖=1 𝑖=1 PROPN iajs-3665 98 14 , , PUNCT iajs-3665 98 15 𝑛 𝑛 PRON iajs-3665 98 16 𝑖=1 𝑖=1 PROPN iajs-3665 98 17 βˆ‘π›Όπ‘–π›·π‘– βˆ‘π›Όπ‘–π›·π‘– PROPN iajs-3665 98 18 ꞌꞌ(πœπ‘— ꞌꞌ(πœπ‘— NOUN iajs-3665 98 19 ) ) PUNCT iajs-3665 98 20 𝑛 𝑛 PRON iajs-3665 98 21 𝑖=1 𝑖=1 PROPN iajs-3665 98 22 ) ) PUNCT iajs-3665 98 23 ; ; PUNCT iajs-3665 98 24 0 0 NUM iajs-3665 98 25 < < X iajs-3665 98 26 𝜁 𝜁 X iajs-3665 98 27 < < X iajs-3665 98 28 1 1 NUM iajs-3665 98 29 , , PUNCT iajs-3665 98 30 0 0 NUM iajs-3665 99 1 < < X iajs-3665 99 2 ∝ ∝ X iajs-3665 99 3 < < X iajs-3665 99 4 2 2 NUM iajs-3665 99 5 , , PUNCT iajs-3665 99 6 ( ( PUNCT iajs-3665 99 7 12 12 NUM iajs-3665 99 8 ) ) PUNCT iajs-3665 99 9 however however ADV iajs-3665 99 10 , , PUNCT iajs-3665 99 11 we we PRON iajs-3665 99 12 get get VERB iajs-3665 99 13 a a DET iajs-3665 99 14 linear linear ADJ iajs-3665 99 15 system system NOUN iajs-3665 99 16 in in ADP iajs-3665 99 17 n n DET iajs-3665 99 18 algebraic algebraic ADJ iajs-3665 99 19 equations equation NOUN iajs-3665 99 20 and and CCONJ iajs-3665 99 21 the the DET iajs-3665 99 22 unknowns unknown NOUN iajs-3665 99 23 are be AUX iajs-3665 99 24 the the DET iajs-3665 99 25 coefficients coefficient NOUN iajs-3665 99 26 𝑐𝑖 𝑐𝑖 VERB iajs-3665 99 27 for for ADP iajs-3665 99 28 𝑖 𝑖 PRON iajs-3665 99 29 = = SYM iajs-3665 99 30 1,2 1,2 NUM iajs-3665 99 31 … … PUNCT iajs-3665 99 32 , , PUNCT iajs-3665 99 33 𝑛. 𝑛. NOUN iajs-3665 99 34 this this DET iajs-3665 99 35 system system NOUN iajs-3665 99 36 is be AUX iajs-3665 99 37 a a DET iajs-3665 99 38 linear linear ADJ iajs-3665 99 39 system system NOUN iajs-3665 99 40 if if SCONJ iajs-3665 99 41 the the DET iajs-3665 99 42 function function NOUN iajs-3665 99 43 𝛷 𝛷 PROPN iajs-3665 99 44 is be AUX iajs-3665 99 45 linear linear ADJ iajs-3665 99 46 . . PUNCT iajs-3665 100 1 the the DET iajs-3665 100 2 coefficient coefficient NOUN iajs-3665 100 3 matrix matrix NOUN iajs-3665 100 4 in in ADP iajs-3665 100 5 this this DET iajs-3665 100 6 system system NOUN iajs-3665 100 7 has have AUX iajs-3665 100 8 given give VERB iajs-3665 100 9 as as SCONJ iajs-3665 100 10 follows follow VERB iajs-3665 100 11 : : PUNCT iajs-3665 100 12 π΄π‘˜π‘™ π΄π‘˜π‘™ PROPN iajs-3665 100 13 = = SYM iajs-3665 100 14 𝛩 𝛩 PROPN iajs-3665 100 15 ( ( PUNCT iajs-3665 100 16 πœπ‘™ πœπ‘™ PROPN iajs-3665 100 17 βƒ—βƒ— βƒ—βƒ— PROPN iajs-3665 100 18 βƒ— βƒ— PROPN iajs-3665 100 19 , , PUNCT iajs-3665 100 20 π›·π‘˜βƒ—βƒ— π›·π‘˜βƒ—βƒ— PROPN iajs-3665 100 21 βƒ—βƒ— βƒ—βƒ— PROPN iajs-3665 100 22 βƒ—βƒ— βƒ—βƒ— PROPN iajs-3665 100 23 ) ) PUNCT iajs-3665 100 24 , , PUNCT iajs-3665 100 25 and and CCONJ iajs-3665 100 26 π‘π‘˜ π‘π‘˜ PRON iajs-3665 100 27 = = SYM iajs-3665 100 28 Ο‰π‘˜ Ο‰π‘˜ ADP iajs-3665 100 29 ( ( PUNCT iajs-3665 100 30 𝑑 𝑑 PROPN iajs-3665 100 31 ) ) PUNCT iajs-3665 100 32 , , PUNCT iajs-3665 100 33 for for ADP iajs-3665 100 34 π‘˜π‘™ π‘˜π‘™ NOUN iajs-3665 100 35 = = SYM iajs-3665 100 36 1,2 1,2 NUM iajs-3665 100 37 , , PUNCT iajs-3665 100 38 . . PUNCT iajs-3665 100 39 . . PUNCT iajs-3665 101 1 , , PUNCT iajs-3665 101 2 𝑛 𝑛 PROPN iajs-3665 101 3 , , PUNCT iajs-3665 101 4 where where SCONJ iajs-3665 101 5 𝛼 𝛼 X iajs-3665 101 6 = = SYM iajs-3665 101 7 ( ( PUNCT iajs-3665 101 8 𝛼1 𝛼1 NOUN iajs-3665 101 9 , , PUNCT iajs-3665 101 10 𝛼2 𝛼2 ADJ iajs-3665 101 11 , , PUNCT iajs-3665 101 12 … … PUNCT iajs-3665 101 13 , , PUNCT iajs-3665 101 14 𝛼𝑛 𝛼𝑛 PROPN iajs-3665 101 15 ) ) PUNCT iajs-3665 101 16 . . PUNCT iajs-3665 102 1 the the DET iajs-3665 102 2 approximated approximate VERB iajs-3665 102 3 of of ADP iajs-3665 102 4 coefficients coefficient NOUN iajs-3665 102 5 are be AUX iajs-3665 102 6 determined determine VERB iajs-3665 102 7 by by ADP iajs-3665 102 8 solving solve VERB iajs-3665 102 9 the the DET iajs-3665 102 10 above above ADJ iajs-3665 102 11 system system NOUN iajs-3665 102 12 of of ADP iajs-3665 102 13 coefficients coefficient NOUN iajs-3665 102 14 𝐴𝛼 𝐴𝛼 PROPN iajs-3665 102 15 = = SYM iajs-3665 102 16 𝑏. 𝑏. NOUN iajs-3665 102 17 ihjpas ihjpas PROPN iajs-3665 102 18 . . PUNCT iajs-3665 103 1 2025 2025 NUM iajs-3665 103 2 , , PUNCT iajs-3665 103 3 38(2 38(2 NUM iajs-3665 103 4 ) ) PUNCT iajs-3665 103 5 343 343 NUM iajs-3665 103 6 3.1.1 3.1.1 NUM iajs-3665 103 7 . . PUNCT iajs-3665 104 1 algorithm algorithm NOUN iajs-3665 104 2 of of ADP iajs-3665 104 3 the the DET iajs-3665 104 4 collocation collocation NOUN iajs-3665 104 5 method method NOUN iajs-3665 104 6 for for ADP iajs-3665 104 7 determining determine VERB iajs-3665 104 8 the the DET iajs-3665 104 9 solution solution NOUN iajs-3665 104 10 of of ADP iajs-3665 104 11 the the DET iajs-3665 104 12 boundary boundary ADJ iajs-3665 104 13 - - PUNCT iajs-3665 104 14 value value NOUN iajs-3665 104 15 issue issue NOUN iajs-3665 104 16 in in ADP iajs-3665 104 17 equations equation NOUN iajs-3665 104 18 ( ( PUNCT iajs-3665 104 19 6 6 NUM iajs-3665 104 20 ) ) PUNCT iajs-3665 104 21 with with ADP iajs-3665 104 22 the the DET iajs-3665 104 23 bcs bcs NOUN iajs-3665 104 24 in in ADP iajs-3665 104 25 equation equation NOUN iajs-3665 104 26 ( ( PUNCT iajs-3665 104 27 7 7 NUM iajs-3665 104 28 ) ) PUNCT iajs-3665 104 29 , , PUNCT iajs-3665 104 30 we we PRON iajs-3665 104 31 give give VERB iajs-3665 104 32 the the DET iajs-3665 104 33 steps step NOUN iajs-3665 104 34 of of ADP iajs-3665 104 35 the the DET iajs-3665 104 36 proposed propose VERB iajs-3665 104 37 numerical numerical ADJ iajs-3665 104 38 method method NOUN iajs-3665 104 39 as as SCONJ iajs-3665 104 40 follows follow VERB iajs-3665 104 41 : : PUNCT iajs-3665 104 42 step step NOUN iajs-3665 104 43 i i PRON iajs-3665 104 44 : : PUNCT iajs-3665 104 45 choose choose VERB iajs-3665 104 46 a a DET iajs-3665 104 47 suitable suitable ADJ iajs-3665 104 48 approximated approximated ADJ iajs-3665 104 49 base base NOUN iajs-3665 104 50 . . PUNCT iajs-3665 105 1 βˆ… βˆ… NOUN iajs-3665 105 2 = = PUNCT iajs-3665 105 3 { { PUNCT iajs-3665 105 4 βˆ…0(𝜁 βˆ…0(𝜁 NOUN iajs-3665 105 5 ) ) PUNCT iajs-3665 105 6 , , PUNCT iajs-3665 105 7 βˆ…1(𝜁 βˆ…1(𝜁 NOUN iajs-3665 105 8 ) ) PUNCT iajs-3665 105 9 , , PUNCT iajs-3665 105 10 . . PUNCT iajs-3665 105 11 . . PUNCT iajs-3665 106 1 , , PUNCT iajs-3665 106 2 βˆ…π‘š(𝜁 βˆ…π‘š(𝜁 NOUN iajs-3665 106 3 ) ) PUNCT iajs-3665 106 4 } } PUNCT iajs-3665 106 5 where where SCONJ iajs-3665 106 6 βˆ…π‘˜(0 βˆ…π‘˜(0 NOUN iajs-3665 106 7 ) ) PUNCT iajs-3665 107 1 = = SYM iajs-3665 107 2 0 0 NUM iajs-3665 108 1 for for ADP iajs-3665 108 2 π‘˜ π‘˜ NOUN iajs-3665 108 3 = = X iajs-3665 108 4 π‘š,π‘š π‘š,π‘š NOUN iajs-3665 108 5 βˆ’ βˆ’ PROPN iajs-3665 108 6 1 1 NUM iajs-3665 108 7 , , PUNCT iajs-3665 108 8 . . PUNCT iajs-3665 108 9 . . PUNCT iajs-3665 109 1 , , PUNCT iajs-3665 109 2 1 1 X iajs-3665 109 3 . . X iajs-3665 109 4 step step PROPN iajs-3665 109 5 ii ii PROPN iajs-3665 109 6 : : PUNCT iajs-3665 109 7 consider consider VERB iajs-3665 109 8 the the DET iajs-3665 109 9 approximated approximate VERB iajs-3665 109 10 solution solution NOUN iajs-3665 109 11 of of ADP iajs-3665 109 12 equation equation NOUN iajs-3665 109 13 ( ( PUNCT iajs-3665 109 14 6 6 NUM iajs-3665 109 15 ) ) PUNCT iajs-3665 109 16 with with ADP iajs-3665 109 17 the the DET iajs-3665 109 18 bcs bcs NOUN iajs-3665 109 19 in in ADP iajs-3665 109 20 equation equation NOUN iajs-3665 109 21 ( ( PUNCT iajs-3665 109 22 7 7 NUM iajs-3665 109 23 ) ) PUNCT iajs-3665 109 24 as as ADP iajs-3665 109 25 in in ADP iajs-3665 109 26 equation equation NOUN iajs-3665 109 27 ( ( PUNCT iajs-3665 109 28 11 11 NUM iajs-3665 109 29 ) ) PUNCT iajs-3665 109 30 step step NOUN iajs-3665 109 31 iii iii NUM iajs-3665 109 32 : : PUNCT iajs-3665 109 33 endpoints endpoints PROPN iajs-3665 109 34 b b NOUN iajs-3665 109 35 ; ; PUNCT iajs-3665 109 36 integer integer NOUN iajs-3665 109 37 n n CCONJ iajs-3665 109 38 ; ; PUNCT iajs-3665 109 39 initial initial ADJ iajs-3665 109 40 condition condition NOUN iajs-3665 109 41 𝛽𝑗 𝛽𝑗 NOUN iajs-3665 109 42 for for ADP iajs-3665 109 43 𝑗 𝑗 NOUN iajs-3665 109 44 = = NOUN iajs-3665 109 45 0,1,2 0,1,2 NUM iajs-3665 109 46 , , PUNCT iajs-3665 109 47 . . PUNCT iajs-3665 109 48 . . PUNCT iajs-3665 110 1 . . PUNCT iajs-3665 111 1 , , PUNCT iajs-3665 112 1 𝑛 𝑛 DET iajs-3665 112 2 βˆ’ βˆ’ NOUN iajs-3665 112 3 1 1 NUM iajs-3665 112 4 . . X iajs-3665 112 5 step step NOUN iajs-3665 112 6 iv iv NUM iajs-3665 112 7 : : PUNCT iajs-3665 112 8 set set VERB iajs-3665 112 9 β„Ž β„Ž X iajs-3665 112 10 = = SYM iajs-3665 112 11 𝑏 𝑏 DET iajs-3665 112 12 𝑁 𝑁 PROPN iajs-3665 112 13 then then ADV iajs-3665 112 14 , , PUNCT iajs-3665 112 15 the the DET iajs-3665 112 16 interval interval NOUN iajs-3665 112 17 [ [ X iajs-3665 112 18 π‘Ž π‘Ž X iajs-3665 112 19 , , PUNCT iajs-3665 112 20 𝑏 𝑏 NOUN iajs-3665 112 21 ] ] PUNCT iajs-3665 112 22 has have VERB iajs-3665 112 23 the the DET iajs-3665 112 24 partition partition NOUN iajs-3665 112 25 𝑑1 𝑑1 NOUN iajs-3665 112 26 , , PUNCT iajs-3665 112 27 𝑑2 𝑑2 PROPN iajs-3665 112 28 , , PUNCT iajs-3665 112 29 𝑑3 𝑑3 PROPN iajs-3665 112 30 , , PUNCT iajs-3665 112 31 … … PUNCT iajs-3665 112 32 , , PUNCT iajs-3665 112 33 𝑑𝑁 𝑑𝑁 PROPN iajs-3665 112 34 ; ; PUNCT iajs-3665 112 35 step step NOUN iajs-3665 112 36 v v NOUN iajs-3665 112 37 : : PUNCT iajs-3665 112 38 put put VERB iajs-3665 112 39 𝑑 𝑑 NOUN iajs-3665 112 40 = = VERB iajs-3665 112 41 𝑑𝑗 𝑑𝑗 PROPN iajs-3665 112 42 in in ADP iajs-3665 112 43 equation equation NOUN iajs-3665 112 44 ( ( PUNCT iajs-3665 112 45 13 13 NUM iajs-3665 112 46 ) ) PUNCT iajs-3665 112 47 to to PART iajs-3665 112 48 obtain obtain VERB iajs-3665 112 49 𝑦(𝑑𝑗 𝑦(𝑑𝑗 NOUN iajs-3665 112 50 ) ) PUNCT iajs-3665 113 1 = = SYM iajs-3665 113 2 βˆ‘π‘Žπ‘–βˆ…π‘–(𝑑𝑗 βˆ‘π‘Žπ‘–βˆ…π‘–(𝑑𝑗 ADJ iajs-3665 113 3 ) ) PUNCT iajs-3665 113 4 𝑁 𝑁 PROPN iajs-3665 113 5 𝑖=0 𝑖=0 PROPN iajs-3665 113 6 . . PUNCT iajs-3665 114 1 ( ( PUNCT iajs-3665 114 2 13 13 NUM iajs-3665 114 3 ) ) PUNCT iajs-3665 114 4 for for ADP iajs-3665 114 5 𝑗 𝑗 NOUN iajs-3665 114 6 = = SYM iajs-3665 114 7 1,2 1,2 NUM iajs-3665 114 8 , , PUNCT iajs-3665 114 9 … … PUNCT iajs-3665 114 10 , , PUNCT iajs-3665 114 11 𝑁 𝑁 PROPN iajs-3665 114 12 step step NOUN iajs-3665 114 13 vi vi NOUN iajs-3665 114 14 : : PUNCT iajs-3665 114 15 substitute substitute NOUN iajs-3665 114 16 equation equation NOUN iajs-3665 114 17 ( ( PUNCT iajs-3665 114 18 13 13 NUM iajs-3665 114 19 ) ) PUNCT iajs-3665 114 20 in in ADP iajs-3665 114 21 equation equation NOUN iajs-3665 114 22 ( ( PUNCT iajs-3665 114 23 6 6 NUM iajs-3665 114 24 ) ) PUNCT iajs-3665 114 25 for for ADP iajs-3665 114 26 𝑗 𝑗 NOUN iajs-3665 114 27 = = SYM iajs-3665 114 28 1,2 1,2 NUM iajs-3665 114 29 , , PUNCT iajs-3665 114 30 … … PUNCT iajs-3665 114 31 , , PUNCT iajs-3665 114 32 𝑁. 𝑁. NOUN iajs-3665 114 33 step step NOUN iajs-3665 114 34 ix ix ADV iajs-3665 114 35 : : PUNCT iajs-3665 114 36 as as ADP iajs-3665 114 37 a a DET iajs-3665 114 38 result result NOUN iajs-3665 114 39 of of ADP iajs-3665 114 40 the the DET iajs-3665 114 41 last last ADJ iajs-3665 114 42 equation equation NOUN iajs-3665 114 43 , , PUNCT iajs-3665 114 44 the the DET iajs-3665 114 45 linear linear ADJ iajs-3665 114 46 system system NOUN iajs-3665 114 47 𝐴π‘₯ 𝐴π‘₯ NOUN iajs-3665 114 48 = = PUNCT iajs-3665 114 49 𝑏 𝑏 NOUN iajs-3665 114 50 is be AUX iajs-3665 114 51 generated generate VERB iajs-3665 114 52 , , PUNCT iajs-3665 114 53 which which PRON iajs-3665 114 54 can can AUX iajs-3665 114 55 be be AUX iajs-3665 114 56 solved solve VERB iajs-3665 114 57 using use VERB iajs-3665 114 58 any any DET iajs-3665 114 59 numerical numerical ADJ iajs-3665 114 60 technique technique NOUN iajs-3665 114 61 for for ADP iajs-3665 114 62 solving solve VERB iajs-3665 114 63 linear linear PROPN iajs-3665 114 64 algebraic algebraic ADJ iajs-3665 114 65 equation equation NOUN iajs-3665 114 66 systems system NOUN iajs-3665 114 67 . . PUNCT iajs-3665 115 1 3.2 3.2 NUM iajs-3665 115 2 main main ADJ iajs-3665 115 3 problem problem NOUN iajs-3665 115 4 in in ADP iajs-3665 115 5 focps focp NOUN iajs-3665 115 6 aims aim VERB iajs-3665 115 7 to to PART iajs-3665 115 8 identify identify VERB iajs-3665 115 9 an an DET iajs-3665 115 10 optimal optimal ADJ iajs-3665 115 11 control control NOUN iajs-3665 115 12 function function NOUN iajs-3665 115 13 . . PUNCT iajs-3665 116 1 in in ADP iajs-3665 116 2 this this DET iajs-3665 116 3 paper paper NOUN iajs-3665 116 4 , , PUNCT iajs-3665 116 5 we we PRON iajs-3665 116 6 present present VERB iajs-3665 116 7 an an DET iajs-3665 116 8 original original ADJ iajs-3665 116 9 numerical numerical ADJ iajs-3665 116 10 approach approach NOUN iajs-3665 116 11 for for ADP iajs-3665 116 12 determining determine VERB iajs-3665 116 13 the the DET iajs-3665 116 14 solutions solution NOUN iajs-3665 116 15 of of ADP iajs-3665 116 16 ( ( PUNCT iajs-3665 116 17 foc foc ADJ iajs-3665 116 18 ) ) PUNCT iajs-3665 116 19 systems system NOUN iajs-3665 116 20 given give VERB iajs-3665 116 21 both both DET iajs-3665 116 22 cases case NOUN iajs-3665 116 23 of of ADP iajs-3665 116 24 ( ( PUNCT iajs-3665 116 25 tt tt NOUN iajs-3665 116 26 ) ) PUNCT iajs-3665 116 27 case case NOUN iajs-3665 117 1 i i PRON iajs-3665 117 2 : : PUNCT iajs-3665 117 3 non non ADJ iajs-3665 117 4 free-(tt free-(tt PROPN iajs-3665 117 5 ) ) PUNCT iajs-3665 117 6 let let VERB iajs-3665 117 7 min min PRON iajs-3665 117 8 π‘₯(𝜍),𝑒(𝜍 π‘₯(𝜍),𝑒(𝜍 NOUN iajs-3665 117 9 ) ) PUNCT iajs-3665 117 10 𝐽 𝐽 PROPN iajs-3665 117 11 ( ( PUNCT iajs-3665 117 12 𝜍 𝜍 X iajs-3665 117 13 , , PUNCT iajs-3665 117 14 π‘₯(𝜍 π‘₯(𝜍 PROPN iajs-3665 117 15 ) ) PUNCT iajs-3665 117 16 , , PUNCT iajs-3665 117 17 𝑒(𝜍 𝑒(𝜍 PROPN iajs-3665 117 18 ) ) PUNCT iajs-3665 117 19 ) ) PUNCT iajs-3665 118 1 = = SYM iajs-3665 118 2 min min PROPN iajs-3665 118 3 π‘₯(𝜍),𝑒(𝜍 π‘₯(𝜍),𝑒(𝜍 NOUN iajs-3665 118 4 ) ) PUNCT iajs-3665 118 5 ∫ ∫ PROPN iajs-3665 118 6 𝑃 𝑃 PROPN iajs-3665 118 7 ( ( PUNCT iajs-3665 118 8 𝜍 𝜍 X iajs-3665 118 9 , , PUNCT iajs-3665 118 10 π‘₯(𝜍 π‘₯(𝜍 PROPN iajs-3665 118 11 ) ) PUNCT iajs-3665 118 12 , , PUNCT iajs-3665 118 13 𝑒(𝜍 𝑒(𝜍 PROPN iajs-3665 118 14 ) ) PUNCT iajs-3665 118 15 ) ) PUNCT iajs-3665 119 1 𝜍1 𝜍1 PROPN iajs-3665 119 2 𝜍0 𝜍0 PROPN iajs-3665 119 3 π‘‘πœ π‘‘πœ ADP iajs-3665 119 4 , , PUNCT iajs-3665 119 5 ( ( PUNCT iajs-3665 119 6 14 14 NUM iajs-3665 119 7 ) ) PUNCT iajs-3665 119 8 as as ADP iajs-3665 119 9 a a DET iajs-3665 119 10 result result NOUN iajs-3665 119 11 of of ADP iajs-3665 119 12 the the DET iajs-3665 119 13 restricted restricted ADJ iajs-3665 119 14 dynamical dynamical ADJ iajs-3665 119 15 - - PUNCT iajs-3665 119 16 system system NOUN iajs-3665 119 17 𝛼 𝛼 NOUN iajs-3665 119 18 οΏ½ οΏ½ PROPN iajs-3665 119 19 ̈ ̈ X iajs-3665 119 20 οΏ½ οΏ½ NOUN iajs-3665 119 21 (𝜍 (𝜍 NOUN iajs-3665 119 22 ) ) PUNCT iajs-3665 120 1 + + SYM iajs-3665 120 2 𝛽 𝛽 NOUN iajs-3665 120 3 οΏ½ οΏ½ PROPN iajs-3665 120 4 Μ‡ Μ‡ PROPN iajs-3665 120 5 οΏ½ οΏ½ PROPN iajs-3665 120 6 (𝜍 (𝜍 NOUN iajs-3665 120 7 ) ) PUNCT iajs-3665 121 1 + + CCONJ iajs-3665 121 2 𝛾𝐷𝛾π‘₯(𝜍 𝛾𝐷𝛾π‘₯(𝜍 ADJ iajs-3665 121 3 ) ) PUNCT iajs-3665 121 4 = = SYM iajs-3665 121 5 πœ€(𝜍)π‘₯(𝜍 πœ€(𝜍)π‘₯(𝜍 X iajs-3665 121 6 ) ) PUNCT iajs-3665 122 1 + + NUM iajs-3665 122 2 𝑓(𝜍)𝑒(𝜍 𝑓(𝜍)𝑒(𝜍 NOUN iajs-3665 122 3 ) ) PUNCT iajs-3665 122 4 + + NUM iajs-3665 122 5 𝑔(𝜍 𝑔(𝜍 NOUN iajs-3665 122 6 ) ) PUNCT iajs-3665 123 1 , , PUNCT iajs-3665 123 2 𝜍0 𝜍0 VERB iajs-3665 123 3 ≀ ≀ NOUN iajs-3665 123 4 𝜍 𝜍 ADP iajs-3665 123 5 ≀ ≀ ADJ iajs-3665 123 6 𝜍1 𝜍1 NOUN iajs-3665 123 7 , , PUNCT iajs-3665 123 8 0 0 NUM iajs-3665 123 9 ≀ ≀ NUM iajs-3665 123 10 𝛾 𝛾 ADP iajs-3665 123 11 ≀ ≀ ADJ iajs-3665 123 12 2 2 NUM iajs-3665 123 13 , , PUNCT iajs-3665 123 14 ( ( PUNCT iajs-3665 123 15 15 15 NUM iajs-3665 123 16 ) ) PUNCT iajs-3665 123 17 and and CCONJ iajs-3665 123 18 the the DET iajs-3665 123 19 restricted restrict VERB iajs-3665 123 20 bcs bc NOUN iajs-3665 123 21 are be AUX iajs-3665 123 22 provided provide VERB iajs-3665 123 23 below below ADP iajs-3665 123 24 : : PUNCT iajs-3665 123 25 π‘₯ π‘₯ PROPN iajs-3665 123 26 ( ( PUNCT iajs-3665 123 27 𝜍0 𝜍0 PROPN iajs-3665 123 28 ) ) PUNCT iajs-3665 123 29 = = SYM iajs-3665 123 30 𝜁 𝜁 PROPN iajs-3665 123 31 , , PUNCT iajs-3665 123 32 π‘₯(𝜍1 π‘₯(𝜍1 NOUN iajs-3665 123 33 ) ) PUNCT iajs-3665 123 34 = = SYM iajs-3665 123 35 πœ‚ πœ‚ NOUN iajs-3665 123 36 , , PUNCT iajs-3665 123 37 ( ( PUNCT iajs-3665 123 38 16 16 NUM iajs-3665 123 39 ) ) PUNCT iajs-3665 123 40 where where SCONJ iajs-3665 123 41 𝛼 𝛼 X iajs-3665 123 42 , , PUNCT iajs-3665 123 43 𝛽 𝛽 NOUN iajs-3665 123 44 β‰  β‰  PROPN iajs-3665 123 45 0 0 NUM iajs-3665 123 46 case case NOUN iajs-3665 123 47 ii ii NOUN iajs-3665 123 48 : : PUNCT iajs-3665 123 49 free--(tt free--(tt ADJ iajs-3665 123 50 ) ) PUNCT iajs-3665 123 51 consider consider VERB iajs-3665 123 52 the the DET iajs-3665 123 53 focp focp PROPN iajs-3665 123 54 in in ADP iajs-3665 123 55 the the DET iajs-3665 123 56 following follow VERB iajs-3665 123 57 equation equation NOUN iajs-3665 123 58 minimum minimum ADJ iajs-3665 123 59 π‘₯(𝜍 π‘₯(𝜍 PROPN iajs-3665 123 60 ) ) PUNCT iajs-3665 123 61 , , PUNCT iajs-3665 123 62 𝑒(𝜍 𝑒(𝜍 PROPN iajs-3665 123 63 ) ) PUNCT iajs-3665 123 64 , , PUNCT iajs-3665 123 65 𝑇 𝑇 PROPN iajs-3665 123 66 𝐽 𝐽 PROPN iajs-3665 123 67 ( ( PUNCT iajs-3665 123 68 𝜍 𝜍 PROPN iajs-3665 123 69 , , PUNCT iajs-3665 123 70 𝑇 𝑇 PROPN iajs-3665 123 71 , , PUNCT iajs-3665 123 72 π‘₯(𝜍 π‘₯(𝜍 PROPN iajs-3665 123 73 ) ) PUNCT iajs-3665 123 74 , , PUNCT iajs-3665 123 75 𝑒(𝜍 𝑒(𝜍 PROPN iajs-3665 123 76 ) ) PUNCT iajs-3665 123 77 ) ) PUNCT iajs-3665 124 1 = = SYM iajs-3665 124 2 minimum minimum ADJ iajs-3665 124 3 π‘₯(𝜍 π‘₯(𝜍 PROPN iajs-3665 124 4 ) ) PUNCT iajs-3665 124 5 , , PUNCT iajs-3665 124 6 𝑒(𝜍 𝑒(𝜍 PROPN iajs-3665 124 7 ) ) PUNCT iajs-3665 124 8 , , PUNCT iajs-3665 124 9 𝑇 𝑇 PROPN iajs-3665 124 10 ∫ ∫ PROPN iajs-3665 124 11 𝑃 𝑃 PROPN iajs-3665 124 12 ( ( PUNCT iajs-3665 124 13 𝜍 𝜍 X iajs-3665 124 14 , , PUNCT iajs-3665 124 15 π‘₯(𝜍 π‘₯(𝜍 PROPN iajs-3665 124 16 ) ) PUNCT iajs-3665 124 17 , , PUNCT iajs-3665 124 18 𝑒(𝜍 𝑒(𝜍 PROPN iajs-3665 124 19 ) ) PUNCT iajs-3665 124 20 ) ) PUNCT iajs-3665 125 1 𝑇 𝑇 PROPN iajs-3665 125 2 𝜍0 𝜍0 NOUN iajs-3665 125 3 π‘‘πœ π‘‘πœ ADP iajs-3665 125 4 , , PUNCT iajs-3665 125 5 ( ( PUNCT iajs-3665 125 6 17 17 NUM iajs-3665 125 7 ) ) PUNCT iajs-3665 125 8 subject subject NOUN iajs-3665 125 9 to to ADP iajs-3665 125 10 : : PUNCT iajs-3665 125 11 π‘₯ π‘₯ X iajs-3665 125 12 ( ( PUNCT iajs-3665 125 13 𝜍0 𝜍0 PROPN iajs-3665 125 14 ) ) PUNCT iajs-3665 125 15 = = SYM iajs-3665 125 16 𝜁 𝜁 PROPN iajs-3665 125 17 , , PUNCT iajs-3665 125 18 π‘₯(𝑇 π‘₯(𝑇 NUM iajs-3665 125 19 ) ) PUNCT iajs-3665 125 20 = = SYM iajs-3665 125 21 πœ‚ πœ‚ NOUN iajs-3665 125 22 , , PUNCT iajs-3665 125 23 ( ( PUNCT iajs-3665 125 24 18 18 NUM iajs-3665 125 25 ) ) PUNCT iajs-3665 125 26 where where SCONJ iajs-3665 125 27 𝑇 𝑇 PROPN iajs-3665 125 28 is be AUX iajs-3665 125 29 a a DET iajs-3665 125 30 free free ADJ iajs-3665 125 31 - - PUNCT iajs-3665 125 32 parameter parameter NOUN iajs-3665 125 33 ? ? PUNCT iajs-3665 126 1 to to PART iajs-3665 126 2 begin begin VERB iajs-3665 126 3 , , PUNCT iajs-3665 126 4 we we PRON iajs-3665 126 5 employ employ VERB iajs-3665 126 6 the the DET iajs-3665 126 7 collocation collocation NOUN iajs-3665 126 8 technique technique NOUN iajs-3665 126 9 to to PART iajs-3665 126 10 approximate approximate VERB iajs-3665 126 11 the the DET iajs-3665 126 12 variable variable ADJ iajs-3665 126 13 π‘₯(𝜍 π‘₯(𝜍 PROPN iajs-3665 126 14 ) ) PUNCT iajs-3665 126 15 with with ADP iajs-3665 126 16 the the DET iajs-3665 126 17 controlvariable controlvariable ADJ iajs-3665 126 18 𝑒(𝜍 𝑒(𝜍 PROPN iajs-3665 126 19 ) ) PUNCT iajs-3665 126 20 using use VERB iajs-3665 126 21 the the DET iajs-3665 126 22 proposed propose VERB iajs-3665 126 23 numerical numerical ADJ iajs-3665 126 24 approach approach NOUN iajs-3665 126 25 , , PUNCT iajs-3665 126 26 where where SCONJ iajs-3665 126 27 𝑒(𝜍 𝑒(𝜍 PROPN iajs-3665 126 28 ) ) PUNCT iajs-3665 126 29 , , PUNCT iajs-3665 126 30 𝑓(𝜍 𝑓(𝜍 PROPN iajs-3665 126 31 ) ) PUNCT iajs-3665 126 32 , , PUNCT iajs-3665 126 33 and and CCONJ iajs-3665 126 34 𝑔(𝜍 𝑔(𝜍 NOUN iajs-3665 126 35 ) ) PUNCT iajs-3665 126 36 are be AUX iajs-3665 126 37 known know VERB iajs-3665 126 38 functions function NOUN iajs-3665 126 39 . . PUNCT iajs-3665 127 1 in in ADP iajs-3665 127 2 this this DET iajs-3665 127 3 study study NOUN iajs-3665 127 4 , , PUNCT iajs-3665 127 5 the the DET iajs-3665 127 6 hooke hooke PROPN iajs-3665 127 7 and and CCONJ iajs-3665 127 8 jeeves jeeve NOUN iajs-3665 127 9 approach approach NOUN iajs-3665 127 10 is be AUX iajs-3665 127 11 utilized utilize VERB iajs-3665 127 12 as as ADP iajs-3665 127 13 a a DET iajs-3665 127 14 search search NOUN iajs-3665 127 15 method method NOUN iajs-3665 127 16 . . PUNCT iajs-3665 128 1 3.3 3.3 NUM iajs-3665 128 2 . . PUNCT iajs-3665 129 1 the the DET iajs-3665 129 2 proposed propose VERB iajs-3665 129 3 method method NOUN iajs-3665 129 4 in in ADP iajs-3665 129 5 this this DET iajs-3665 129 6 subsection subsection NOUN iajs-3665 129 7 , , PUNCT iajs-3665 129 8 the the DET iajs-3665 129 9 proposed propose VERB iajs-3665 129 10 method method NOUN iajs-3665 129 11 for for ADP iajs-3665 129 12 solving solve VERB iajs-3665 129 13 ( ( PUNCT iajs-3665 129 14 foc foc ADJ iajs-3665 129 15 ) ) PUNCT iajs-3665 129 16 problems problem NOUN iajs-3665 129 17 with with ADP iajs-3665 129 18 cases case NOUN iajs-3665 129 19 of of ADP iajs-3665 129 20 ( ( PUNCT iajs-3665 129 21 tt tt PROPN iajs-3665 129 22 ) ) PUNCT iajs-3665 129 23 is be AUX iajs-3665 129 24 discussed discuss VERB iajs-3665 129 25 . . PUNCT iajs-3665 130 1 the the DET iajs-3665 130 2 algorithm algorithm NOUN iajs-3665 130 3 of of ADP iajs-3665 130 4 the the DET iajs-3665 130 5 proposed propose VERB iajs-3665 130 6 approach approach NOUN iajs-3665 130 7 for for ADP iajs-3665 130 8 determining determine VERB iajs-3665 130 9 the the DET iajs-3665 130 10 solutions solution NOUN iajs-3665 130 11 of of ADP iajs-3665 130 12 ( ( PUNCT iajs-3665 130 13 foc foc ADJ iajs-3665 130 14 ) ) PUNCT iajs-3665 130 15 problems problem NOUN iajs-3665 130 16 with with ADP iajs-3665 130 17 freeand freeand PROPN iajs-3665 130 18 non non ADJ iajs-3665 130 19 - - ADJ iajs-3665 130 20 free free ADJ iajs-3665 130 21 ( ( PUNCT iajs-3665 130 22 tt tt NOUN iajs-3665 130 23 ) ) PUNCT iajs-3665 130 24 is be AUX iajs-3665 130 25 explained explain VERB iajs-3665 130 26 . . PUNCT iajs-3665 131 1 the the DET iajs-3665 131 2 algorithm algorithm NOUN iajs-3665 131 3 has have VERB iajs-3665 131 4 two two NUM iajs-3665 131 5 cases case NOUN iajs-3665 131 6 : : PUNCT iajs-3665 131 7 ο‚· ο‚· PRON iajs-3665 131 8 non non ADJ iajs-3665 131 9 - - ADJ iajs-3665 131 10 free free ADJ iajs-3665 131 11 tt tt PROPN iajs-3665 131 12 ihjpas ihjpa VERB iajs-3665 131 13 . . PUNCT iajs-3665 132 1 2025 2025 NUM iajs-3665 132 2 , , PUNCT iajs-3665 132 3 38(2 38(2 NUM iajs-3665 132 4 ) ) PUNCT iajs-3665 132 5 344 344 NUM iajs-3665 132 6 1 1 NUM iajs-3665 132 7 . . PUNCT iajs-3665 132 8 construct construct VERB iajs-3665 132 9 an an DET iajs-3665 132 10 approximate approximate ADJ iajs-3665 132 11 solution solution NOUN iajs-3665 132 12 of of ADP iajs-3665 132 13 ( ( PUNCT iajs-3665 132 14 foc foc PROPN iajs-3665 132 15 ) ) PUNCT iajs-3665 132 16 as as ADP iajs-3665 132 17 in in ADP iajs-3665 132 18 equation equation NOUN iajs-3665 132 19 ( ( PUNCT iajs-3665 132 20 11 11 NUM iajs-3665 132 21 ) ) PUNCT iajs-3665 132 22 whenever whenever SCONJ iajs-3665 132 23 the the DET iajs-3665 132 24 bcs bc NOUN iajs-3665 132 25 are be AUX iajs-3665 132 26 homogenous homogenous ADJ iajs-3665 132 27 otherwise otherwise ADV iajs-3665 132 28 the the DET iajs-3665 132 29 approximated approximated ADJ iajs-3665 132 30 solution solution NOUN iajs-3665 132 31 of of ADP iajs-3665 132 32 ( ( PUNCT iajs-3665 132 33 foc foc PROPN iajs-3665 132 34 ) ) PUNCT iajs-3665 132 35 has have VERB iajs-3665 132 36 the the DET iajs-3665 132 37 following follow VERB iajs-3665 132 38 form form NOUN iajs-3665 132 39 , , PUNCT iajs-3665 132 40 π‘₯(𝑑 π‘₯(𝑑 NOUN iajs-3665 132 41 ) ) PUNCT iajs-3665 132 42 = = SYM iajs-3665 132 43 1 1 NUM iajs-3665 132 44 𝜏1 𝜏1 NOUN iajs-3665 132 45 βˆ’ βˆ’ PROPN iajs-3665 132 46 𝜏0 𝜏0 PROPN iajs-3665 132 47 ( ( PUNCT iajs-3665 132 48 πœ‚(𝑑 πœ‚(𝑑 NOUN iajs-3665 132 49 βˆ’ βˆ’ NUM iajs-3665 132 50 𝜏0 𝜏0 PROPN iajs-3665 132 51 ) ) PUNCT iajs-3665 132 52 βˆ’ βˆ’ PROPN iajs-3665 132 53 𝜁(𝑑 𝜁(𝑑 PROPN iajs-3665 132 54 βˆ’ βˆ’ PROPN iajs-3665 132 55 𝜏1 𝜏1 NOUN iajs-3665 132 56 ) ) PUNCT iajs-3665 132 57 ) ) PUNCT iajs-3665 133 1 + + CCONJ iajs-3665 134 1 𝑐0βˆ…0(𝑑 𝑐0βˆ…0(𝑑 PUNCT iajs-3665 134 2 ) ) PUNCT iajs-3665 134 3 + + NUM iajs-3665 134 4 𝑐1βˆ…1(𝑑 𝑐1βˆ…1(𝑑 NOUN iajs-3665 134 5 ) ) PUNCT iajs-3665 134 6 + + NUM iajs-3665 134 7 β‹―+ β‹―+ NOUN iajs-3665 134 8 π‘π‘›βˆ…π‘›(𝑑 π‘π‘›βˆ…π‘›(𝑑 NOUN iajs-3665 134 9 ) ) PUNCT iajs-3665 134 10 . . PUNCT iajs-3665 135 1 ( ( PUNCT iajs-3665 135 2 19 19 NUM iajs-3665 135 3 ) ) PUNCT iajs-3665 135 4 in in ADP iajs-3665 135 5 equations equation NOUN iajs-3665 135 6 ( ( PUNCT iajs-3665 135 7 14)-(15 14)-(15 NUM iajs-3665 135 8 ) ) PUNCT iajs-3665 135 9 that that SCONJ iajs-3665 135 10 , , PUNCT iajs-3665 135 11 given give VERB iajs-3665 135 12 the the DET iajs-3665 135 13 approximated approximate VERB iajs-3665 135 14 base base NOUN iajs-3665 135 15 , , PUNCT iajs-3665 135 16 satisfies satisfy VERB iajs-3665 135 17 the the DET iajs-3665 135 18 boundary boundary ADJ iajs-3665 135 19 requirements requirement NOUN iajs-3665 135 20 in in ADP iajs-3665 135 21 equation equation NOUN iajs-3665 135 22 ( ( PUNCT iajs-3665 135 23 000 000 NUM iajs-3665 135 24 ) ) PUNCT iajs-3665 135 25 . . PUNCT iajs-3665 136 1 2 2 X iajs-3665 136 2 . . X iajs-3665 136 3 if if SCONJ iajs-3665 136 4 the the DET iajs-3665 136 5 de de X iajs-3665 136 6 in in ADP iajs-3665 136 7 equation equation NOUN iajs-3665 136 8 ( ( PUNCT iajs-3665 136 9 15 15 NUM iajs-3665 136 10 ) ) PUNCT iajs-3665 136 11 is be AUX iajs-3665 136 12 specified specify VERB iajs-3665 136 13 explicitly explicitly ADV iajs-3665 136 14 in in ADP iajs-3665 136 15 the the DET iajs-3665 136 16 control control NOUN iajs-3665 136 17 function function NOUN iajs-3665 136 18 u(t u(t NOUN iajs-3665 136 19 ) ) PUNCT iajs-3665 136 20 , , PUNCT iajs-3665 136 21 the the DET iajs-3665 136 22 function function NOUN iajs-3665 136 23 u(t u(t NOUN iajs-3665 136 24 ) ) PUNCT iajs-3665 136 25 should should AUX iajs-3665 136 26 be be AUX iajs-3665 136 27 evaluated evaluate VERB iajs-3665 136 28 . . PUNCT iajs-3665 137 1 3 3 X iajs-3665 137 2 . . X iajs-3665 137 3 substitute substitute VERB iajs-3665 137 4 the the DET iajs-3665 137 5 approximated approximate VERB iajs-3665 137 6 formulas formula NOUN iajs-3665 137 7 for for ADP iajs-3665 137 8 the the DET iajs-3665 137 9 functions function NOUN iajs-3665 137 10 x(t x(t PROPN iajs-3665 137 11 ) ) PUNCT iajs-3665 137 12 and and CCONJ iajs-3665 137 13 u(t u(t NOUN iajs-3665 137 14 ) ) PUNCT iajs-3665 137 15 in in ADP iajs-3665 137 16 equation equation NOUN iajs-3665 137 17 ( ( PUNCT iajs-3665 137 18 14 14 NUM iajs-3665 137 19 ) ) PUNCT iajs-3665 137 20 . . PUNCT iajs-3665 138 1 4 4 X iajs-3665 138 2 . . X iajs-3665 138 3 use use VERB iajs-3665 138 4 a a DET iajs-3665 138 5 suitable suitable ADJ iajs-3665 138 6 minimizing minimize VERB iajs-3665 138 7 search search NOUN iajs-3665 138 8 approach approach NOUN iajs-3665 138 9 , , PUNCT iajs-3665 138 10 such such ADJ iajs-3665 138 11 as as ADP iajs-3665 138 12 the the DET iajs-3665 138 13 hooke hooke NOUN iajs-3665 138 14 - - PUNCT iajs-3665 138 15 jeeves jeeve NOUN iajs-3665 138 16 method method NOUN iajs-3665 138 17 , , PUNCT iajs-3665 138 18 to to PART iajs-3665 138 19 determine determine VERB iajs-3665 138 20 the the DET iajs-3665 138 21 minimal minimal ADJ iajs-3665 138 22 parameter(s parameter(s NOUN iajs-3665 138 23 ) ) PUNCT iajs-3665 138 24 in in ADP iajs-3665 138 25 equation equation NOUN iajs-3665 138 26 ( ( PUNCT iajs-3665 138 27 14 14 NUM iajs-3665 138 28 ) ) PUNCT iajs-3665 138 29 . . PUNCT iajs-3665 139 1 ο‚· ο‚· PROPN iajs-3665 139 2 free free ADJ iajs-3665 139 3 ( ( PUNCT iajs-3665 139 4 tt tt NOUN iajs-3665 139 5 ) ) PUNCT iajs-3665 139 6 1 1 NUM iajs-3665 139 7 . . PUNCT iajs-3665 139 8 steps step NOUN iajs-3665 139 9 1 1 NUM iajs-3665 139 10 - - SYM iajs-3665 139 11 4 4 NUM iajs-3665 139 12 of of ADP iajs-3665 139 13 the the DET iajs-3665 139 14 previous previous ADJ iajs-3665 139 15 method method NOUN iajs-3665 139 16 should should AUX iajs-3665 139 17 be be AUX iajs-3665 139 18 followed follow VERB iajs-3665 139 19 . . PUNCT iajs-3665 140 1 2 2 X iajs-3665 140 2 . . X iajs-3665 140 3 the the DET iajs-3665 140 4 optimal optimal ADJ iajs-3665 140 5 parameters parameter NOUN iajs-3665 140 6 ( ( PUNCT iajs-3665 140 7 minimum minimum NOUN iajs-3665 140 8 ) ) PUNCT iajs-3665 140 9 , , PUNCT iajs-3665 140 10 including include VERB iajs-3665 140 11 the the DET iajs-3665 140 12 parameter parameter NOUN iajs-3665 140 13 t t PROPN iajs-3665 140 14 in in ADP iajs-3665 140 15 equation equation NOUN iajs-3665 140 16 ( ( PUNCT iajs-3665 140 17 14 14 NUM iajs-3665 140 18 ) ) PUNCT iajs-3665 140 19 , , PUNCT iajs-3665 140 20 can can AUX iajs-3665 140 21 be be AUX iajs-3665 140 22 determined determine VERB iajs-3665 140 23 using use VERB iajs-3665 140 24 appropriate appropriate ADJ iajs-3665 140 25 minimizing minimizing NOUN iajs-3665 140 26 search search NOUN iajs-3665 140 27 methods method NOUN iajs-3665 140 28 , , PUNCT iajs-3665 140 29 such such ADJ iajs-3665 140 30 as as ADP iajs-3665 140 31 hooke hooke NOUN iajs-3665 140 32 - - PUNCT iajs-3665 140 33 jeeves jeeve NOUN iajs-3665 140 34 approach approach NOUN iajs-3665 140 35 . . PUNCT iajs-3665 141 1 3.4 3.4 NUM iajs-3665 141 2 . . PUNCT iajs-3665 141 3 dual dual ADJ iajs-3665 141 4 discreet discreet ADJ iajs-3665 141 5 problem problem NOUN iajs-3665 141 6 in in ADP iajs-3665 141 7 case case NOUN iajs-3665 141 8 of of ADP iajs-3665 141 9 non non ADJ iajs-3665 141 10 - - ADJ iajs-3665 141 11 free free ADJ iajs-3665 141 12 ( ( PUNCT iajs-3665 141 13 tt tt PROPN iajs-3665 141 14 ) ) PUNCT iajs-3665 141 15 , , PUNCT iajs-3665 141 16 the the DET iajs-3665 141 17 control control NOUN iajs-3665 141 18 function function NOUN iajs-3665 141 19 is be AUX iajs-3665 141 20 obtained obtain VERB iajs-3665 141 21 as as ADP iajs-3665 141 22 optimal optimal ADJ iajs-3665 141 23 problem problem NOUN iajs-3665 141 24 π‘šπ‘–π‘›π‘–π‘šπ‘’π‘š π‘šπ‘–π‘›π‘–π‘šπ‘’π‘š NOUN iajs-3665 141 25 βˆ… βˆ… NOUN iajs-3665 141 26 ( ( PUNCT iajs-3665 141 27 𝑐0 𝑐0 NOUN iajs-3665 141 28 , , PUNCT iajs-3665 141 29 𝑐1 𝑐1 NOUN iajs-3665 141 30 , , PUNCT iajs-3665 141 31 𝑐2 𝑐2 NOUN iajs-3665 141 32 , , PUNCT iajs-3665 141 33 … … PUNCT iajs-3665 141 34 , , PUNCT iajs-3665 141 35 𝑐𝑛 𝑐𝑛 PROPN iajs-3665 141 36 , , PUNCT iajs-3665 141 37 βˆ…0(𝑑 βˆ…0(𝑑 NUM iajs-3665 141 38 ) ) PUNCT iajs-3665 141 39 , , PUNCT iajs-3665 141 40 βˆ…1(𝑑 βˆ…1(𝑑 NOUN iajs-3665 141 41 ) ) PUNCT iajs-3665 141 42 , , PUNCT iajs-3665 141 43 … … PUNCT iajs-3665 141 44 , , PUNCT iajs-3665 141 45 βˆ…π‘›(𝑑 βˆ…π‘›(𝑑 NOUN iajs-3665 141 46 ) ) PUNCT iajs-3665 141 47 ) ) PUNCT iajs-3665 142 1 whilst whilst SCONJ iajs-3665 142 2 in in ADP iajs-3665 142 3 the the DET iajs-3665 142 4 case case NOUN iajs-3665 142 5 of of ADP iajs-3665 142 6 non non ADJ iajs-3665 142 7 - - ADJ iajs-3665 142 8 free free ADJ iajs-3665 142 9 ( ( PUNCT iajs-3665 142 10 tt tt PROPN iajs-3665 142 11 ) ) PUNCT iajs-3665 142 12 , , PUNCT iajs-3665 142 13 the the DET iajs-3665 142 14 control control NOUN iajs-3665 142 15 function function NOUN iajs-3665 142 16 is be AUX iajs-3665 142 17 obtained obtain VERB iajs-3665 142 18 as as ADP iajs-3665 142 19 optimal optimal ADJ iajs-3665 142 20 problem problem NOUN iajs-3665 142 21 π‘šπ‘–π‘›π‘–π‘šπ‘’π‘š π‘šπ‘–π‘›π‘–π‘šπ‘’π‘š NOUN iajs-3665 142 22 βˆ… βˆ… NOUN iajs-3665 142 23 ( ( PUNCT iajs-3665 142 24 𝑐0 𝑐0 NOUN iajs-3665 142 25 , , PUNCT iajs-3665 142 26 𝑐1 𝑐1 NOUN iajs-3665 142 27 , , PUNCT iajs-3665 142 28 𝑐2 𝑐2 NOUN iajs-3665 142 29 , , PUNCT iajs-3665 142 30 … … PUNCT iajs-3665 142 31 , , PUNCT iajs-3665 142 32 𝑐𝑛 𝑐𝑛 PROPN iajs-3665 142 33 , , PUNCT iajs-3665 142 34 βˆ…0 βˆ…0 NOUN iajs-3665 142 35 ( ( PUNCT iajs-3665 142 36 𝑑 𝑑 NOUN iajs-3665 142 37 ) ) PUNCT iajs-3665 142 38 , , PUNCT iajs-3665 142 39 βˆ…1(𝑑 βˆ…1(𝑑 NOUN iajs-3665 142 40 ) ) PUNCT iajs-3665 142 41 , , PUNCT iajs-3665 142 42 … … PUNCT iajs-3665 142 43 , , PUNCT iajs-3665 142 44 βˆ…π‘›(𝑑 βˆ…π‘›(𝑑 PROPN iajs-3665 142 45 ) ) PUNCT iajs-3665 142 46 , , PUNCT iajs-3665 142 47 𝑇 𝑇 PROPN iajs-3665 142 48 ) ) PUNCT iajs-3665 142 49 4 4 NUM iajs-3665 142 50 . . PUNCT iajs-3665 143 1 numerical numerical ADJ iajs-3665 143 2 implementations implementation NOUN iajs-3665 143 3 in in ADP iajs-3665 143 4 this this DET iajs-3665 143 5 part part NOUN iajs-3665 143 6 , , PUNCT iajs-3665 143 7 we we PRON iajs-3665 143 8 have have AUX iajs-3665 143 9 examined examine VERB iajs-3665 143 10 two two NUM iajs-3665 143 11 types type NOUN iajs-3665 143 12 of of ADP iajs-3665 143 13 ( ( PUNCT iajs-3665 143 14 foc foc ADJ iajs-3665 143 15 ) ) PUNCT iajs-3665 143 16 systems system NOUN iajs-3665 143 17 where where SCONJ iajs-3665 143 18 u(t u(t NOUN iajs-3665 143 19 ) ) PUNCT iajs-3665 143 20 is be AUX iajs-3665 143 21 the the DET iajs-3665 143 22 control control NOUN iajs-3665 143 23 function function NOUN iajs-3665 143 24 as as SCONJ iajs-3665 143 25 known know VERB iajs-3665 143 26 in in ADP iajs-3665 143 27 the the DET iajs-3665 143 28 ( ( PUNCT iajs-3665 143 29 foc foc ADJ iajs-3665 143 30 ) ) PUNCT iajs-3665 143 31 problem problem NOUN iajs-3665 143 32 . . PUNCT iajs-3665 144 1 example example NOUN iajs-3665 145 1 1 1 NUM iajs-3665 145 2 . . X iajs-3665 145 3 consider consider VERB iajs-3665 145 4 the the DET iajs-3665 145 5 following follow VERB iajs-3665 145 6 non non ADJ iajs-3665 145 7 - - ADJ iajs-3665 145 8 free free ADJ iajs-3665 145 9 tt tt PROPN iajs-3665 145 10 system system NOUN iajs-3665 145 11 with with ADP iajs-3665 145 12 min min NOUN iajs-3665 145 13 π‘₯(𝜎),𝑒(𝜎 π‘₯(𝜎),𝑒(𝜎 PROPN iajs-3665 145 14 ) ) PUNCT iajs-3665 145 15 𝐽 𝐽 PROPN iajs-3665 145 16 ( ( PUNCT iajs-3665 145 17 𝜎 𝜎 PROPN iajs-3665 145 18 , , PUNCT iajs-3665 145 19 π‘₯(𝜎 π‘₯(𝜎 PROPN iajs-3665 145 20 ) ) PUNCT iajs-3665 145 21 , , PUNCT iajs-3665 145 22 𝑒(𝜎 𝑒(𝜎 PROPN iajs-3665 145 23 ) ) PUNCT iajs-3665 145 24 ) ) PUNCT iajs-3665 146 1 = = SYM iajs-3665 146 2 min min NOUN iajs-3665 146 3 π‘₯(𝜎),𝑒(𝜎 π‘₯(𝜎),𝑒(𝜎 NUM iajs-3665 146 4 ) ) PUNCT iajs-3665 146 5 ∫ ∫ PROPN iajs-3665 146 6 ( ( PUNCT iajs-3665 146 7 πœŽπ‘’(𝜎 πœŽπ‘’(𝜎 NUM iajs-3665 146 8 ) ) PUNCT iajs-3665 146 9 βˆ’ βˆ’ PROPN iajs-3665 146 10 ( ( PUNCT iajs-3665 146 11 𝛾 𝛾 NOUN iajs-3665 146 12 + + X iajs-3665 146 13 2)π‘₯(𝜎 2)π‘₯(𝜎 NUM iajs-3665 146 14 ) ) PUNCT iajs-3665 146 15 ) ) PUNCT iajs-3665 146 16 2 2 NUM iajs-3665 146 17 1 1 NUM iajs-3665 146 18 0 0 NUM iajs-3665 146 19 π‘‘πœŽ π‘‘πœŽ NOUN iajs-3665 146 20 , , PUNCT iajs-3665 146 21 ( ( PUNCT iajs-3665 146 22 20 20 NUM iajs-3665 146 23 ) ) PUNCT iajs-3665 146 24 subject subject NOUN iajs-3665 146 25 to to ADP iajs-3665 146 26 𝐷𝛾π‘₯(𝜎 𝐷𝛾π‘₯(𝜎 VERB iajs-3665 146 27 ) ) PUNCT iajs-3665 146 28 + + CCONJ iajs-3665 146 29 οΏ½ οΏ½ PROPN iajs-3665 146 30 Μ‡ Μ‡ PROPN iajs-3665 146 31 οΏ½ οΏ½ PROPN iajs-3665 146 32 (𝜎 (𝜎 PROPN iajs-3665 146 33 ) ) PUNCT iajs-3665 146 34 = = SYM iajs-3665 146 35 𝑒(𝜎 𝑒(𝜎 PROPN iajs-3665 146 36 ) ) PUNCT iajs-3665 147 1 βˆ’ βˆ’ PROPN iajs-3665 147 2 𝜎2 𝜎2 NOUN iajs-3665 147 3 βˆ’ βˆ’ PROPN iajs-3665 148 1 ( ( PUNCT iajs-3665 148 2 π‘Ž π‘Ž X iajs-3665 148 3 + + X iajs-3665 148 4 π‘Žπ›Ύ)π‘’π‘ŽπœŽ π‘Žπ›Ύ)π‘’π‘ŽπœŽ ADJ iajs-3665 148 5 , , PUNCT iajs-3665 148 6 𝜎0 𝜎0 NOUN iajs-3665 148 7 ≀ ≀ PUNCT iajs-3665 148 8 𝜎 𝜎 PROPN iajs-3665 148 9 ≀ ≀ NOUN iajs-3665 148 10 𝜎1 𝜎1 ADJ iajs-3665 148 11 , , PUNCT iajs-3665 148 12 0 0 NUM iajs-3665 148 13 ≀ ≀ NUM iajs-3665 148 14 𝛾 𝛾 ADP iajs-3665 148 15 ≀ ≀ ADJ iajs-3665 148 16 2 2 NUM iajs-3665 148 17 , , PUNCT iajs-3665 148 18 ( ( PUNCT iajs-3665 148 19 21 21 NUM iajs-3665 148 20 ) ) PUNCT iajs-3665 148 21 with with ADP iajs-3665 148 22 the the DET iajs-3665 148 23 bcs bcs NOUN iajs-3665 148 24 π‘₯(0 π‘₯(0 PROPN iajs-3665 148 25 ) ) PUNCT iajs-3665 148 26 = = SYM iajs-3665 148 27 1 1 X iajs-3665 148 28 , , PUNCT iajs-3665 148 29 π‘₯(1 π‘₯(1 NOUN iajs-3665 148 30 ) ) PUNCT iajs-3665 148 31 = = PUNCT iajs-3665 149 1 βˆ’π‘’π‘Ž βˆ’π‘’π‘Ž NOUN iajs-3665 149 2 + + CCONJ iajs-3665 149 3 1 1 NUM iajs-3665 149 4 , , PUNCT iajs-3665 149 5 ( ( PUNCT iajs-3665 149 6 22 22 NUM iajs-3665 149 7 ) ) PUNCT iajs-3665 149 8 where where SCONJ iajs-3665 149 9 the the DET iajs-3665 149 10 problem problem NOUN iajs-3665 149 11 has have VERB iajs-3665 149 12 the the DET iajs-3665 149 13 exact exact ADJ iajs-3665 149 14 - - PUNCT iajs-3665 149 15 solution solution NOUN iajs-3665 149 16 ( ( PUNCT iajs-3665 149 17 es es NOUN iajs-3665 149 18 ) ) PUNCT iajs-3665 149 19 as as SCONJ iajs-3665 149 20 follows follow VERB iajs-3665 149 21 : : PUNCT iajs-3665 149 22 ( ( PUNCT iajs-3665 149 23 π‘₯(𝜎 π‘₯(𝜎 PROPN iajs-3665 149 24 ) ) PUNCT iajs-3665 149 25 , , PUNCT iajs-3665 149 26 𝑒(𝜎 𝑒(𝜎 PROPN iajs-3665 149 27 ) ) PUNCT iajs-3665 149 28 ) ) PUNCT iajs-3665 150 1 = = PUNCT iajs-3665 150 2 ( ( PUNCT iajs-3665 150 3 𝜎2 𝜎2 PROPN iajs-3665 150 4 βˆ’ βˆ’ PROPN iajs-3665 150 5 π‘’π‘ŽπœŽ π‘’π‘ŽπœŽ PROPN iajs-3665 150 6 , , PUNCT iajs-3665 150 7 𝜎2 𝜎2 PROPN iajs-3665 150 8 ) ) PUNCT iajs-3665 150 9 . . PUNCT iajs-3665 151 1 using use VERB iajs-3665 151 2 the the DET iajs-3665 151 3 b b NOUN iajs-3665 151 4 - - PUNCT iajs-3665 151 5 spline spline NOUN iajs-3665 151 6 approximation approximation NOUN iajs-3665 151 7 base base NOUN iajs-3665 151 8 , , PUNCT iajs-3665 151 9 then then ADV iajs-3665 151 10 , , PUNCT iajs-3665 151 11 the the DET iajs-3665 151 12 approximation approximation NOUN iajs-3665 151 13 formula formula NOUN iajs-3665 151 14 of of ADP iajs-3665 151 15 π‘₯(𝜎 π‘₯(𝜎 NOUN iajs-3665 151 16 ) ) PUNCT iajs-3665 151 17 as as ADP iajs-3665 151 18 in in ADP iajs-3665 151 19 the the DET iajs-3665 151 20 equation equation NOUN iajs-3665 151 21 ( ( PUNCT iajs-3665 151 22 19 19 NUM iajs-3665 151 23 ) ) PUNCT iajs-3665 151 24 . . PUNCT iajs-3665 152 1 then then ADV iajs-3665 152 2 , , PUNCT iajs-3665 152 3 we we PRON iajs-3665 152 4 have have VERB iajs-3665 152 5 𝑒(𝑑 𝑒(𝑑 NOUN iajs-3665 152 6 ) ) PUNCT iajs-3665 152 7 = = SYM iajs-3665 152 8 βˆ… βˆ… NOUN iajs-3665 152 9 ( ( PUNCT iajs-3665 152 10 𝑐0 𝑐0 NOUN iajs-3665 152 11 , , PUNCT iajs-3665 152 12 𝑐1 𝑐1 NOUN iajs-3665 152 13 , , PUNCT iajs-3665 152 14 𝑐2 𝑐2 NOUN iajs-3665 152 15 , , PUNCT iajs-3665 152 16 … … PUNCT iajs-3665 152 17 , , PUNCT iajs-3665 152 18 𝑐𝑛 𝑐𝑛 PROPN iajs-3665 152 19 , , PUNCT iajs-3665 152 20 βˆ…0(𝑑 βˆ…0(𝑑 NUM iajs-3665 152 21 ) ) PUNCT iajs-3665 152 22 , , PUNCT iajs-3665 152 23 βˆ…1(𝑑 βˆ…1(𝑑 NOUN iajs-3665 152 24 ) ) PUNCT iajs-3665 152 25 , , PUNCT iajs-3665 152 26 … … PUNCT iajs-3665 152 27 , , PUNCT iajs-3665 152 28 βˆ…π‘›(𝑑 βˆ…π‘›(𝑑 NOUN iajs-3665 152 29 ) ) PUNCT iajs-3665 152 30 ) ) PUNCT iajs-3665 152 31 . . PUNCT iajs-3665 153 1 substitute substitute VERB iajs-3665 153 2 the the DET iajs-3665 153 3 equation equation NOUN iajs-3665 153 4 ( ( PUNCT iajs-3665 153 5 19 19 NUM iajs-3665 153 6 ) ) PUNCT iajs-3665 153 7 in in ADP iajs-3665 153 8 equation equation NOUN iajs-3665 153 9 ( ( PUNCT iajs-3665 153 10 21 21 NUM iajs-3665 153 11 ) ) PUNCT iajs-3665 153 12 to to PART iajs-3665 153 13 obtain obtain VERB iajs-3665 153 14 the the DET iajs-3665 153 15 optimal optimal ADJ iajs-3665 153 16 parameters parameter NOUN iajs-3665 153 17 and and CCONJ iajs-3665 153 18 the the DET iajs-3665 153 19 non non ADJ iajs-3665 153 20 - - ADJ iajs-3665 153 21 freeparameter freeparameter ADJ iajs-3665 153 22 𝑇. 𝑇. PROPN iajs-3665 153 23 as as ADP iajs-3665 153 24 a a DET iajs-3665 153 25 result result NOUN iajs-3665 153 26 , , PUNCT iajs-3665 153 27 the the DET iajs-3665 153 28 problem problem NOUN iajs-3665 153 29 is be AUX iajs-3665 153 30 approximated approximate VERB iajs-3665 153 31 in in ADP iajs-3665 153 32 figure figure NOUN iajs-3665 153 33 2-(a 2-(a NUM iajs-3665 153 34 ) ) PUNCT iajs-3665 153 35 using use VERB iajs-3665 153 36 the the DET iajs-3665 153 37 hooke hooke NOUN iajs-3665 153 38 - - PUNCT iajs-3665 153 39 jeeves jeeve NOUN iajs-3665 153 40 technique technique NOUN iajs-3665 153 41 . . PUNCT iajs-3665 154 1 example example NOUN iajs-3665 154 2 2 2 NUM iajs-3665 154 3 . . X iajs-3665 154 4 consider consider VERB iajs-3665 154 5 the the DET iajs-3665 154 6 following follow VERB iajs-3665 154 7 non non ADJ iajs-3665 154 8 - - ADJ iajs-3665 154 9 free free ADJ iajs-3665 154 10 tt tt PROPN iajs-3665 154 11 problem problem NOUN iajs-3665 154 12 . . PUNCT iajs-3665 155 1 min min NOUN iajs-3665 155 2 π‘₯(𝜎),𝑒(𝜎),𝑇 π‘₯(𝜎),𝑒(𝜎),𝑇 PROPN iajs-3665 155 3 𝐽 𝐽 PROPN iajs-3665 155 4 ( ( PUNCT iajs-3665 155 5 𝜎 𝜎 PROPN iajs-3665 155 6 , , PUNCT iajs-3665 155 7 π‘₯(𝜎 π‘₯(𝜎 PROPN iajs-3665 155 8 ) ) PUNCT iajs-3665 155 9 , , PUNCT iajs-3665 155 10 𝑒(𝜎 𝑒(𝜎 PROPN iajs-3665 155 11 ) ) PUNCT iajs-3665 155 12 , , PUNCT iajs-3665 155 13 𝑇 𝑇 PROPN iajs-3665 155 14 ) ) PUNCT iajs-3665 155 15 = = SYM iajs-3665 155 16 min min PROPN iajs-3665 155 17 π‘₯(𝜎),𝑒(𝜎),𝑇 π‘₯(𝜎),𝑒(𝜎),𝑇 NUM iajs-3665 155 18 ∫ ∫ NOUN iajs-3665 155 19 ( ( PUNCT iajs-3665 155 20 πœŽπ‘’(𝜎 πœŽπ‘’(𝜎 NUM iajs-3665 155 21 ) ) PUNCT iajs-3665 155 22 βˆ’ βˆ’ NOUN iajs-3665 155 23 2π‘₯(𝜎 2π‘₯(𝜎 NUM iajs-3665 155 24 ) ) PUNCT iajs-3665 155 25 ) ) PUNCT iajs-3665 155 26 2 2 NUM iajs-3665 155 27 𝑇 𝑇 PROPN iajs-3665 155 28 0 0 NUM iajs-3665 155 29 π‘‘πœŽ π‘‘πœŽ PROPN iajs-3665 155 30 , , PUNCT iajs-3665 155 31 ( ( PUNCT iajs-3665 155 32 23 23 NUM iajs-3665 155 33 ) ) PUNCT iajs-3665 155 34 subject subject NOUN iajs-3665 155 35 to to ADP iajs-3665 155 36 condition condition NOUN iajs-3665 155 37 ihjpas ihjpa NOUN iajs-3665 155 38 . . PUNCT iajs-3665 155 39 2025 2025 NUM iajs-3665 155 40 , , PUNCT iajs-3665 155 41 38(2 38(2 NUM iajs-3665 155 42 ) ) PUNCT iajs-3665 155 43 345 345 NUM iajs-3665 155 44 οΏ½ οΏ½ PROPN iajs-3665 155 45 ̈ ̈ X iajs-3665 155 46 οΏ½ οΏ½ NOUN iajs-3665 155 47 (𝜎 (𝜎 X iajs-3665 155 48 ) ) PUNCT iajs-3665 156 1 + + CCONJ iajs-3665 156 2 οΏ½ οΏ½ PROPN iajs-3665 156 3 Μ‡ Μ‡ PROPN iajs-3665 156 4 οΏ½ οΏ½ PROPN iajs-3665 156 5 (𝜎 (𝜎 NOUN iajs-3665 156 6 ) ) PUNCT iajs-3665 156 7 = = SYM iajs-3665 156 8 1 1 NUM iajs-3665 156 9 βˆ’ βˆ’ PROPN iajs-3665 156 10 𝜎2 𝜎2 NOUN iajs-3665 156 11 + + CCONJ iajs-3665 156 12 𝑒(𝜎 𝑒(𝜎 PROPN iajs-3665 156 13 ) ) PUNCT iajs-3665 156 14 , , PUNCT iajs-3665 156 15 ( ( PUNCT iajs-3665 156 16 24 24 NUM iajs-3665 156 17 ) ) PUNCT iajs-3665 156 18 with with ADP iajs-3665 156 19 the the DET iajs-3665 156 20 ( ( PUNCT iajs-3665 156 21 bcs bcs NOUN iajs-3665 156 22 ) ) PUNCT iajs-3665 156 23 π‘₯(0 π‘₯(0 PROPN iajs-3665 156 24 ) ) PUNCT iajs-3665 156 25 = = SYM iajs-3665 156 26 0 0 NUM iajs-3665 156 27 , , PUNCT iajs-3665 156 28 π‘₯(1 π‘₯(1 PROPN iajs-3665 156 29 ) ) PUNCT iajs-3665 156 30 = = SYM iajs-3665 156 31 0 0 NUM iajs-3665 156 32 , , PUNCT iajs-3665 156 33 ( ( PUNCT iajs-3665 156 34 25 25 NUM iajs-3665 156 35 ) ) PUNCT iajs-3665 156 36 where where SCONJ iajs-3665 156 37 the the DET iajs-3665 156 38 es es NOUN iajs-3665 156 39 is be AUX iajs-3665 156 40 has have VERB iajs-3665 156 41 the the DET iajs-3665 156 42 form form NOUN iajs-3665 156 43 ( ( PUNCT iajs-3665 156 44 π‘₯(𝜎 π‘₯(𝜎 PROPN iajs-3665 156 45 ) ) PUNCT iajs-3665 156 46 , , PUNCT iajs-3665 156 47 𝑒(𝜎 𝑒(𝜎 PROPN iajs-3665 156 48 ) ) PUNCT iajs-3665 156 49 ) ) PUNCT iajs-3665 157 1 = = SYM iajs-3665 157 2 ( ( PUNCT iajs-3665 157 3 𝜎(βˆ’πœŽ 𝜎(βˆ’πœŽ X iajs-3665 157 4 + + NUM iajs-3665 157 5 1 1 NUM iajs-3665 157 6 ) ) PUNCT iajs-3665 157 7 , , PUNCT iajs-3665 157 8 2 2 NUM iajs-3665 157 9 + + NUM iajs-3665 157 10 2𝜎 2𝜎 NUM iajs-3665 157 11 βˆ’ βˆ’ PROPN iajs-3665 157 12 𝜎2 𝜎2 NOUN iajs-3665 157 13 ) ) PUNCT iajs-3665 157 14 . . PUNCT iajs-3665 158 1 from from ADP iajs-3665 158 2 the the DET iajs-3665 158 3 approximated approximate VERB iajs-3665 158 4 solution solution NOUN iajs-3665 158 5 which which PRON iajs-3665 158 6 expressed express VERB iajs-3665 158 7 in in ADP iajs-3665 158 8 equation equation NOUN iajs-3665 158 9 ( ( PUNCT iajs-3665 158 10 11 11 NUM iajs-3665 158 11 ) ) PUNCT iajs-3665 158 12 , , PUNCT iajs-3665 158 13 we we PRON iajs-3665 158 14 get get VERB iajs-3665 158 15 𝑒(𝑑 𝑒(𝑑 NOUN iajs-3665 158 16 ) ) PUNCT iajs-3665 158 17 = = SYM iajs-3665 158 18 βˆ… βˆ… NOUN iajs-3665 158 19 ( ( PUNCT iajs-3665 158 20 𝑐0 𝑐0 NOUN iajs-3665 158 21 , , PUNCT iajs-3665 158 22 𝑐1 𝑐1 NOUN iajs-3665 158 23 , , PUNCT iajs-3665 158 24 𝑐2 𝑐2 NOUN iajs-3665 158 25 , , PUNCT iajs-3665 158 26 … … PUNCT iajs-3665 158 27 , , PUNCT iajs-3665 158 28 𝑐𝑛 𝑐𝑛 PROPN iajs-3665 158 29 , , PUNCT iajs-3665 158 30 βˆ…0(𝑑 βˆ…0(𝑑 NUM iajs-3665 158 31 ) ) PUNCT iajs-3665 158 32 , , PUNCT iajs-3665 158 33 βˆ…1(𝑑 βˆ…1(𝑑 NOUN iajs-3665 158 34 ) ) PUNCT iajs-3665 158 35 , , PUNCT iajs-3665 158 36 … … PUNCT iajs-3665 158 37 , , PUNCT iajs-3665 158 38 βˆ…π‘›(𝑑 βˆ…π‘›(𝑑 NOUN iajs-3665 158 39 ) ) PUNCT iajs-3665 158 40 ) ) PUNCT iajs-3665 158 41 . . PUNCT iajs-3665 159 1 substitute substitute NOUN iajs-3665 159 2 equation equation NOUN iajs-3665 159 3 ( ( PUNCT iajs-3665 159 4 19 19 NUM iajs-3665 159 5 ) ) PUNCT iajs-3665 159 6 in in ADP iajs-3665 159 7 equation equation NOUN iajs-3665 159 8 ( ( PUNCT iajs-3665 159 9 23 23 NUM iajs-3665 159 10 ) ) PUNCT iajs-3665 159 11 to to PART iajs-3665 159 12 obtain obtain VERB iajs-3665 159 13 the the DET iajs-3665 159 14 parameter parameter NOUN iajs-3665 159 15 𝑇 𝑇 PROPN iajs-3665 159 16 = = PROPN iajs-3665 159 17 0.997 0.997 NUM iajs-3665 159 18 . . PUNCT iajs-3665 160 1 hence hence ADV iajs-3665 160 2 , , PUNCT iajs-3665 160 3 π‘₯(𝜎 π‘₯(𝜎 NUM iajs-3665 160 4 ) ) PUNCT iajs-3665 160 5 = = PUNCT iajs-3665 160 6 𝜎(βˆ’πœŽ 𝜎(βˆ’πœŽ X iajs-3665 160 7 + + NOUN iajs-3665 160 8 2 2 X iajs-3665 160 9 ) ) PUNCT iajs-3665 160 10 and and CCONJ iajs-3665 160 11 then then ADV iajs-3665 160 12 , , PUNCT iajs-3665 160 13 plotted plot VERB iajs-3665 160 14 in in ADP iajs-3665 160 15 figure figure NOUN iajs-3665 160 16 2-(b 2-(b NUM iajs-3665 160 17 ) ) PUNCT iajs-3665 160 18 . . PUNCT iajs-3665 160 19 example example NOUN iajs-3665 161 1 3 3 X iajs-3665 161 2 . . X iajs-3665 161 3 consider consider VERB iajs-3665 161 4 the the DET iajs-3665 161 5 following follow VERB iajs-3665 161 6 focss focss PROPN iajs-3665 161 7 with with ADP iajs-3665 161 8 non non ADJ iajs-3665 161 9 - - ADJ iajs-3665 161 10 free free ADJ iajs-3665 161 11 tt tt PROPN iajs-3665 161 12 induced induce VERB iajs-3665 161 13 by by ADP iajs-3665 161 14 al al PROPN iajs-3665 161 15 - - PUNCT iajs-3665 161 16 shaher shaher PROPN iajs-3665 161 17 et et PROPN iajs-3665 161 18 al al PROPN iajs-3665 161 19 . . PROPN iajs-3665 162 1 ( ( PUNCT iajs-3665 162 2 12 12 NUM iajs-3665 162 3 ) ) PUNCT iajs-3665 162 4 min min NOUN iajs-3665 162 5 π‘₯(𝜎),𝑒(𝜎),𝑇 π‘₯(𝜎),𝑒(𝜎),𝑇 PROPN iajs-3665 162 6 𝐽 𝐽 PROPN iajs-3665 162 7 ( ( PUNCT iajs-3665 162 8 𝜎 𝜎 PROPN iajs-3665 162 9 , , PUNCT iajs-3665 162 10 π‘₯(𝜎 π‘₯(𝜎 PROPN iajs-3665 162 11 ) ) PUNCT iajs-3665 162 12 , , PUNCT iajs-3665 162 13 𝑒(𝜎 𝑒(𝜎 PROPN iajs-3665 162 14 ) ) PUNCT iajs-3665 162 15 , , PUNCT iajs-3665 162 16 𝑇 𝑇 PROPN iajs-3665 162 17 ) ) PUNCT iajs-3665 162 18 = = SYM iajs-3665 162 19 min min PROPN iajs-3665 162 20 π‘₯(𝜎),𝑒(𝜎),𝑇 π‘₯(𝜎),𝑒(𝜎),𝑇 NUM iajs-3665 162 21 ∫ ∫ NOUN iajs-3665 162 22 ( ( PUNCT iajs-3665 162 23 πœŽπ‘’(𝜎 πœŽπ‘’(𝜎 NUM iajs-3665 162 24 ) ) PUNCT iajs-3665 162 25 βˆ’ βˆ’ PROPN iajs-3665 162 26 ( ( PUNCT iajs-3665 162 27 𝛾 𝛾 NOUN iajs-3665 162 28 + + X iajs-3665 162 29 2)π‘₯(𝜎 2)π‘₯(𝜎 NUM iajs-3665 162 30 ) ) PUNCT iajs-3665 162 31 ) ) PUNCT iajs-3665 162 32 2 2 NUM iajs-3665 162 33 𝑇 𝑇 PROPN iajs-3665 162 34 0 0 NUM iajs-3665 162 35 π‘‘πœŽ π‘‘πœŽ PROPN iajs-3665 162 36 , , PUNCT iajs-3665 162 37 ( ( PUNCT iajs-3665 162 38 26 26 NUM iajs-3665 162 39 ) ) PUNCT iajs-3665 162 40 subject subject ADJ iajs-3665 162 41 to to ADP iajs-3665 162 42 : : PUNCT iajs-3665 162 43 𝐷𝜎 𝐷𝜎 ADP iajs-3665 162 44 𝛾 𝛾 ADP iajs-3665 162 45 π‘₯(𝜎 π‘₯(𝜎 NUM iajs-3665 162 46 ) ) PUNCT iajs-3665 162 47 + + CCONJ iajs-3665 162 48 οΏ½ οΏ½ PROPN iajs-3665 162 49 Μ‡ Μ‡ PROPN iajs-3665 162 50 οΏ½ οΏ½ PROPN iajs-3665 162 51 (𝜎 (𝜎 PROPN iajs-3665 162 52 ) ) PUNCT iajs-3665 162 53 = = SYM iajs-3665 162 54 𝜎2 𝜎2 PROPN iajs-3665 162 55 + + CCONJ iajs-3665 162 56 𝑒(𝜎 𝑒(𝜎 PROPN iajs-3665 162 57 ) ) PUNCT iajs-3665 162 58 , , PUNCT iajs-3665 162 59 ( ( PUNCT iajs-3665 162 60 27 27 NUM iajs-3665 162 61 ) ) PUNCT iajs-3665 162 62 with with ADP iajs-3665 162 63 bcs bcs NOUN iajs-3665 162 64 π‘₯(0 π‘₯(0 PROPN iajs-3665 162 65 ) ) PUNCT iajs-3665 162 66 = = SYM iajs-3665 162 67 0 0 NUM iajs-3665 162 68 , , PUNCT iajs-3665 162 69 π‘₯(𝑇 π‘₯(𝑇 NOUN iajs-3665 162 70 ) ) PUNCT iajs-3665 162 71 = = SYM iajs-3665 162 72 1 1 X iajs-3665 162 73 . . PUNCT iajs-3665 163 1 ( ( PUNCT iajs-3665 163 2 28 28 NUM iajs-3665 163 3 ) ) PUNCT iajs-3665 163 4 we we PRON iajs-3665 163 5 get get VERB iajs-3665 163 6 the the DET iajs-3665 163 7 approximation approximation NOUN iajs-3665 163 8 of of ADP iajs-3665 163 9 π‘₯(𝜏 π‘₯(𝜏 PROPN iajs-3665 163 10 ) ) PUNCT iajs-3665 163 11 as as ADP iajs-3665 163 12 in in ADP iajs-3665 163 13 equation equation NOUN iajs-3665 163 14 ( ( PUNCT iajs-3665 163 15 19 19 NUM iajs-3665 163 16 ) ) PUNCT iajs-3665 163 17 using use VERB iajs-3665 163 18 the the DET iajs-3665 163 19 b b PROPN iajs-3665 163 20 - - PUNCT iajs-3665 163 21 spline spline NOUN iajs-3665 163 22 approximation approximation NOUN iajs-3665 163 23 base base NOUN iajs-3665 163 24 , , PUNCT iajs-3665 163 25 which which PRON iajs-3665 163 26 satisfied satisfy VERB iajs-3665 163 27 the the DET iajs-3665 163 28 bcs bcs NOUN iajs-3665 163 29 in in ADP iajs-3665 163 30 equation equation NOUN iajs-3665 163 31 ( ( PUNCT iajs-3665 163 32 28 28 NUM iajs-3665 163 33 ) ) PUNCT iajs-3665 163 34 then then ADV iajs-3665 163 35 , , PUNCT iajs-3665 163 36 we we PRON iajs-3665 163 37 get get VERB iajs-3665 163 38 the the DET iajs-3665 163 39 function function NOUN iajs-3665 163 40 𝑒(𝑑 𝑒(𝑑 NOUN iajs-3665 163 41 ) ) PUNCT iajs-3665 163 42 = = SYM iajs-3665 163 43 βˆ… βˆ… NOUN iajs-3665 163 44 ( ( PUNCT iajs-3665 163 45 𝑐0 𝑐0 NOUN iajs-3665 163 46 , , PUNCT iajs-3665 163 47 𝑐1 𝑐1 NOUN iajs-3665 163 48 , , PUNCT iajs-3665 163 49 𝑐2 𝑐2 NOUN iajs-3665 163 50 , , PUNCT iajs-3665 163 51 … … PUNCT iajs-3665 163 52 , , PUNCT iajs-3665 163 53 𝑐𝑛 𝑐𝑛 PROPN iajs-3665 163 54 , , PUNCT iajs-3665 163 55 βˆ…0(𝑑 βˆ…0(𝑑 NUM iajs-3665 163 56 ) ) PUNCT iajs-3665 163 57 , , PUNCT iajs-3665 163 58 βˆ…1(𝑑 βˆ…1(𝑑 NOUN iajs-3665 163 59 ) ) PUNCT iajs-3665 163 60 , , PUNCT iajs-3665 163 61 … … PUNCT iajs-3665 163 62 , , PUNCT iajs-3665 163 63 βˆ…π‘›(𝑑 βˆ…π‘›(𝑑 NOUN iajs-3665 163 64 ) ) PUNCT iajs-3665 163 65 ) ) PUNCT iajs-3665 163 66 . . PUNCT iajs-3665 164 1 to to PART iajs-3665 164 2 identify identify VERB iajs-3665 164 3 the the DET iajs-3665 164 4 optimal optimal ADJ iajs-3665 164 5 solution solution NOUN iajs-3665 164 6 of of ADP iajs-3665 164 7 this this DET iajs-3665 164 8 problem problem NOUN iajs-3665 164 9 using use VERB iajs-3665 164 10 the the DET iajs-3665 164 11 hooke hooke NOUN iajs-3665 164 12 - - PUNCT iajs-3665 164 13 jeeves jeeve NOUN iajs-3665 164 14 technique technique NOUN iajs-3665 164 15 as as SCONJ iajs-3665 164 16 shown show VERB iajs-3665 164 17 in in ADP iajs-3665 164 18 figure figure NOUN iajs-3665 164 19 2-(c 2-(c NUM iajs-3665 164 20 ) ) PUNCT iajs-3665 164 21 , , PUNCT iajs-3665 164 22 substitute substitute NOUN iajs-3665 164 23 equation equation NOUN iajs-3665 164 24 ( ( PUNCT iajs-3665 164 25 19 19 NUM iajs-3665 164 26 ) ) PUNCT iajs-3665 164 27 for for ADP iajs-3665 164 28 minimizing minimize VERB iajs-3665 164 29 the the DET iajs-3665 164 30 challenge challenge NOUN iajs-3665 164 31 in in ADP iajs-3665 164 32 equation equation NOUN iajs-3665 164 33 ( ( PUNCT iajs-3665 164 34 26 26 NUM iajs-3665 164 35 ) ) PUNCT iajs-3665 164 36 . . PUNCT iajs-3665 165 1 ( ( PUNCT iajs-3665 165 2 a a X iajs-3665 165 3 ) ) PUNCT iajs-3665 165 4 ( ( PUNCT iajs-3665 165 5 b b X iajs-3665 165 6 ) ) PUNCT iajs-3665 165 7 ( ( PUNCT iajs-3665 165 8 c c X iajs-3665 165 9 ) ) PUNCT iajs-3665 165 10 figure figure NOUN iajs-3665 165 11 2 2 NUM iajs-3665 165 12 . . PUNCT iajs-3665 166 1 a a DET iajs-3665 166 2 numerical numerical ADJ iajs-3665 166 3 comparison comparison NOUN iajs-3665 166 4 with with ADP iajs-3665 166 5 the the DET iajs-3665 166 6 exact exact ADJ iajs-3665 166 7 - - PUNCT iajs-3665 166 8 solutions solution NOUN iajs-3665 166 9 for for ADP iajs-3665 166 10 the the DET iajs-3665 166 11 examples example NOUN iajs-3665 166 12 ( ( PUNCT iajs-3665 166 13 a a X iajs-3665 166 14 ) ) PUNCT iajs-3665 166 15 1 1 NUM iajs-3665 166 16 , , PUNCT iajs-3665 166 17 ( ( PUNCT iajs-3665 166 18 b b NOUN iajs-3665 166 19 ) ) PUNCT iajs-3665 166 20 2 2 NUM iajs-3665 166 21 , , PUNCT iajs-3665 166 22 and and CCONJ iajs-3665 166 23 ( ( PUNCT iajs-3665 166 24 c c NOUN iajs-3665 166 25 ) ) PUNCT iajs-3665 166 26 3 3 NUM iajs-3665 166 27 ihjpas ihjpa NOUN iajs-3665 166 28 . . PUNCT iajs-3665 167 1 2025 2025 NUM iajs-3665 167 2 , , PUNCT iajs-3665 167 3 38(2 38(2 NUM iajs-3665 167 4 ) ) PUNCT iajs-3665 167 5 346 346 NUM iajs-3665 167 6 5 5 NUM iajs-3665 167 7 . . PUNCT iajs-3665 167 8 discussion discussion NOUN iajs-3665 167 9 from from ADP iajs-3665 167 10 the the DET iajs-3665 167 11 numerical numerical ADJ iajs-3665 167 12 comparison comparison NOUN iajs-3665 167 13 with with ADP iajs-3665 167 14 the the DET iajs-3665 167 15 exact exact ADJ iajs-3665 167 16 solutions solution NOUN iajs-3665 167 17 of of ADP iajs-3665 167 18 the the DET iajs-3665 167 19 examples example NOUN iajs-3665 167 20 ( ( PUNCT iajs-3665 167 21 a)1 a)1 PROPN iajs-3665 167 22 , , PUNCT iajs-3665 167 23 ( ( PUNCT iajs-3665 167 24 b b X iajs-3665 167 25 ) ) PUNCT iajs-3665 167 26 2 2 NUM iajs-3665 167 27 , , PUNCT iajs-3665 167 28 and and CCONJ iajs-3665 167 29 ( ( PUNCT iajs-3665 167 30 c c X iajs-3665 167 31 ) ) PUNCT iajs-3665 167 32 3 3 NUM iajs-3665 167 33 in in ADP iajs-3665 167 34 figure figure NOUN iajs-3665 167 35 2 2 NUM iajs-3665 167 36 ( ( PUNCT iajs-3665 167 37 a a DET iajs-3665 167 38 - - PUNCT iajs-3665 167 39 c c NOUN iajs-3665 167 40 ) ) PUNCT iajs-3665 167 41 respectively respectively ADV iajs-3665 167 42 , , PUNCT iajs-3665 167 43 we we PRON iajs-3665 167 44 can can AUX iajs-3665 167 45 conclude conclude VERB iajs-3665 167 46 that that SCONJ iajs-3665 167 47 the the DET iajs-3665 167 48 numerical numerical ADJ iajs-3665 167 49 results result NOUN iajs-3665 167 50 of of ADP iajs-3665 167 51 the the DET iajs-3665 167 52 implementations implementation NOUN iajs-3665 167 53 show show VERB iajs-3665 167 54 that that SCONJ iajs-3665 167 55 the the DET iajs-3665 167 56 proposed propose VERB iajs-3665 167 57 approach approach NOUN iajs-3665 167 58 can can AUX iajs-3665 167 59 be be AUX iajs-3665 167 60 used use VERB iajs-3665 167 61 to to PART iajs-3665 167 62 solve solve VERB iajs-3665 167 63 both both DET iajs-3665 167 64 types type NOUN iajs-3665 167 65 of of ADP iajs-3665 167 66 this this DET iajs-3665 167 67 problem problem NOUN iajs-3665 167 68 with with ADP iajs-3665 167 69 freeor freeor PROPN iajs-3665 167 70 non non ADJ iajs-3665 167 71 - - ADJ iajs-3665 167 72 free free ADJ iajs-3665 167 73 terminal terminal ADJ iajs-3665 167 74 time time NOUN iajs-3665 167 75 . . PUNCT iajs-3665 168 1 furthermore furthermore ADV iajs-3665 168 2 , , PUNCT iajs-3665 168 3 from from ADP iajs-3665 168 4 the the DET iajs-3665 168 5 numerical numerical ADJ iajs-3665 168 6 results result NOUN iajs-3665 168 7 , , PUNCT iajs-3665 168 8 there there PRON iajs-3665 168 9 is be VERB iajs-3665 168 10 a a DET iajs-3665 168 11 high high ADJ iajs-3665 168 12 level level NOUN iajs-3665 168 13 of of ADP iajs-3665 168 14 agreement agreement NOUN iajs-3665 168 15 between between ADP iajs-3665 168 16 numerical numerical ADJ iajs-3665 168 17 and and CCONJ iajs-3665 168 18 analytical analytical ADJ iajs-3665 168 19 solutions solution NOUN iajs-3665 168 20 . . PUNCT iajs-3665 169 1 as as ADP iajs-3665 169 2 a a DET iajs-3665 169 3 result result NOUN iajs-3665 169 4 , , PUNCT iajs-3665 169 5 the the DET iajs-3665 169 6 new new ADJ iajs-3665 169 7 strategy strategy NOUN iajs-3665 169 8 is be AUX iajs-3665 169 9 effective effective ADJ iajs-3665 169 10 , , PUNCT iajs-3665 169 11 and and CCONJ iajs-3665 169 12 the the DET iajs-3665 169 13 results result NOUN iajs-3665 169 14 are be AUX iajs-3665 169 15 promising promise VERB iajs-3665 169 16 . . PUNCT iajs-3665 170 1 in in ADP iajs-3665 170 2 the the DET iajs-3665 170 3 future future NOUN iajs-3665 170 4 , , PUNCT iajs-3665 170 5 the the DET iajs-3665 170 6 scope scope NOUN iajs-3665 170 7 underlying underlie VERB iajs-3665 170 8 this this DET iajs-3665 170 9 research research NOUN iajs-3665 170 10 could could AUX iajs-3665 170 11 be be AUX iajs-3665 170 12 broadened broaden VERB iajs-3665 170 13 in in ADP iajs-3665 170 14 new new ADJ iajs-3665 170 15 ways way NOUN iajs-3665 170 16 , , PUNCT iajs-3665 170 17 for for ADP iajs-3665 170 18 as as ADP iajs-3665 170 19 improved improve VERB iajs-3665 170 20 numerical numerical ADJ iajs-3665 170 21 methods method NOUN iajs-3665 170 22 . . PUNCT iajs-3665 171 1 6 6 X iajs-3665 171 2 . . X iajs-3665 171 3 conclusion conclusion VERB iajs-3665 171 4 the the DET iajs-3665 171 5 main main ADJ iajs-3665 171 6 objective objective NOUN iajs-3665 171 7 of of ADP iajs-3665 171 8 this this DET iajs-3665 171 9 study study NOUN iajs-3665 171 10 is be AUX iajs-3665 171 11 to to PART iajs-3665 171 12 provide provide VERB iajs-3665 171 13 a a DET iajs-3665 171 14 numerical numerical ADJ iajs-3665 171 15 solution solution NOUN iajs-3665 171 16 for for ADP iajs-3665 171 17 dealing deal VERB iajs-3665 171 18 with with ADP iajs-3665 171 19 two two NUM iajs-3665 171 20 situations situation NOUN iajs-3665 171 21 of of ADP iajs-3665 171 22 focps focp NOUN iajs-3665 171 23 with with ADP iajs-3665 171 24 freeand freeand PROPN iajs-3665 171 25 non non ADJ iajs-3665 171 26 - - ADJ iajs-3665 171 27 free free ADJ iajs-3665 171 28 terminal terminal ADJ iajs-3665 171 29 time time NOUN iajs-3665 171 30 utilizing utilize VERB iajs-3665 171 31 the the DET iajs-3665 171 32 collocation collocation NOUN iajs-3665 171 33 method method NOUN iajs-3665 171 34 using use VERB iajs-3665 171 35 a a DET iajs-3665 171 36 b b NOUN iajs-3665 171 37 - - PUNCT iajs-3665 171 38 cubic cubic ADJ iajs-3665 171 39 spline spline NOUN iajs-3665 171 40 base base NOUN iajs-3665 171 41 . . PUNCT iajs-3665 172 1 accordingly accordingly ADV iajs-3665 172 2 , , PUNCT iajs-3665 172 3 we we PRON iajs-3665 172 4 aim aim VERB iajs-3665 172 5 to to PART iajs-3665 172 6 study study VERB iajs-3665 172 7 the the DET iajs-3665 172 8 findings finding NOUN iajs-3665 172 9 of of ADP iajs-3665 172 10 the the DET iajs-3665 172 11 solutions solution NOUN iajs-3665 172 12 of of ADP iajs-3665 172 13 the the DET iajs-3665 172 14 fractionalorder fractionalorder NOUN iajs-3665 172 15 systems system NOUN iajs-3665 172 16 of of ADP iajs-3665 172 17 optimal optimal ADJ iajs-3665 172 18 control control NOUN iajs-3665 172 19 using use VERB iajs-3665 172 20 a a DET iajs-3665 172 21 b b NOUN iajs-3665 172 22 - - PUNCT iajs-3665 172 23 cubic cubic ADJ iajs-3665 172 24 spline spline NOUN iajs-3665 172 25 base base NOUN iajs-3665 172 26 . . PUNCT iajs-3665 173 1 these these DET iajs-3665 173 2 numerical numerical ADJ iajs-3665 173 3 solutions solution NOUN iajs-3665 173 4 for for ADP iajs-3665 173 5 ( ( PUNCT iajs-3665 173 6 focps focp NOUN iajs-3665 173 7 ) ) PUNCT iajs-3665 173 8 are be AUX iajs-3665 173 9 very very ADV iajs-3665 173 10 important important ADJ iajs-3665 173 11 in in ADP iajs-3665 173 12 the the DET iajs-3665 173 13 context context NOUN iajs-3665 173 14 of of ADP iajs-3665 173 15 science science NOUN iajs-3665 173 16 , , PUNCT iajs-3665 173 17 engineering engineering NOUN iajs-3665 173 18 , , PUNCT iajs-3665 173 19 and and CCONJ iajs-3665 173 20 operations operation NOUN iajs-3665 173 21 research research NOUN iajs-3665 173 22 due due ADP iajs-3665 173 23 to to ADP iajs-3665 173 24 the the DET iajs-3665 173 25 various various ADJ iajs-3665 173 26 applications application NOUN iajs-3665 173 27 in in ADP iajs-3665 173 28 this this DET iajs-3665 173 29 area area NOUN iajs-3665 173 30 . . PUNCT iajs-3665 174 1 in in ADP iajs-3665 174 2 addition addition NOUN iajs-3665 174 3 , , PUNCT iajs-3665 174 4 we we PRON iajs-3665 174 5 try try VERB iajs-3665 174 6 to to PART iajs-3665 174 7 compare compare VERB iajs-3665 174 8 the the DET iajs-3665 174 9 numerical numerical ADJ iajs-3665 174 10 results result NOUN iajs-3665 174 11 obtained obtain VERB iajs-3665 174 12 by by ADP iajs-3665 174 13 the the DET iajs-3665 174 14 proposed propose VERB iajs-3665 174 15 method method NOUN iajs-3665 174 16 using use VERB iajs-3665 174 17 a a DET iajs-3665 174 18 b b NOUN iajs-3665 174 19 - - PUNCT iajs-3665 174 20 cubic cubic ADJ iajs-3665 174 21 spline spline NOUN iajs-3665 174 22 base base NOUN iajs-3665 174 23 to to ADP iajs-3665 174 24 the the DET iajs-3665 174 25 exact exact ADJ iajs-3665 174 26 solutions solution NOUN iajs-3665 174 27 for for ADP iajs-3665 174 28 three three NUM iajs-3665 174 29 test test NOUN iajs-3665 174 30 problems problem NOUN iajs-3665 174 31 in in ADP iajs-3665 174 32 two two NUM iajs-3665 174 33 types type NOUN iajs-3665 174 34 of of ADP iajs-3665 174 35 free free ADJ iajs-3665 174 36 and and CCONJ iajs-3665 174 37 non non ADJ iajs-3665 174 38 - - ADJ iajs-3665 174 39 free free ADJ iajs-3665 174 40 terminal terminal ADJ iajs-3665 174 41 time time NOUN iajs-3665 174 42 . . PUNCT iajs-3665 175 1 acknowledgment acknowledgment NOUN iajs-3665 175 2 our our PRON iajs-3665 175 3 researcher researcher NOUN iajs-3665 175 4 extends extend VERB iajs-3665 175 5 his his PRON iajs-3665 175 6 sincere sincere ADJ iajs-3665 175 7 thanks thank NOUN iajs-3665 175 8 to to ADP iajs-3665 175 9 the the DET iajs-3665 175 10 editor editor NOUN iajs-3665 175 11 and and CCONJ iajs-3665 175 12 members member NOUN iajs-3665 175 13 of of ADP iajs-3665 175 14 the the DET iajs-3665 175 15 preparatory preparatory PROPN iajs-3665 175 16 committee committee NOUN iajs-3665 175 17 of of ADP iajs-3665 175 18 the the DET iajs-3665 175 19 ibn ibn PROPN iajs-3665 175 20 al al PROPN iajs-3665 175 21 - - PUNCT iajs-3665 175 22 haitham haitham PROPN iajs-3665 175 23 journal journal PROPN iajs-3665 175 24 of of ADP iajs-3665 175 25 pure pure ADJ iajs-3665 175 26 and and CCONJ iajs-3665 175 27 applied applied ADJ iajs-3665 175 28 sciences science NOUN iajs-3665 175 29 . . 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PUNCT iajs-3665 178 7 partial partial ADJ iajs-3665 178 8 differential differential ADJ iajs-3665 178 9 equations equation NOUN iajs-3665 178 10 for for ADP iajs-3665 178 11 scientists scientist NOUN iajs-3665 178 12 and and CCONJ iajs-3665 178 13 engineers engineer NOUN iajs-3665 178 14 . . PUNCT iajs-3665 179 1 courier courier PROPN iajs-3665 179 2 dover dover PROPN iajs-3665 179 3 publications publication NOUN iajs-3665 179 4 ; ; PUNCT iajs-3665 179 5 2012 2012 NUM iajs-3665 179 6 . . PUNCT iajs-3665 180 1 2 2 X iajs-3665 180 2 . . 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