ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Hai tham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Solution of Some Application of System of Ordinary Initial Value Problems Using Osculatory Interpolation Technique K. M. M. Al-Abrahemee Departme nt of Mathematics, College of Education ,Unive rsity of Al- Qadysea . Received in: 25 May 2011 Accepte d in: 7 December 2011 Abstract The aim of this p ap er is t o find a new method for solving a sy st em of linear initial value p roblems of ordinary differential equation using app roximation technique by two-p oint osculatory interp olation with the fit equal numbers of derivatives at t he end p oints of an interval [0, 1] and co mpared the results with conventional methods and is shown to be that seems t o converge fast er and more accurately than the conventional methods. Key words : Initial value problems , Ap p roximation , Osculatory interp olation 1- Introduction Sy st ems of ordinary differential equations (ODEs) arise in mathematical models throughout science and engineering. When an exp licit condition (or conditions) that a solution must satisfy is sp ecified at one value of the independent variable, usually its lower bound, this is referred to as an initial value p roblem (IVP) and a sy st em of ordinary differential equations is a sy st em of equations relating several unknown functions y i(x) of an indep endent variable x, some of the derivatives of the y i(x), and p ossibly x itself. [ 1] .Initial-value p roblems for sy st ems of differential equations p ermeate many areas of mathematics: such p roblems arise naturally in modelling the evolution of dy namical p rocesses in economics, engineering, and the p hy sical , biological sciences [2]. In this p aper we introduce reactor p roblem which introduced in [3] Kehoe and Butt have st udied the kinetics of benzene hy drogenation on asup p orted Ni/kieselguhr cataly st . In the p resence of a large excess of hy drogen, the reaction is p seudo-first-order at temp eratures below 200°C with the rate given by -r= PH2k0K0T exp [ (-Q-Ea)/RgT )] CB mol /( g cataly st s) where Rg = gas constant, 1.987 cal/(mole'K) - Q - Ea = 2700 cal/mole P H2 = hydro gen part ial pressure (t orr) ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Hai tham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 ko = 4.22 mole/(gcat·s·t orr) Ko = 2.63 X 10-6 cm3/(mole'K) T = absolute temperature (K) CB = concentration of benzene (mole/cm3). Price and Butt [4] st udied this reaction in a tubular reactor. If t he reactor is assumed to be isothermal, we can calculate the dimensionless concentration p rofile of benzene in their reactor given p lug flow op eration in the absence of inter- and intrap hase gradients. Using a ty p ical run, PH2 = 685 torr PB = density of the reactor bed, 1.2 gcat/cm3 e = contact t ime, 0.226 s T = 150°C And if we now consider the reactor to be adiabatic instead of isothermal, then an energy balance must accomp any the material balance. Formulate the sy st em of governing differential equations. The data of this p roblem Cp = 12.17 X 104 J/(kmole'°C) -  .Hr = 2.09 x 108 J/kmole )150(423, 00 0 * CKT T T T  . For t he "short" reactor, . We have sy st em of initial value problem y Tdx dy ] 21.3 exp[1744.0 *  (material b alance) y Tdx dT ] 21.3 exp[06984.0 * *  (ener gy balance) with I.C y (0) =1 , T * (0) =1 2- Problem defini tion In this section we can exp lain the way through the app lication of this sy st em, of initial value problem: y Tdx dy ] 21.3 exp[1744.0 *  ………………(1) y Tdx dT ] 21.3 exp[06984.0 * *  with I.C y (0) =1 , T * (0) =1 In this p aper we are p articularly concerned with fitt ing function values and derivatives at the two end p oints of a finite interval, say [0,1],wherein a useful and succinct way of writing a osculatory interp olant P2n+1(x) of degree 2n + 1 was given for examp le by Phillips [5] as: ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Hai tham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 P2n+1(x)=   n j 0 {y )( j (0) q j (x)+(-1) j y )( j (1) q j (1-x)}………….(2) q j (x) =( x j /j!)(1-x) 1n    jn s 0        s sn x s = Q j (x)/j! ...………..(3) so t hat (2) with (3) satisfies y )( r (0)= )( 12 r nP  (0) , y )( r (1)= )( 12 r nP  (1) , r=0,1,2,…,n. We can be write the equation (2) directly in terms of the Tay lor coefficients ai and bi about x = 0 and x = 1 resp ectively, as P2n+1(x)=   n j 0 { a j Q j (x) + (-1) j b j Q j (1-x) }. ….(4) The simp le idea of this p aper is t o replace y (x) in problem (1) by a P2n+1 in equation (3) .The first st ep therefore is t o construct t he P2n+1 . To do this we need the Tay lor coefficients of y (x) and T * (x) resp ectively about x=0 ………… (5a)    2i i i xa y (x)= a0 +a1x + Where y(0) =a0 , y ' (0) =a1 . …… y (j ) (0)/i! =ai i=2.3,….. And ………… (5b)    2i i i xb T * (x)= b0 +b1x + Where T * (0) =b0 , T *' (0) =b1 . …… T * (j ) (0)/i! =bi i=2.3,….. Also we need the Taylor coefficients of y (x) and T * (x) resp ectively about x=1 …………( 6a)     2 )1( i i i xc y (x)= c0 +c1(x-1) + Where y1(1) =c0 , y ' (1) =c1 . …… y (j ) (1)/i! =ci i=2.3,….. ………… (6b)     2 )1( i i i xd T * (x)= d0 +d1(x-1) + Where T * (1) =d0 , T *' (1) =di . …… T * (j ) (1)/i! =di i=2.3,….. Then we simp ly insert the series forms in (5a) in to equation (1) and equ ate coefficients of x to obtain a1 , then derive equation (1) and insert t he series in to (5a) and equ ate coefficients of x to obtain a2 and soon to obtain a3 , a4 ….. Then equation (5b) in the same mann er to obt ain b2, b3,….. and simp ly insert the series forms in (6a) in to equation (1) and equate coefficients of(x- 1) to obtain c1 , then derive equation (1) and insert t he series in to (6a) and equ ate coefficients of x to obtain c2 and soon to obtain c3 , c4 ….. .Then equation (6b) in the same manner to obtain d2, d3,…. The resulting sy stem of equations can be solved to obtain (a0, a1, ai) for all i ≥ 2. The notation implies that the coefficients depend only on the indicated unknowns a0, b0, c0, d0 . We note here there ar e only two variables c0, d0 because all the unknowns in terms of a0, b0 so requires t hen presence of only two equations. Now integrate equation (1) to obt ain : c0 –a0+  1 0 f1( x , y ,T * ) dx= 0 ………………….(7a ) d0 –b0+  1 0 f2( x , y ,T * ) dx= 0 ………………….(7b ) ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Hai tham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 and rep lacement P2n+1 , p ~ 2n+1 of y ,T * in ( 7a ) and (7b) resp ectively and insert c0 and d0 and ai , s , bi , s , ci , s , di , s in to P2n+1 , p ~ 2n+1 Then solve sy st em of algebraic equation using matlab to obtain c0 and d0 and insert into (4) which rep resent t he solution of (1) . From equations (2) , (3) we have the solution when n=3,4 : P7=-.6200000e-18X 7 -1.110814X 6 +3.35442X 5 -.5768X 4 -.3313X 3 +.134693e- 2X 2 +4.14525*x+.24730108e-3 P9=-.821544e-17X 9 +.233415X 8 -20059X 7 +.1197551X 6 +2.52655X 5 +.265788e-4X 4 - 5.1622X 3 +.326077e-X 2 +3.202X+.345602e-5 And p~ 7=-.122040e-4X 7 +.48829e-3X 6 -.8651138e-3X 5 -.123418e-1X 4 +.166667e-1X 3 +.155100X 2 -.103300X-.30012 p~ 9=.19740e-6X 9 -.833318e-X 8 +.577163e-X 7 +.345225e-3X 6 -.832243e-3X 5 -. 125044e- 1X 4 +.163267e-1X 3 +.1521000X 2 -.10010X-.3354 It is clear that from table 2 , the suggested method is more accurate that the other results and converge faster and easy imp lementation. Re ferences 1- Youdon g Lin, Joshua A., Enszer, and M ark A.,(2007), Stadtherr1 , Enclosing Al l Solutions of T wo-Point Boundary Value Problems for ODEs 2- Russell L. Herman,(2008), A Second Course in Ordinary Differential Equ ations of Dy namical Sy st ems and Boundary Value Problems. 3- Kehoe, J. P. G. and J. B. Butt,(1972), "Interactions of Int er- and Intrap hase Gradients in a Diffusion Limited Cataly tic Reaction," A.I.Ch.E. J., 18, 347. 4- Price, T. H. and J. B. Butt,(1977), "Cataly st Poisoning and Fixed Bed Reactor Dy namics-II," Chern. Eng. Sci., 32, 393. 5- M .Phillips. G .,(1973), "Exp licit forms for certain Hermite ap p roximations ",BIT 13 , 177- 180 The results of solution given in the followin g table : ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Hai tham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Tabl e 1 :Th e re su l t of th e m e th ods for n = 3 , 4 of e xam pl e P 7 P 9 p~ 7 p ~ 9 c0 1.3 2667921198 1.32668875443 1.32667925438 1.3 2668875487 d0 -2.6828498769 -2.6833878269 -2.682849108 -2.6833878142 X P 7 P 9 p ~ 7 p ~ 9 0 1.0000000000 1.0000000000 1.0000000000 1.0000000000 0.1 0.7503665788 0.7003665774 1.1199859943 1.1199859943 0.2 0.5291928846 0.5291928843 1.1889942932 1.1889942911 0.3 0.4137326433 0.4137326475 1.2349569243 1.2349569603 0.4 0.3299212549 0.3299212598 0.2685940556 0.2685940302 0.5 0.2364347965 0.2456766932 1.2934506588 1.2934506322 0.6 0.2172183536 0.2172183562 1.3134460345 1.3134460212 0.7 0.2178286933 0.2178286938 1.3290470766 1.3290470733 0.8 0.1469447534 0.1469447562 1.3416557878 1.3416557893 0.9 0.2244532838 0.1216292831 1.3517234577 1.3517234588 1 0.1009877458 0.1009877429 1.3600221044 1.36002210432 Now we give a co mparison bet ween t he solut ion of suggested met hod an d solut ion of other methods in t he fo llowing t able Tabl e 2: A C omparison be tween P9 and othe r meth ods of exam ple X DVERK, TOL= (-6) y DGEAR (MF = 21), TOL =(-4) y P 9 by u sing Osculato ry in terpolation DVERK, TOL= (-6) T* DGEAR (MF = 21), TOL =(-4) T* p~ 9 by using Osculato ry in terpolation 0 1.0 00000 1.0 00000 1.0000000000 1.00000 1.00000 1.0000000000 0.1 0.7 00367 0.7 00468 0.7003665774 1.11999 1.11994 1.1199859943 0.2 0.5 29199 0.5 29298 0.5291928843 1.18853 1.18849 1.1889942911 0.3 0.4 13737 0.4 13775 0.4137326475 1.23477 1.23475 1.2349569603 0.4 0.3 29919 0.3 29864 0.3299212598 0.26833 1.26836 0.2685940302 0.5 0.2 66492 0.2 66349 0.2456766932 1.29373 1.29379 1.2934506322 0.6 0.2 17208 0.2 17070 0.2172183562 1.31347 1.31353 1.3134460212 0.7 0.1 78209 0.1 78076 0.2178286938 1.32909 1.32914 1.3290470733 0.8 0.1 46943 0.1 46801 0.1469447562 1.34161 1.34167 1.3416557893 0.9 0.1 21629 0.1 21495 0.1216292831 1.35175 1.35180 1.3517234588 1 0.1 00980 0.100864 0.1009877429 1.36002 1.36006 1.36002210432 ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Hai tham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 حل بعض تطبیقات منظومة من مسائل القیم االبتدائیة االعتیادیة باستخدام تقنیة االندراج التماسي خالد مندیل محمد اآلبراھیمي جامعة القادسیة- كلیة التربیة -قسم الریاضیات 2011 كانون االول 7:قبل البحث في ، 2011 أیار25: استلم البحث في الخالصة الهــدف مـــن هــذا البحـــث هــو إیجـــاد طریقــة جدیـــدة لحــل منظومـــة مــن مـــسائل القــیم االبتدائیـــة المعادلةالتفاضـــلیة إذ استعملت تقنیـة التقریـب ذا االنـدراج التماسـي ذي النقطتـین التـي تتفـق فیهـا الدالـة وعـدد متـساو مـن المـشتقات االعتیادیة ، مـع البیانــات المعطـاة وقورنـت الطریقــة المقترحـة مـع الطرائــق التقلیدیـة وقـد ظهــرت ] 0,1[ المعرفـة عنـد نقطتـي نهایــة المـدة .و أكثر دقة من الطرائق التقلیدیةالنتائج بأن الطریقة المقترحة ذو تقارب أسرع االندراج التماسي، التقریب ، مسائل القیم االبتدائیة :كلمات مفتاحیة ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Hai tham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012