مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for 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 Department of Mathematics, College of Education ,University of Al- Qadysea . Received in: 25 May 2011 Accepted in: 7 December 2011 Abstract The aim of this paper is to find a new method for solving a system of linear initial value problems of ordinary differential equation using approximation technique by two-point osculatory interpolation with the fit equal numbers of derivatives at the end points of an interval [0, 1] and compared the results with conventional methods and is shown to be that seems to converge faster and more accurately than the conventional methods. Key words : Initial value problems , Approximation , Osculatory interpolation 1- Introduction Systems of ordinary differential equations (ODEs) arise in mathematical models throughout science and engineering. When an explicit condition (or conditions) that a solution must satisfy is specified at one value of the independent variable, usually its lower bound, this is referred to as an initial value problem (IVP) and a system of ordinary differential equations is a system of equations relating several unknown functions yi(x) of an independent variable x, some of the derivatives of the yi(x), and possibly x itself. [ 1] .Initial-value problems for systems of differential equations permeate many areas of mathematics: such problems arise naturally in modelling the evolution of dynamical processes in economics, engineering, and the physical , biological sciences [2]. In this paper we introduce reactor problem which introduced in [3] Kehoe and Butt have studied the kinetics of benzene hydrogenation on asupported Ni/kieselguhr catalyst. In the presence of a large excess of hydrogen, the reaction is pseudo-first-order at temperatures below 200°C with the rate given by -r= PH2k0K0T exp[ (-Q-Ea)/RgT )] CB mol /( g catalyst s) where Rg = gas constant, 1.987 cal/(mole'K) - Q - Ea = 2700 cal/mole P H2 = hydrogen part ial pressure (torr) مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 ko = 4.22 mole/(gcat·s·torr) Ko = 2.63 X 10-6 cm3/(mole'K) T = absolute temperature (K) CB = concentration of benzene (mole/cm3). Price and Butt [4] studied this reaction in a tubular reactor. If the reactor is assumed to be isothermal, we can calculate the dimensionless concentration profile of benzene in their reactor given plug flow operation in the absence of inter- and intraphase gradients. Using a typical run, PH2 = 685 torr PB = density of the reactor bed, 1.2 gcat/cm3 e = contact time, 0.226 s T = 150°C And if we now consider the reactor to be adiabatic instead of isothermal, then an energy balance must accompany the material balance. Formulate the system of governing differential equations. The data of this problem Cp = 12.17 X 104 J/(kmole'°C) -  .Hr = 2.09 x 108 J/kmole )150(423, 00 0 * CKT T T T  . For the "short" reactor, . We have system of initial value problem y Tdx dy ] 21.3 exp[1744.0 *  (material balance) y Tdx dT ] 21.3 exp[06984.0 * *  (energy balance) with I.C y(0) =1 , T * (0) =1 2- Problem definition In this section we can explain the way through the application of this system, 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 paper we are particularly concerned with fitting function values and derivatives at the two end points of a finite interval, say [0,1],wherein a useful and succinct way of writing a osculatory interpolant P2n+1(x) of degree 2n + 1 was given for example by Phillips [5] as: مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for 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 that (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 Taylor coefficients ai and bi about x = 0 and x = 1 respectively, as P2n+1(x)=  n j 0 { a j Q j (x) + (-1) j b j Q j (1-x) }. ….(4) The simple idea of this paper is to replace y(x) in problem (1) by a P2n+1 in equation (3) .The first step therefore is to construct the P2n+1 . To do this we need the Taylor coefficients of y (x) and T *(x) respectively about x=0 ………… (5a)    2i i ixa y (x)= a0 +a1x + Where y(0) =a0 , y '(0) =a1 . …… y (j ) (0)/i! =ai i=2.3,….. And ………… (5b)    2i i ixb 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) respectively 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 simply insert the series forms in (5a) in to equation (1) and equate coefficients of x to obtain a1 , then derive equation (1) and insert the series in to (5a) and equate coefficients of x to obtain a2 and soon to obtain a3 , a4 ….. Then equation (5b) in the same manner to obtain b2, b3,….. and simply 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 the series in to (6a) and equate 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 system 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 are only two variables c0, d0 because all the unknowns in terms of a0, b0 so requires then presence of only two equations. Now integrate equation (1) to obtain : c0 –a0+  1 0 f1( x , y ,T * ) dx= 0 ………………….(7a ) d0 –b0+  1 0 f2( x , y ,T * ) dx= 0 ………………….(7b ) مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 and replacement P2n+1 , p~ 2n+1 of y ,T * in ( 7a ) and (7b) respectively and insert c0 and d0 and ai , s , bi , s , ci , s , di , s in to P2n+1 , p~ 2n+1 Then solve system of algebraic equation using matlab to obtain c0 and d0 and insert into (4) which represent the solution of (1) . From equations (2) , (3) we have the solution when n=3,4 : P7=-.6200000e-18X7-1.110814X6+3.35442X5-.5768X4-.3313X3 +.134693e- 2X2+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-X2+3.202X+.345602e-5 And p~ 7=-.122040e-4X7+.48829e-3X6-.8651138e-3X5-.123418e-1X4 +.166667e-1X3+.155100X2-.103300X-.30012 p~ 9=.19740e-6X9-.833318e-X8+.577163e-X7+.345225e-3X6 -.832243e-3X5-. 125044e- 1X4+.163267e-1X3+.1521000X2-.10010X-.3354 It is clear that from table 2 , the suggested method is more accurate that the other results and converge faster and easy implementation. References 1- Youdong Lin, Joshua A., Enszer, and Mark A.,(2007), Stadtherr1 , Enclosing All Solutions of Two-Point Boundary Value Problems for ODEs 2- Russell L. Herman,(2008), A Second Course in Ordinary Differential Equations of Dynamical Systems and Boundary Value Problems. 3- Kehoe, J. P. G. and J. B. Butt,(1972), "Interactions of Inter- and Intraphase Gradients in a Diffusion Limited Catalytic Reaction," A.I.Ch.E. J., 18, 347. 4- Price, T. H. and J. B. Butt,(1977), "Catalyst Poisoning and Fixed Bed Reactor Dynamics-II," Chern. Eng. Sci., 32, 393. 5- M.Phillips. G .,(1973), "Explicit forms for certain Hermite approximations ",BIT 13 , 177- 180 The results of solution given in the following table : مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Table 1 :The re sult of the methods for n = 3, 4 of example P7 P9 p~ 7 p~ 9 c0 1.32667921198 1.32668875443 1.32667925438 1.32668875487 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 comparison between the solution of suggested method and solution of other methods in the following table Table 2: A Comparison between P9 and other methods of example X DVERK, TOL= (-6) y DGEAR (MF = 21), TOL =(-4) y P 9 by using Osculatory in terpolation DVERK, TOL= (-6) T* DGEAR (MF = 21), TOL =(-4) T* p~ 9 by using Osculatory in terpolation 0 1.000000 1.000000 1.0000000000 1.00000 1.00000 1.0000000000 0.1 0.700367 0.700468 0.7003665774 1.11999 1.11994 1.1199859943 0.2 0.529199 0.529298 0.5291928843 1.18853 1.18849 1.1889942911 0.3 0.413737 0.413775 0.4137326475 1.23477 1.23475 1.2349569603 0.4 0.329919 0.329864 0.3299212598 0.26833 1.26836 0.2685940302 0.5 0.266492 0.266349 0.2456766932 1.29373 1.29379 1.2934506322 0.6 0.217208 0.217070 0.2172183562 1.31347 1.31353 1.3134460212 0.7 0.178209 0.178076 0.2178286938 1.32909 1.32914 1.3290470733 0.8 0.146943 0.146801 0.1469447562 1.34161 1.34167 1.3416557893 0.9 0.121629 0.121495 0.1216292831 1.35175 1.35180 1.3517234588 1 0.100980 0.100864 0.1009877429 1.36002 1.36006 1.36002210432 مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for 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 ، ة عنـد نقطتـي نهایــة المـدة ة مـع الطرائــق التقلیدیـة وقـد ظهــرت ] 0,1[ المعرفـ مـع البیانــات المعطـاة وقورنـت الطریقــة المقترحـ .و أكثر دقة من الطرائق التقلیدیةالنتائج بأن الطریقة المقترحة ذو تقارب أسرع االندراج التماسي، التقریب ، مسائل القیم االبتدائیة :كلمات مفتاحیة مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012