Studies on the Method of Orthogonal Collocation III. The Use of Jacobi Orthogonal Polynomials for the Solution of Boundary Value Problems

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1 J. Kng Saud Unv., Vol., Eng. Sc. (), pp. 9-, (A.H. 49/999) Studes on the Method of Orthogonal Collocaton III. The Use of Jacob Orthogonal Polynomals for the Soluton of Boundary Value Problems Mostafa Al Solman and A.A. Ibrahm Department of Chemcal Engneerng, College of Engneerng, Kng Saud Unversty, P.O. Bo 8, Ryadh 4, Saud Araba (Receved on December, 997; accepted for publcaton September, 998) Abstract. Prevous work dentfed the knd of Jocob polynomals sutable to solve boundary value problems of the dffuson type n a slab, cylnder or sphere. In ths paper we dentfy the general form of boundary value problems for whch Jacob orthogonal polynomals are optmal when they are used wthn the framework of the method of orthogonal collocaton. Introducton For boundary value problems epressng dffuson wth frst order reacton, Vlladsen and Mchelsen [] compared dfferent methods of weghted resduals for ther soluton and concluded that Galerkn method s the best. A collocaton method where the resdual s requred to be zero at the zeros of the proper Jacob polynomal could gve the same result as the Galerkn method. Vlladsen and Mchelsen [] dentfed the proper Jacob polynomal for a slab, cylndrcal and sphercal geometry. The queston arses of what the general form of boundary value problems, for whch a general Jacob polynomal s the optmal choce. We address ths queston n ths paper and gve some general forms. umercal results confrm the effcency of the method of collocaton as appled to ths general form. In the prevous two parts of ths seres [,3] we appled the method of orthogonal collocaton to the transent heat conducton problem. 9

2 9 M.A. Solman and A.A. Ibrahm Jacob Orthogonal Polynomals Jacob polynomals Pn ( ) satsfy the orthogonalty condton α α, α, ( ) Pn ( ) Pm ( ) =. m n o m = n, m =,,... n α, > [, ]... () For brevty we wll wrte, Pn( ) = Pn ( ) () They satsfy the followng dfferental Eq. [4]. d P n dp n ( ) + [( + ) ( α+ + ) ] = nn ( + α+ + ) Pn ( ) (3) whch can be put n the followng form d + α d α ( ) {( ) Pn ( )} = [ -n(n + α + + ) - α( + )] Pn() (4) Dfferental Equatons for whch Jacob Polynomals are Optmal Vlladsen and Mchelsen [] concluded that Jacob polynomals are optmal for problems of dffuson and chemcal reacton wth α =, = - /,, / for a slab, cylnder and sphere, respectvely. The queston arses f there are dfferental equatons for whch other values of α and are optmal. Let us have a look at the general dfferental equaton d + α d α u c u φ + ( ) ( ) {( ) ( )} = u or d u [( + ) ( ) ( α ) ] du +

3 and where c s a constant. If we wrte the soluton u() as Studes on the Method of Orthogonal Collecton III. 93 [( c ( α ) ( + )) ( ) ( α ) ] (5) + ( u ) = φ u ( ) u() = (6) where u() = + (-) = ap( ) (7) P( ) P ( ), = α and a s are constants then substtute Eq. (7) nto Eq. (5), we obtan d d + ( α) α ( ) ( ) ap c a P ( ) = + = = φ + ( ) ap( ) = (8) By vrtue of Eq. (4), the left hand sde of Eq. (8) can be wrtten as d d + ( α) α ( ) ( ) ap c ap ( ) = + ( ) = = = bp( ) (9) where the b's are constants related to the a s. Thus the terms contanng P () n both the left hand sde and the rght hand sde of Eq. (8) become zero f we use the zeros of P () as collocaton ponts. Ths s the desred property whch makes ths specfc polynomal optmal snce the soluton wll be accurate at the collocaton ponts up to whatever the value of a. otce that f α =, c =, Eq. (5) reduces to that of dffuson wth frst order reacton. An ntegral whch can be evaluated accurately usng Radau quadrature and whch s smlar to the effectveness factor s

4 94 M.A. Solman and A.A. Ibrahm where Snce α η= S ( ) u( ) Γ ( α+ + ) S = [ Γ( α) Γ( + )] α η = S ( ) u( ) α = S ( ) [ + ( ) b P ( )] = () α α = S [( ) + ( ) b P ( )] The term contanng P () wll be zero f we use the zeros of P () for the evaluaton of the quadrature. Another general system of dfferental equatons for whch Jacob polynomals are sutable s the followng: = or d + ( ) d ( ) [( ) α α [ u ( ) ( g+ f( ))] + c = φ u ( ) ( u ( g+ f ( )) ] ( ) d u [( )( ) ( α ) ] du + g+ f ( ) ( α ) ( ) ( α) c ( ) ( ) ( ) () *[ u ( g+ f ( ))] = φ u wth u() = g () and u(o) = f (3) If we wrte the soluton u() as

5 Studes on the Method of Orthogonal Collecton III. 95 u() = g + f(-) + (-) = b P ( ) (4) and make use of Eq. (3) we wll note that both sdes of Eq. () wll have the term contanng P () zero f the collocaton ponts are the zeros of P (). The followng ntegral wll be evaluated accurately usng Lobatto quadrature, where Snce α η = S ( ) u( ) (5) Γ ( α+ ) S = [ Γ( α) Γ( )] α η = S ( ) [ a+ d( ) + ( ) b P ( )] and the term contanng P () wll be zero. For problems of ths knd, an arbtrary etra collocaton pont can be used to mprove the profle. Also the dead zone method [] can be used for large values of φ. In ths case there wll be a boundary layer near = and another one near =. The dead zone wll be n the mddle. A perturbaton analyss of Eqs. (-3) for small φ wll ndcate that φ term contans a cubc polynomal n and φ 4 term contans a polynomal of ffth order n. Ths s n contrary to the system (5-6) where the order of the polynomal ncreases by one as the order of φ ncreases by, (see append A) Thus we need more collocaton ponts for the system (-3) to get the same accuracy n φ as for system (5 and 6). Thus t s desrable to dentfy cases for whch Eqs. (-3) can be transformed nto equatons (5 and 6). The followng two cases are dentfed. (a) f =, c =, and α =. The followng transformaton u() = v() y = =

6 96 M.A. Solman and A.A. Ibrahm wll transform Eqs. (-3) nto v() = g () g = f =, and α = In ths case we use the transformaton 4y d v dv + ( + ) = φ v dy dy to obtan y = ( - ) y d u y du + + c + ( ( ) ( )( ) + 5.) + dy y dy 4 ( y) ( ) y φ + = u ( y) 6 Fnally the system of dfferental equaton d u du ( ) + [ + ) ( α+ + ) ] = φ u ( ) (6) wll have a soluton other than the trval soluton u() = f and φ = n (n+α++) (7) u() = b Pn() (8) Thus φ wll be the egenvalues and P n() are the egenvectors. The followng ntegral α η = ( ) u ( ) (9) equal zero (f n ) and ths wll be true f the collocaton ponts are taken as the zeros of Pn() and we use Gauss quadrature.

7 Studes on the Method of Orthogonal Collecton III. 97 umercal Results In ths secton we gve numercal results for three eamples. The results show that we can get hgh accuracy wth low order polynomals. Eamples () and () are lnear boundary value problems. Although the development for ths paper s for a lnear system, eample (3) shows that good results can also be obtaned when we apply the collocaton method to a non-lnear problem. Eample For Eq. (5) and the case of c = and ) α =, = 4 ) α = 4, = ) α = 3, = 3 and φ = 8, obtan the numercal soluton for u and η for = to 4. In the followng table, we lst the values of u (o) and η = = =3 =4 α=, =4 u(o) η α=4, = u(o) η α=3, =3 u(o) η As epected η s calculated accurately wth few collocaton ponts. Eample A frst order reacton of a gas dffusng n a lqud flm s gven by The soluton s gven by d u =φ u() = g u() = f u

8 98 M.A. Solman and A.A. Ibrahm gsnh φ+ f snh φ( ) u = snh φ compare the numercal soluton of u, = D = du and I = u wth the analytcal soluton for the case of φ =, g =, f =.. We frst note that the above dfferental equaton s obtaned from the general dfferental Eq. () by lettng c =, α = and =. The followng results are obtaned: U Calculated u Analytcal = = 4 = I D Eample 3 In ths eample we treat a non-lnear problem where the R.H.S of Eq. (5) equals to u. For the case of c = and () α =, = 4; () α = 4, = ; () α = 3, = 3, we obtan the numercal soluton for u () and η for = to =. In the followng table, we lst the values of u() and η.

9 Studes on the Method of Orthogonal Collecton III. 99 = = =3 =4 =5 =6 =7 =8 =9 = α = u() = 4 η α = 4 u() = η α = 3 u() = 3 η We notce that η s obtaned accurately for = 3. However a large number of collocaton ponts are needed ( 9) to obtan accurate value for u(). Certanly the effcency of the method wll be affected by the degree of non-lnearty as wth any other numercal method. Acknowledgment. The authors gratefully acknowlegde the Research Center at College of Engneerng, Kng Saud Unversty, Ryadh Saud Araba for ts support of the project. References [] Vlladsen, J. and Mchelsen, M.L. Soluton of Dfferental Equaton Models by Polynomal Appromaton., Englewood Clffs, ew Jersey: Prentce-Hall, 978. [] Chhara, T.S. An Introducton to Orthogonal Polynomals. ew York: Gordon & Breach, 978. [3] Solman, M.A. and Ibrahm, A.A., Studes on the Method of Orthogonal Collocaton I. A One -Pont Collocaton Method for the Transent Heat Conducton Problem. J. Kng Saud Unv., Vol., Eng. Sc. (), 48 H, [4] Solman, M.A. and Ibrahm A.A. Studes on the Method of Orthogonal Collocaton II. An Effcent umercal Method For the soluton of Transent Heat Conducton Problem. J. Kng Saud Unv., Vol., Eng. Sc. () 49 H, (In Press). Append A Substtutng an epanson of the form 4 u= u + φ u + φ u +... (A.) n Eqs. (5,6) and collectng terms of the same order n φ, we obtan

10 M.A. Solman and A.A. Ibrahm 4 + α+ c φ ( )( ( ) ) φ ( ) ( + ) α c u= + (( + ) α c) (( + ) α c) (( + ) ( α+ ) c) (A.) As can be seen the term contanng φ s frst order n and the term contanng φ 4 s second order n. The root of P α, ( a ), a s gven by + X a = + α (A.3) If we substtute a for n Eq (A.), we obtan, 4 ua = φ ( α+ ) φ ( α+ ) + ( α+ + )(( + ) α c) ( α+ + ) (( + ) α c) where u a s the value of u at a. We notce that the frst three terms form a geometrc sequence. If we assume that further terms wll follow the same pattern, we can sum the geometrc seres to obtan u a = φ ( α+ ) + ( α+ + )(( + ) α c) On the other hand f we substtute an epresson of the form ( uc )( ) u = + ( ) a (A.5) n Eq. (5) and apply the collocaton method at the root of P α, (), a, we obtan u c = φ ( α+ ) + ( α+ + )(( + ) α c) (A.6) We notce that u c from (A.6) s the same as u a n (A.4). Thus the one-pont collocaton s equvalent to summng up the regular perturbaton epanson of u at the collocaton pont as f t s a geometrc seres and for a problem of the type of Eq. (5).

11 Studes on the Method of Orthogonal Collecton III. ow substtute the regular perturbaton epanson (A.) n Eqs. (-3) and collectng terms of the same order n φ, we obtan ( g f)( ) f ( α) + c ( + α+ ) u= g+ f ( ) φ ( )[( c ( + ) α ( g f) +... ( α) + c ( + α+ ) (A.7) We notce here that the zero order term s frst order n and the term contanng φ s thrd order n. Smlarly the term contanng φ 4 wll be ffth order n, and so on.

12 M.A. Solman and A.A. Ibrahm دراسات على طريقة التنظيم المتعامد - ٣ استخدام تنظيم آثيرات الحدود المتعامد لجاآوبي لحل المساي ل ذات القيمة الحدية مصطفى علي سليمان و أحمد عيديد إبراهيم قسم الهندسة الكيمياي ية آلية الهندسة جامعة الملك سعود ص. ب ٨٠٠ الرياض ١١٤٢١ المملكة العربية السعودية (استلم في ١٩٩٧/١٢/١٠ م وقبل للنشر في ١٩٩٨/٩/٢٢ م) ملخص البحث. تم في أعمال سابقة تحديد نوع آثيرات الحدود لجاآوبي المناسبة لحل مساي ل ذات قيم حدية من نوع الانتشار في جسم على شكل لوح أو أسطوانة أو آ رة بينما يتم في هذه المقالة تحديد الشكل العام للمساي ل ذات القيم الحدية التي يكون حلها مناسبا باستخدام طريقة التنظيم المتعامد وآثيرات حدود جاآوبي.