Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method on Using Legendre Polynomials

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1 IOSR Joural of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume 11, Issue 6 Ver. IV (Nov. - Dec. 15), PP Numerical Solutios of Secod Order Boudary Value Problems by Galeri Residual Method o Usig Legedre Polyomials M. B. Hossai 1*, M. J. Hossai, M. M. Rahama 3, M. M. H. Sidar 4 M.A.Rahama 5 1, 3 Departmet of Mathematics,,5 Departmet of CIT, 4 Departmet of Statistics Patuahali Sciece ad Techology Uiversity, Dumi, Patuahali-86 Abstract: I this paper, a aalysis is preseted to fid the umerical solutios of the secod order liear ad oliear differetial equatios with Robi, Neuma, Cauchy ad Dirichlet boudary coditios. We use the Legedre ecewise polyomials to the approimate solutios of secod order boudary value problems. Here the Legedre polyomials over the iterval [,1] are chose as trial fuctios to satisfy the correspodig homogeeous form of the Dirichlet boudary coditios i the Galeri weighted residual method. I additio to that the give differetial equatio over arbitrary fiite domai [a,b] ad the boudary coditios are coverted ito its equivalet form over the iterval [,1]. Numerical eamples are cosidered to verify the effectiveess of the derivatios. The umerical solutios i this study are compared with the eact solutios ad also with the solutios of the eistig methods. A reliable good accuracy is obtaied i all cases. Keywords: Galeri Method, Liear ad Noliear VBP, Legedre polyomials I. Itroductio I order to fid out the umerical solutios of may liear ad oliear problems i sciece ad egieerig, amely secod order differetial equatios, we have see that there are may methods to solve aalytically but a few methods for solvig umerically with various types of boudary coditios. I the literature of umerical aalysis solvig a secod order boudary value problem of differetial equatios, may authors have attempted to obtai higher accuracy radly by usig umerical methods. Amog various umerical techiques, fiite differece method has bee widely used but it taes more computatioal costs to get higher accuracy. I this method, a large umber of parameters are required ad it ca ot be used to evaluate the value of the desired poits betwee two grid poits. For this reaso, Galeri weighted residual method is widely used to fid the approimate solutios to ay poit i the domai of the problem. Cotiuous or ecewise polyomials are icredibly useful as mathematical tools sice they are precisely defied ad ca be differetiated ad itegrated easily. They ca be approimated ay fuctio to ay accuracy desired [1], splie fuctios have bee studied etesively i [-9]. Solvig boudary value problems oly with Dirichlet boudary coditios has bee attempted i [4] while Berstei polyomials [1, 11] have bee used to solve the two poit boudary value problems very recetly by the authors Bhatti ad Brace [1] rigorously by the Galeri method. But it is limited to the secod order boudary value problems with Dirichlet boudary coditios ad to first order oliear differetial equatio. O the other had, Ramada et al. [] has studied liear boudary value problems with Neuma boudary coditios usig quadratic cubic polyomial splies ad opolyomial splies. We have also foud that the liear boudary value problems with Robi boudary coditios have bee solved usig fiite differece method [1] ad Sic-Collocatio method [13], respectively. Thus ecept [9], little cocetratio has bee give to solve the secod order oliear boudary value problems with dirichlet, Neuma ad Robi boudary coditios. Therefore, the aim of this paper is to preset the Galeri weighted residual method to solve both liear ad oliear secod order boudary value problems with all types of boudary coditios. But oe has attempted, to the owledge of the preset authors, usig these polyomials to solve the secod order boudary value problems. Thus i this paper, we have give our attetio to solve some liear ad oliear boudary value problems umerically with differet types of boudary coditios though it is origiated i [1]. However, i this paper, we have solved secod order differetial equatios with various types of boudary coditios umerically by the techique of very well-ow Galeri method [15] ad Legedre ecewise polyomials [14] are used as trial fuctio i the basis. Idividual formulas for each boudary value problem cosistig of Dirichlet, Neuma, Robi ad Cauchy boudary coditios are derived respectively. Numerical eamples of both liear ad oliear boudary value problems are cosidered to verify the effectiveess of the derived formulas ad are also compared with the eact solutios. All derivatios are performed by MATLAB programmig laguage. DOI: 1.979/ Page

2 Numerical Solutios of Secod Order Boudary Value Problems by Galeri Residual Method II. Legedre Polyomials The solutio of the Legedre s equatio is called the Legedre polyomial of degree ad is deoted by p (. The p ( N r ( 1) where N for eve ad N 1 for odd The first few Legedre polyomials are p1 ( 1 p ( (3 1) 1 3 p3( ( p 4 ( (35 3 3) p5 ( ( p ( ( r ( r)! r!( r)!( r)! p7 ( etc 16 Graphs of first few Legedre polyomials 5) r p1 p p3 p4 p5 p6 p Shifted Legedre polyomials Here the "shiftig" fuctio (i fact, it is a affie trasformatio) is chose such that it bijectively maps the iterval [, 1] to the iterval [ 1, 1], implyig that the polyomials are A eplicit epressio for the shifted Legedre polyomials is give by orthogoal o [, 1]: DOI: 1.979/ Page

3 Numerical Solutios of Secod Order Boudary Value Problems by Galeri Residual Method ~ p( ( 1) ( The aalogue of Rodrigues' formula for the shifted Legedre polyomials is ~ 1 d p ( (! To satisfy the coditio p ( ) p(1), 1, we modified the shifted Legedre polyomials give above i the followig form 1 d p ( ( ( 1) ( 1).! Sice Legedre polyomials have special properties at ad 1: p ( ) ad p ( 1), 1 respectively, so that they ca be used as set of basis fuctio to satisfy the correspodig homogeeous form of the Dirichlet boudary coditios to derive the matri formulatio of secod order BVP over the iterval [,1]. III. Formulatio Of Secod Order Bvp We cosider the geeral secod order liear BVP [15]: d du p( q( u r(, a b (1a) u( a) 1u( a) c1, u( b) 1u( b) c (1b) where p (, q( ad r are specified cotiuous fuctios ad, 1,, 1, c1, c are specified umbers. Sice our aim is to use the Legedre polyomials as trial fuctios which are derived over the iterval [,1], so the BVP (1) is to be coverted to a equivalet problem o [,1] by replacig by ( b a) a, ad thus we have: d ~ du p( q~ ( u r(, 1 (a) 1 1 u( ) u() c1, u(1) u(1) c b a b a (b) where ~ 1 p ( p(( b a) a), q~ ( q(( b a) a), ~ r ( r(( b a) a) ( b a) Usig Legedre polyomials, p i ( we assume a approimate solutio i a form, u~ ( ai (, 1 (3) i1 Now the Galeri weighted residual equatios correspodig to the differetial equatio (1a) is give by 1 d du~ ~ p q~ u ~ ~ ( ) ( r ( p j (, j 1,,, (4) After mior simplificatio, from () we ca obtai 1 dp dp j b a p p i i p j b a p p j ~ p q~ ( ) ~ (1) (1) (1) ( ) ~ () () () ( ) ( ( p j ( ai i ~ c ( b a) ~ p(1) p j (1) c1( b a) ~ p() p j () r ( p j ( (5) 1 1 Or, equivaletly i matri form DOI: 1.979/ Page

4 Numerical Solutios of Secod Order Boudary Value Problems by Galeri Residual Method Ki, jai Fj, j 1,,3,, (6a) i1 1 where ~ dp ( ) ~ (1) (1) (1) ( ) ~ () () (), ( ) i dp j ~ b a p p j b a p p j Ki j p q ( ( p j( (6b) c b a p p c b a p p F ~ ( ) ~ (1) j (1) 1( ) ~ () j () j r ( p j (, j 1,,, (6c) 1 1 Solvig the system (6a), we fid the values of the parameters a i ( i 1,,, ) ad the substitutig these parameters ito eq. (3), we get the approimate solutio of the boudary value problem (). If we replace a by i u ~ ( ), the we get the desired approimate solutio of the boudary value problem (1). b a Now we discuss the differet types of boudary value problems usig various types of boudary coditios as follows: Case 1: The matri formulatio with the Robi boudary coditios (, 1,, 1 ), are already discussed i equatio (6). Case : The matri formulatio of the differetial equatio (1a) with the Dirichlet boudary coditios ( i. e.,, 1,, 1 ) is give by Ki, j ai Fj, j 1,,, (7a) i1 where 1, ~ dp dp K p( ~ q ( ( p j (, i, j 1,,, (7b) 1 ~ ~ d dp j F ~ j r ( p j ( p ( q ( ( p j (, j 1,, (7c) Case 3: The approimate solutio of the differetial equatio (1a) cosistig of Neuma boudary coditios ( i. e.,, 1,, 1 ) is give by Ki, j ai Fj, j 1,,, (8a) i1 where 1, ~ dp dp K p( ~ q ( ( p j (, i, j 1,,, (8b) 1 c b a p p c b a p p F ~ ( ) ~ (1) j (1) 1( ) ~ () j () j r ( p j (, j 1,,, (8c) 1 1 Case 4(i): The approimate solutio of the differetial equatio (1a) cosistig of Cauchy boudary coditios ( i. e., 1, 1 ) is give by Ki, j ai Fj, j 1,,, (9a) i1 where 1, ~ ~ dp dp j ~ p () p () p () K p( i q ( ( p j (, i, j 1,,, (9b) 1 1 ~ ~ ~ ~ d dp j 1 ~ c p () p j () p () () p j () F j r ( p j ( p ( q ( ( p j ( (9c) 1 1 Case 4(ii): The matri formulatio with the Cauchy boudary coditios ( i. e., 1, 1 ) is give by DOI: 1.979/ Page

5 Numerical Solutios of Secod Order Boudary Value Problems by Galeri Residual Method Ki, j ai Fj, j 1,,, (1a) i1 where 1, ~ ~ dp dp j ~ p (1) p (1) p (1) K p( i q ( ( p j (, i, j 1,,, (1b) 1 1 ~ ~ ~ ~ d dp j ~ c p (1) p j (1) p (1) (1) p j (1) Fj r ( p j ( p ( q ( ( p j (, j 1,, (1c) 1 1 Similar calculatio for oliear boudary value problems usig the Legedre polyomials ca be derived which will be discussed through umerical eamples i et portio. IV. Numerical Eamples I this sectio, we eplai four liear ad two oliear boudary value problems which are available i the eistig literatures, cosiderig four types of boudary coditios to verify the effectiveess of the preset formulatios described i the previous sectios. The covergece of each liear boudary value problem is calculated by E u 1( u( where u ( represets the approimate solutio by the proposed method usig -th degree polyomial approimatio. The covergece of oliear boudary value problem is assumed whe the absolute error of two cosecutive iteratios is recorded below the covergece criterio such that ~ N 1 ~ N u u where N is the Newto s iteratio umber ad varies from 1-8. Eample1. First we cosider the boudary value problem with Robi boudary coditios [15]: d u u cos, (11a) u 3u 1, u( ) 4u( ) 4 (11b) The eact solutio is u( cos. The boudary value problem (11) over [, 1] is equivalet to the BVP 1 d u u cos, 1 u() 3u() 1, u(1) 4u(1) 4 Usig the formula derived i Case-1, equatio (6) ad usig differet umber of Legedre polyomials, the approimate solutios are show i Table 1. It is observe that the accuracy is foud early the order 1-5, 1-6, 1-6, 1-7 o usig 6, 7, 8, ad 9 Legedre polyomials respectively. DOI: 1.979/ Page

6 Numerical Solutios of Secod Order Boudary Value Problems by Galeri Residual Method Table 1: Eact, approimate solutios ad absolute differeces for the eample 1 π/ 11π/ 3π/5 13π/ 7π/1 3π/4 4π/5 17π/ 9π/1 19π/ π π/ 11π/ 3π/5 13π/ 7π/1 3π/4 4π/5 17π/ 9π/1 19π/ π Eact Approimate Error Approimate Error 6 Legedre polyomials 7 Legedre polyomials E E E E E E E E E E E E E E E E E E Legedre polyomials 9 Legedre polyomials E E E E E E E E E E E E E E E E E E-7 Eample. We cosider the boudary value problem with Dirichlet boudary coditios [1]: d u u e, 1 (1a) u ( ), u(1) (1b) 1 The eact solutio is: e 1 e cos e The boudary value problem (1) is similar to the VBP cot1 11cos ec1 si 1 e 1 d u 1 u e, 1 1 (13a) u ( ) u(1) (13b) Usig the formula calculated i Case-, equatio (7), the approimate solutios are summarized i Table. It is show that the accuracy up to 3, 5, 6 ad 8 sigificat digits are obtaied for 8, 1, 1 ad 15 Legedre polyomials respectively. Table : Eact, approimate solutios ad absolute differeces for the eample Eact Approimate Error Approimate Error 8 Legedre Polyomials 1 Legedre Polyomials E E E E E E E E E E-3 E E E E E E E E E E E-5 E+ 1 Legedre Polyomials 15 Legedre Polyomials DOI: 1.979/ Page

7 Numerical Solutios of Secod Order Boudary Value Problems by Galeri Residual Method E E E E E E E E E E-6 E E E E E E E E E E E-9 E+ Eample3. I this case we cosider the boudary value problem with Neuma boudary coditios []: d u u 1, 1 (14a) 1 cos1 1 cos1 u (), u(1) (14b) si1 si1 1 cos1 whose eact solutio is, u ( cos si 1. si1 Applyig the formula derived i Case-3,equatio (8), the approimate solutios, give i Table 3, are obtaied o usig 5, 7, 8 ad 1 Legedre polyomials with the remarable accuracy early the order of 1-11, 1-13, 1-13 ad O the other had, Ramada et al. [6] has foud early the accuracy of order 1-6 ad 1-6, ad 1-8 o usig quadratic ad cubic polyomial splies, ad opolyomial splie respectively with h=1/18 where h= (b-a)/n, a ad b are the edpoits of the domai ad N is umber of subdivisio of itervals [a,b]. Table 3: Eact, approimate solutios ad absolute differeces for the eample Eact Approimate Error Approimate Error 5 Legedre Polyomials 7 Legedre Polyomials E+ E E E E E E E E E E E E E E E E E E E-13 E+ E+ 8 Legedre Polyomials 1 Legedre Polyomials E+ E E E E E E E E E E E E E E E E E E E-16 E+ E+ Eample4. We cosider the Cauchy (mied) boudary value problem [4]: d u 3( u 5), (15a) with mied boudary coditios u( ) 15 ( Dirichlet ), u() ( Neuma ) (15b) DOI: 1.979/ Page

8 Numerical Solutios of Secod Order Boudary Value Problems by Galeri Residual Method 3 3 whose eact solutio is 3 15e 15e u( 5 15e e 1 e Usig the formula illustrated i Case-4(ii), equatio (1), ad usig differet umber of Legedre polyomials, the approimate solutios are show i Table 4. It is observe that the accuracy is foud early the order 1-5, 1-7, 1-9 ad 1-14 o usig 8, 1, 1, ad 15 Legedre polyomials respectively Table 4: Eact, approimate solutios ad absolute differeces for the eample 4 Eact Approimate Error Approimate Error 8 Legedre polyomials 1 Legedre polyomials 15 E+ 15 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+ 1 Legedre polyomials 15 Legedre polyomials 15 E+ 15 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+ We ow also apply the procedure described i sectio 3, formulatio of secod order liear BVP, to fid the umerical solutios of oe oliear secod order boudary value problem. Eample5. We cosider a oliear BVP with Dirichlet boudary coditios [16] d u 1 du 1 4 u, (16a) 43 u ( 1) 17 ad u ( 3) 3 (16b) 16 The eact solutio of the problem is give by u( DOI: 1.979/ Page

9 Numerical Solutios of Secod Order Boudary Value Problems by Galeri Residual Method To implemet the Legedre polyomials, first we covert the BVP (16) to a equivalet BVP o the iterval [,1] by replacig by 1 such that d u 1 du u 16 ( 1) 3, 1 (17a) 4 43 u ( ) 17 ad u ( 1) (17b) 3 Suppose that the approimate solutio of the boudary value problem (17) applyig the Legedre polyomials is give by u ~ ( ( ai (, 1 (18) i1 8 Where ( 17 is specified by the dirichlet boudary coditios at ad 1ad 3 p i ( ) p i (1) for each i 1,,,. The weighted residual equatios of (17a) correspodig to the approimatio solutio (18), give by 1 d u~ 1 ~ ~ du 3 u 16 ( 1) p (, 1,,, (19) 4 Eploitig itegratio by parts with mior simplificatios, we have 1 dp i dp 1 dp i d dp j p 1 p a j p ai i j d dp 1 d 16 1 p p, 1,,, () 4 The above equatio () is equivalet to the matri form ( D C) A G (1a) where the elemets of the matri A, C, D ad G are, c d ad g respectively, give by a i i,, i, 1 dp dp 1 dp d d i i i, p p (1b) dp j ci, a j p (1c) 4 j1 1 3 d dp 1 d g 16 1 p p, 1,,, (1d) 4 The iitial values of these coefficiets ai are obtaied by applyig the Galeri method to the BVP eglectig the oliear term i (17a). Therefore, to fid the iitial coefficiets, we will solve the system DA G (a) where the elemets of the matrices are give by dp dp d i i, (b) g d dp 16 1 p, 1,,, Oce the iitial values of the parameters ai are obtaied from equatio (a), they are substituted ito equatio (1a) to obtai ew estimates for the values of a i. This iteratio process cotiues util the coverged values of the uows are obtaied. Puttig the fial values of coefficiets ito equatio (18), we obtai a (c) DOI: 1.979/ Page

10 Numerical Solutios of Secod Order Boudary Value Problems by Galeri Residual Method approimate solutio of the BVP (17), ad if we replace by 1 i this solutio we will get the approimate solutio of the give BVP (16). Usig first 1 ad 15 Legedre polyomials with 1 iteratios, the absolute differeces betwee eact ad the approimate solutios are give i Table 5. It is observed that the accuracy is foud of the order early 1-6 ad 1-9 o usig 1 ad 15 Legedre polyomials respectively. Table 5: Eact, approimate solutios ad absolute differeces of eample 5 usig 1 iteratios Eact Approimate Error Approimate Error 1 Legedre polyomials 15 Legedre polyomials E E E E E E E E E-6.674E E E E E E E E E-6 5.6E E-6.E E E E E E E E E E E E E E E E E E E E E-1.E- V. Coclusios I this paper, the formulatio of oe dimesioal liear ad oliear secod order boudary value problems have bee discussed i details by the Galeri weighted residual method applyig Legedre polyomials as the basis fuctios i the approimatio. The proposed method is applied to solve some umerical eamples both liear ad oliear. The computed results are compared with the eact solutios ad we have foud a good agreemet with the eact solutio. All the mathematical formulatios ad umerical computatios have bee doe by MATLAB-1 code. Refereces [1]. M. I. Bhatti ad P. Brace, Solutios of Differetial Equatios i a Berstei Polyomial Basis, Joural of Computatioal ad Applied Mathematics, Vol. 5, No.1, 7, pp.7-8. doi:1.116/j.cam []. M. A. Ramada, I. F. Lashie ad W. K. Zahra, Polyomial ad Nopolyomial Splie Approaches to the Numerical Solutio of Secod Order Boudary Value Problem, Applied Mathematics ad Computatio, Vol.184, No., 7, pp doi:1.116/j.amc [3]. R. A. Usmai ad M. Saai, A Coectio betwee Quatric Splie ad Numerov Solutio of a Boudary value Problem, Iteratioal Joural of Computer Mathematics, Vol. 6, No. 3, 1989, pp doi:1.18/ [4]. Arshad Kha, Parametric Cubic Splie Solutio of Two Poit Boudary Value Problems, Applied Mathematics ad Computatio, Vol. 154, No. 1, 4, pp doi:1.116/s9-33(3)71-x. [5]. E. A. Al-Said, Cubic Splie Method for Solvig Two Poit Boudary Value Problems, Korea Joural of Computatioal ad Applied Mathematics, Vol. 5, 1998, pp [6]. E. A. Al-Said, Quadratic Splie Solutio of Two Poit Boudary Value Problems, Joural of Natural Geometry, Vol. 1, 1997, pp [7]. D. J. Fyfe, The Use of Cubic splies i the Solutio of Two Poit Boudary Value Problems, The Computer Joural, Vol. 1, No., 1969, pp doi:1.193/comjl/ [8]. A.K. Khalifa ad J. C. Eilbec, Collocatio with Quadratic ad Cubic Splies, The IMA Joural of Numerical Aalysis, Vol., No. 1, 198, pp doi:1.193/imaum/ [9]. G. Mulleheim, Solvig Two-Poit Boudary Value Problems with Splie Fuctios, The IMA Joural of Numerical Aalysis, Vol. 1, No. 4, 199, pp doi:1.193/imaum/ [1]. J. Reiehof, Differetiatio ad Itegratio Usig Berstei s Polyomials, Iteratioal Joural for Numerical Methods i Egieerig, Vol. 11, No. 1, 1977, pp doi:1.1/me [11]. E. Kreyszig, Berstei Polyomials ad Numerical Itegratio, Iteratioal Joural for Numerical Methods i Egieerig, Vol. 14, No., 1979, pp doi:1.1/me [1]. R. A. Usmai, Bouds for the Solutio of a Secod Order Differetial Equatio with Mied Boudary Coditios, Joural of Egieerig Mathematics, Vol. 9, No. 1975, pp doi:1.17/bf [13]. B. Bialeci, Sic-Collocatio methods for Two Poit Boudary Value Problems, The IMA Joural of Numerical Aalysis, Vol. 11, No. 3, 1991, pp , doi:1.193/imaum/ DOI: 1.979/ Page

11 Numerical Solutios of Secod Order Boudary Value Problems by Galeri Residual Method [14]. N. Sara, S. D. Sharma ad T. N. Trivedi, Special Fuctios, Seveth Editio, Pragati Praasha,. [15]. P. E. Lewis ad J. P. Ward, The Fiite elemet Method, Priciples ad Applicatios, Addiso-Wesley, Bosto, [16]. R. L. Burde ad J. D. Faires, Numerical Aalysis, Boos/Cole Publishig Co. Pacific Grove, 199. [17]. M. K. Jai, Numerical Solutio of Differetial Equatios, d Editio, New Age Iteratioal, New Delhi,. [18]. J. Stephe Chapma, MATLAB Programmig for Egieers, Third Editio, Thomso Learig, 4. [19]. C. Steve Chapra, Applied Numerical Methods with MATLAB for Egieers ad Scietists, Secod Editio, Tata McGraw-Hill, 7. DOI: 1.979/ Page