Application of B-Spline to Numerical Solution of a System of Singularly Perturbed Problems

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1 Mathematca Aeterna, Vol. 1, 011, no. 06, Applcaton of B-Splne to Numercal Soluton of a System of Sngularly Perturbed Problems Yogesh Gupta Department of Mathematcs Unted College of Engneerng & Management Allahabad-11010(U.P.) Inda Panka Kumar Srvastava Department of Mathematcs Jaypee Insttute of Informaton Technology, Noda-01301(U.P.) Inda Mano Kumar Department of Mathematcs Motlal Nehru Natonal Insttute of Technology Allahabad-11004(U.P.) Inda Abstract Present paper recaptulates a numercal method based on cubc B-splne to solve boundary value problems for a system of sngularly perturbed second order ordnary dfferental equatons. The method utlzes the values of cubc B-splne and ts dervatves at nodal ponts together wth the equatons of the gven system and boundary condtons, ensung nto the lnear matrx equaton. Selected numercal examples of perturbed systems for dfferent cases of perturbaton parameters from the lterature are presented, whch demonstrate the effcency of present method and also confrm how the developed algorthm s better than exstng numercal methods. Key words: Sngularly perturbed problem, Cubc B-splne, Nodal ponts. AMS Mathematcs Subect Classfcaton: 34D15 65L10

2 406 Yogesh Gupta, Panka Kumar Srvastava and Mano Kumar 1 Introducton Sngularly perturbed boundary value problems for ordnary dfferental equatons arse n varous felds of applcaton such as flud dynamcs, quantum mechancs, elastcty, chemcal reactor theory, gas porous electrodes theory, etc. The desgn and analyss of approprate numercal methods for sngularly perturbed dfferental equatons s an area of current nterest. An assortment of numercal methods has been developed for such problems. The systems of two sngularly perturbed ordnary dfferental equatons have applcatons n electro-analytcal chemstry. The parameters multplyng the hghest dervatves characterze the dffuson coeffcent of the substances. The predator-prey populaton dynamcs also ncorporate applcaton of such systems. However, lttle lterature pertanng to numercal soluton of systems of sngularly perturbed dfferental equatons s found. Some of sgnfcant contrbutors are G.I. Shshkn, S. Matthews, J.J.H. Mller, E.O. Rordan, S. Bellew, N. Madden, T. Valanarasu and N. Ramanuam. [1,, 1-15, 18, 19]. Consder a system of two sngularly perturbed ordnary dfferental equatons of the form, d r 1 0 dx r r ur Lu( x) u( x) Au( x) f ( x), x D [ a, b], d 0 dx (1) r p q u( a) r, u( b). r s () where 1, are small postve parameters and r u1( x) a11( x) a1 ( x) ur f1( x) u( x), A, f ( x) u( x) a1( x) a( x) f( x) The functons a11( x), a1( x), a1( x), a( x), f1( x), f( x) are suffcently smooth and satsfyng the nequaltes a11 ( x) a1 ( x), a( x) a1( x), a1 ( x) 0, a1( x) 0 x D a, b. (3) Maxmum prncple and stablty result of the above system can be found n [15]. For such systems three separate cases were dentfed n [18]: () 1, () 1, 1 and () 1, arbtrary. The frst case has been examned n [14], [19]; whle [15] explans method for second case; and thrd case has been thrashed out n [1]. A consderable amount of work has been done for development of numercal methods for boundary value problems (BVPs) usng varous splnes. In partcular, Cubc B-splne methods represent an mportant class of numercal methods used for soluton of boundary value problems[4], sngular boundary

3 Applcaton of B-Splne to Numercal Soluton value problems [11], sngular perturbaton problems [6-10], and system of boundary value problems [3, 5]. The present paper employs the B-splne method to solve system of two sngularly perturbed equatons of the form (1) & (). Remanng part of ths paper s organzed as follows: Secton descrbes the defnton of cubc splne and values of ts dervatves at nodal ponts. In Secton 3, the Cubc B-splne method for perturbed system s developed. Secton 4 of the paper conssts of numercal results for two problems, one wth constant coeffcents and another wth varable coeffcents. Fnally, Paper s concluded n secton 5. Cubc B-splne The thrd degree B-Splnes are defned as 3 ( x x ), f x [ x, x 1] 3 h 3 h ( x x 1) 3 h( x x 1) 3 3( x x 1), f x [ x 1, x ] 1 3 B ( x) 3 ( 3 h h x 3 x) 3 h( x3 x) 6h 3 3( x 3 x), f x [ x, x 3] 3 ( x4 x), f x [ x 3, x4 ] 0, otherwse. Let s( x) be the B-splne nterpolatng functon at the nodal ponts. Then s( x) can be wrtten as Where n1 s( x) c B ( x) (4) 3 c ' s are unknown coeffcents and B ( x ) s are thrd degree B-Splne functons. To solve second order boundary value problems, B ( x), B( x) and B ( x) evaluated at nodal ponts are needed, whch are summarzed n Table1. Table 1. Values of B ( x), B( x) and B ( x) at nodal ponts B ( x ) B ( x) B ( ) x x x / h 6 / h x / h x / h 6 / h x

4 408 Yogesh Gupta, Panka Kumar Srvastava and Mano Kumar 3 Descrpton of method For development of the B-splne method, consder the system (1)-() n the form d u a ( x) u ( x) a ( x) u ( x) f ( x) dx d u 1 1 a ( x) u ( x) a ( x) u ( x) f ( x) dx where u1( a) p, u1( b) q, u( a) r, u( b) s. (5) (6) (7) Let n1 u1 ( x) c B ( x) 3 n1 u( x) d B ( x) 3 (8) (9) be the approxmate soluton of gven system (1-), where c ' s and d ' s are unknown coeffcents and B ( x ) s are thrd degree B-Splne functons. Further, let x0, x1, x,..., x n be grd ponts n the ntervala, b, so b a that x ah( 0,1,... n) ; x0 a, xn b, h. n Now, the approxmate soluton gven by (8) and (9) must satsfy the gven system at the ponts x x. For, puttng values from (8) and (9) n (5-6), we get n1 n1 n1 c B ( x ) c a ( x ) B ( x ) d a ( x ) B ( x ) f ( x ) n1 n1 n1 d B( x ) c a ( x ) B ( x ) d a ( x ) B ( x ) f ( x ) (10) (11) 0,1,,..., n and boundary condtons(7) gve

5 Applcaton of B-Splne to Numercal Soluton n1 3 n1 3 n1 3 n1 3 c B ( x) p, for x a c B ( x) q, for x b d B ( x) r, for x a d B ( x) s, for x b (1) (13) (14) (15) The values of splne functon at the knots are determned usng table 1 and substtutng n (10)-(15), a system of ( n 3) lnear equatons n ( n 3) unknowns c 3, c,..., cn 1, d 3, d,..., dn 1 s obtaned. Ths system can be wrtten n matrx vector form as AE F Where E c, c,..., c, d, d,..., d 3 n1 3 n1 F p, f ( x ),..., f ( x ), q, r, f ( x ),..., f ( x ), s n 0 T n T (16) (17) (18) And Where A A A A A ( x0 ) 1( x0 ) 1( x0 ) ( x1 ) 1( x1 ) 1( x1 ) A ( xn) 1( xn ) 1( xn ) (19)

6 410 Yogesh Gupta, Panka Kumar Srvastava and Mano Kumar A ( x0 ) ( x0) ( x0 ) ( x1 ) ( x1 ) ( x1 ) ( xn) ( xn ) ( xn ) ( x0 ) 3( x0) 3( x0) ( x1 ) 3( x1 ) 3( x1 ) A ( xn ) 3( xn) 3( xn ) A ( x0 ) 4( x0) 4( x0 ) ( x1 ) 4( x1 ) 4( x1 ) ( xn) 4( xn ) 4( xn ) Where elements of A are gven by, ( 0,1,..., n ) s 61 1( x ) a 11( x ) 1( x ), ( x ) a1( x ) ( x ), h 6 3( x ) a ( x ) 3( x ) 4( x ) a1( x ) 4( x ) h 11 1( x ) 4 a 11( x ), ( x ) 4 a1( x ) h

7 Applcaton of B-Splne to Numercal Soluton ( x ) 4 a ( x ), 4( x ) 4 a1( x ) h The splne soluton of gven system s obtaned by solvng the above matrx equaton. 4 Computatonal results In ths secton, we present two numercal examples from the lterature to llustrate the applcablty and effcency of the method presented n secton 3. Frst perturbed system nvolves BVPs wth constant coeffcents whereas second contans varable coeffcents perturbed BVPs. The numercal results presented here by our algorthm are compared wth those of Valanarasu et. al. [19] and Matthews et. al. [15]. For the sake of comparson of results, dfferent modes of expressng the results for the two problems are consdered. Table corresponds to the numercal results of frst problem, where pont wse errors on nodes for u 1 and u are gven. Numercal results for problem are tabulated n table 3 and table 4. For ths problem maxmum absolute errors for u1 and u are shown for dfferent choces of n and. Example 1. (For 1 ; constant coeffcents) [19] d 0 dx r 4 r 1 u( x) u( x), x D [0,1], 1 3 d 0 (0) dx r 0 r 0 u(0), u(1). 0 0 (1) Example. (Varable coeffcents problem wth 1, 1) [15] d u1 3 x ( x 1) u1( x) (1 x ) u( x) e, dx d u1 x 1 x u 1 x e u x x cos( ) ( ). ( ) 10 1, dx 4 () wth the boundary condtons

8 41 Yogesh Gupta, Panka Kumar Srvastava and Mano Kumar u (0) u (1) u (0) u (1) (3) 4 5 Table : Numercal soluton of Example 1 wth 10, h 10. Error n u1 Error n u Nodes Method by Valanarasu [19] Present method Method by Valanarasu [19] Present method 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 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 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+00

9 Applcaton of B-Splne to Numercal Soluton Table 3: Approxmate errors for u1 of the soluton of Example n 4 6 Method by Present Method by Present Matthews[15] method Matthews[15] method E E E E E E E E E-04.9E E E E E E E E E-06.67E E-06 Table 4: Approxmate errors for u of the soluton of Example n 4 6 Method by Present Method by Present Matthews[15] method Matthews[15] method E E E E E E E E E E E E E E E E E-07.53E E E-07 5 Concluson In ths paper, a numercal technque for a system of sngularly perturbed boundary value problems usng B-splne functons s derved. Smplcty of the adaptaton of B-splnes and obtanng acceptable solutons can be noted as advantages of gven numercal methods. The results obtaned usng ths method are better than those usng the stated exstng methods wth the same number of knots and values of. References [1] G.M. Amralyev, The convergence of a fnte dfference method on layeradapted mesh for a sngularly perturbed system, Appled Mathematcs and Computaton 16 (005) [] S. Bellewa, E. O Rordan, A parameter robust numercal method for a system of two sngularly perturbed convecton dffuson equatons, Appled Numercal Mathematcs 51 (004)

10 414 Yogesh Gupta, Panka Kumar Srvastava and Mano Kumar [3] N. Caglar, H. Caglar, B-splne method for solvng lnear system of second order boundary value problems, Computers and Mathematcs wth Applcatons 57 (009) [4] H. Caglar, N. Caglar, K. Elfatur, B-splne nterpolaton compared wth fnte dfference, fnte element and fnte volume methods whch appled to two pont boundary value problems, Appled Mathematcs and Computaton 175 (006) [5] M. Dehghan, M. Lakestan, Numercal soluton of nonlnear system of second-order boundary value problems usng cubc B-splne scalng functons, Internatonal Journal of Computer Mathematcs, 85(9)(008) [6] M.K. Kadalbaoo, V. Gupta, Numercal soluton of sngularly perturbed convecton-dffuson problem usng parameter unform B-splne collocaton method, Journal of Mathematcal Analyss and Applcatons, 355 (009) [7] M.K. Kadalbaoo, V. Gupta, A. Awasth, A unformly convergent B-splne collocaton method on a non-unform mesh for sngularly perturbed onedmensonal tme-dependent lnear convecton-dffuson problem, Journal of Computatonal and Appled Mathematcs 0 (008) [8] M.K. Kadalbaoo, Puneet Arora, B-splne collocaton method for the sngular perturbaton problem usng artfcal vscosty, Computers and Mathematcs wth Applcatons, 57 (009) [9] M.K. Kadalbaoo, V.K. Aggarwal, Ftted mesh B-splne collocaton method for solvng self-adont sngularly perturbed boundary value problems, Appled Mathematcs and Computaton 161 (3) (005) [10] M.K. Kadalbaoo, V.K. Aggarwal, Ftted mesh B-splne method for solvng a class of sngular sngularly perturbed boundary value problems, Internatonal Journal of Computer Mathematcs 8 (005) [11] M. Kumar, Y. Gupta, Methods for solvng sngular boundary value problems usng splnes: a revew. Journal of Appled Mathematcs and Computng, 3(010) [1] N. Madden, Numercal methods for wave-current nteractons, Ph.D., Natonal Unversty of Ireland, Cork, 000. [13] N. Madden, M. Stynes, A unformly convergent numercal method for a coupled system of two sngularly perturbed lnear reacton dffuson problems, IMA Journal of Numercal Analyss 3 (4) (003) [14] S. Matthews, J.J.H. Mller, E. O Rordan, G.I. Shshkn, Parameter-robust numercal methods for a system of reacton dffuson problems wth boundary layers, n: J.J.H. Mller, G.I. Shshkn, L. Vulkov(Eds.), Analytcal and Numercal Methods for Convecton-Domnated and Sngularly Perturbed Problems, Nova Scence Publshers, Inc., New York, USA, 000, pp

11 Applcaton of B-Splne to Numercal Soluton [15] S. Matthews, E. O Rordan, G.I. Shshkn, A numercal method for a system of sngularly perturbed reacton dffuson equatons, Journal of Computatonal and Appled Mathematcs 145 (00) [16] S.C.S. Rao, M. Kumar, Optmal B-splne collocaton method for selfadont Sngularly Perturbed Boundary Value Problems, Appled Mathematcs and Computaton 188 (007) [17] S.C.S. Rao, M. Kumar, B-splne collocaton method for nonlnear sngularly-perturbed two-pont boundary-value problem, Journal of Optmzaton Theory and Applcatons, 134 (007) [18] G.I. Shshkn, Mesh approxmaton of sngularly perturbed boundaryvalue problems for systems of ellptc and parabolc equatons, Computatonal Mathematcs and Mathematcal Physcs 35 (4) (1995) [19] T. Valanarasu, N. Ramanuam, An asymptotc ntal value method for boundary value problems for a system of sngularly perturbed second order ordnary dfferental equatons, Appled Mathematcs and Computaton 147 (004) 7 40.

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