Parametric Spline Method for Solving Bratu s Problem

Transcription

1 ISSN print, online International Journal of Nonlinear Science Vol4202 No,pp3-0 Parametric Spline Metod for Solving Bratu s Problem M Zarebnia, Z Sarvari 2,2 Department of Matematics, University of Moageg Ardabili, , Ardabil, Iran Received 2 December 20, accepted 6 July 202 Abstract: In tis paper, parametric spline metod is introduced for solving Bratu s problem Te convergence analysis of te presented metod is discussed Te metod is illustrated wit two numerical examples and te results sow tat te metod converges rapidly and approximates te exact solution very accurately Keywords: Numerical metod; Parametric spline; Quintic spline; Convergence analysis; Bratu s problem; Boundary-value problem Introduction Consider te Liouville-Bratu-Gelfand equation[-3]: { ut + λe ut = 0, t Ω, ut = 0, t Ω, were λ > 0, and Ω is a bounded domain Te Bratu-type model[2-4] in one-dimensional planar coordinates is of te form: u t + λe ut = 0, 0 t, 2 u0 = u = 0, 3 and is used to model a combustion problem in a numerical slab Te Bratu-type models appear in a number of applications suc as te fuel ignition of te termal combustion teory and in te Candrasekar model of te expansion of te universe It stimulates a termal reaction process in a rigid material were te process depends on te balance between cemically generated eat and eat transfer by conduction, see [5-9] and te references tere in Several numerical metods for approximating te solution of Bratu s problem are known Laplace transform decomposition numerical algoritm is used for solving Bratu s problem [0] Te Perturbation-iteration algoritm [], applied to Bratu-type equationsmosen etal [2] introduced new smooter to enance multigrid-based metods for Bratu problem One-point pseudospectral collocation metod as been used for te solution of te one-dimensional Bratu equation [3] Te main purpose of te present paper is to use parametric quintic spline metod [4-6] for te numerical solution of nonlinear boundary value problem 2 Te metod consists of reducing te problem to a set of nonlinear algebraic equations Te outline of te paper is as follows First, in Section 2 we introduce parametric quintic spline metod and describe te basic formulation of spline approximation required for our subsequent development Section 3 outlines convergence analysis of te parametric spline metod for solution of Bratu s problem Finally numerical examples are given in section 4 to illustrate te efficiency of te presented metod Corresponding autor address: zarebnia@umaacir Copyrigt c World Academic Press, World Academic Union IJNS202085/632

2 4 International Journal of Nonlinear Science, Vol4202, No, pp Description of te metod Consider te partition of [a, b] R Let S k denote te set of piecewise polynomials of degree k on subinterval I i = [t i, t i ] of partition In tis work, we consider parametric quintic spline metod for finding approximate solution of variational problems Consider te grid points t i on te interval [a, b] as follows: a = t 0 < t < t 2 < < t n < t n = b, t i = t 0 + i, i = 0,, 2,, n, = b a n, were n is a positive integer Let S t, τ be quintic spline function of class C 4 [a, b] tat interpolates ut at te grid points {t i } n i=0 Also, S t, τ depends on a parameter τ > 0 tat is called a parametric spline function also, S t, τ reduces to a ordinary quintic spline as τ 0 By considering parametric quintic spline S t, τ = S t, te spline function S t satisfies in te following equation: S 4 t + τ 2 S 2 t = S 4 t i + τ 2 S 2 t i [ t t i ] 4 + S t i + τ 2 S 2 t i [ t i t], 4 were t [t i, t i ], S t i = ut i, and = t i t i Te Eq4 is a inomogeneous ordinary differential equation We solve te Eq4 and obtain te constants of integration by using interpolation conditions at te endpoints of te interval [t i, t i ], ten we get: were S t = t t i ui + t i t [ ui + 2 3! t i t 3 t i t + M i t t i t t i sin w t i t sin w sin w t t i 3 t t i M i ] + w 4[ w 2 t t i 3 t t i 3! ] F i + w 4[ w 2 t i t 3 t i t 3! t i t sin w ] F i, 5 S t i = ut i = u i, S t i = M i, S 4 t i = F i, w = τ, τ > 0 6 were We use te continuity of first and tird derivatives of spline function 5 at t i, and obtain te following result: M i+ + 4M i + M i = 6 2 ui+ 2u i + u i 6 2 α F i+ + 2β F i + α F i, 7 M i+ 2M i + M i = 2 αf i+ + 2βF i + αf i, 8 α = w 2 w csc w, β = w 2 w cot w, 9 α = w 2 6 α, β = w 2 3 β 0 Considering Eqs 7, 8 and also some simple calculations, we can obtain te value of F i as follows: [ F i = 2 2 α + 6α M i+ + M i + 4α 2α M i α β αβ 6α ] ui+ 2 2u i + u i IJNS for contribution: editor@nonlinearscienceorguk

3 M Zarebnia, Z Sarvari: Parametric Spline Metod for Solving Bratu s Problem 5 Having used Eq and replaced F i, F i and F i+ in Eq 8, te following result is obtained: p 2 M i+2 + M i2 + 2 sm i + 2 q M i+ + M i = α ui+2 + u i2 were +2β α u i+ + u i + 2α 4β ui, i = 2, 3,, n 2, 2 p = α [ 6 + α ] [ ], q = 2 2α + β α β, s = 2 α + 4β + α 2β For a numerical solution of te Bratu s problem 2 and 3, te interval [0, ] is divided into a set of grid points wit step size Setting t = t i = t 0 + i, in Eq 2, we obtain: by using te assumption S t i = M {i} we get: u t i = λe ut i, 4 M i = λe uti 5 Te following equation is obtained by replacing M i+j, j = 2,, 0,, 2 as Eq 5 in Eq2, after simplification, we ave: 2[ p λe ui+2 λe ui2 + q λe ui+ λe ui sλe ui ] = α u i+2 + u i2 + 2β α ui+ + u i + 2α 4β ui, 6 αu i2 + pλ 2 e u i2 + 2β αu i + qλ 2 e u i + 2α 4β u i + sλ 2 e u i + 2β αu i+ + qλ 2 e u i+ + αu i+2 + pλ 2 e u i+2 = 0, i = 2, 3,, n 2, 7 were u 0 = 0, u n = 0 Using Taylor s series for Eq 7, we can obtain local truncation error as follows: t i = 4[ ] 7α + β 4p + q u [ 0080 In Eq 8, if α = 2 and β = 5 2, ten 27α + β 360 i + 6[ ] 3α + β 6p + q u 6 i 80 2 ] 64p + q u 8 i + O 9 8 p = 360, q = , t i = O 8, i = 2, 3,, n 2 9 Te nonlinear system 7 consists of n 3 equation wit n unknowns u i, i =, 2,, n To obtain unique solution, we need two more equations wic are supplied by using metod of undetermined coefficients 4u 0 7u + 2u 2 + u 3 = 2[ u u u u u 4 + ] 60 u 5, 20 4u n 7u n + 2u n2 + u n3 = 2[ u n u n u n2 + 3 u n u n4 + ] 60 u n5 2 Considering tat u 0 = 0, u n = 0 and u i = λeu i, terefore we can rewrite te Eqs 2 and 2 as follows: 7u λ2 e u + 2u λ2 e u 2 + u λ2 e u λ2 e u λ2 e u 5 + λ 2 7 = 0, u n λ2 e u n +2u n λ2 e u n2 +u n3 + 3 λ2 e u n λ2 e u n λ2 e u n5 +λ 2 7 = Te above nonlinear system consists of n equations wit n unknowns u i, i =,, n Solving tis nonlinear system by Newton s metod, we can obtain an approximation to te solution of 2 IJNS omepage: ttp://wwwnonlinearscienceorguk/

4 6 International Journal of Nonlinear Science, Vol4202, No, pp Convergence analysis Now we discuss te convergence of te parametric spline metod for te Bratu s problem 2 We consider te Eqs 7, 2 and 23 and ten rewrite tese equations in te matrix form wic is te nonlinear system: A 0U + λ 2 BF U = 0, 24 were U = T u, u 2,, u n Also A 0 = [ ] [ ] a ij, B = bij are n n -dimensional and define as follows: A =, B = and F U = diag e u i, i =, 2,, n Teorem [7,Teorem 77] Let M be a matrix suc tat M <, and let I denote te unit matrix Ten I+M exists, and I + M < M Consider te matrix A 0 defined by 25, ten we can write: 2A 0 = A 0 A + M, 27 were and 2, i = j, A 0 =, i j =, 0, ow, M = 4, i = j, A =, i j =, 0, ow, We know tat te inverse of A 0 exists and is bounded as follows [8]: and also for A we ave [4]: A b a , 30 A 2 3 IJNS for contribution: editor@nonlinearscienceorguk

5 M Zarebnia, Z Sarvari: Parametric Spline Metod for Solving Bratu s Problem 7 Considering te Eq 29, we get M = 0 From Eq 27 we can write A 0 = 2 A 0A I + A A 0 M 32 Now by using te following teorems, we sow te inverse of A 0, defined by Eq 32, exists and is bounded Teorem 2 [8,Teorem 72] A five-diagonal matrix D = [d ij ] is irreducible, if and only if d i,i 0 i = 2, 3, n, d i,i+ 0 i =, 2, n, d i,i2 0 i = 3, 4, n, d i,i+2 0 i =, 2, n 2 Teorem 3 [8,Teorem 74] Let te matrix M = [m ij ] be irreducible and satisfy te conditions ten M is monotone i m ij 0, i j, i, j =, 2,, n, n { 0, i =, 2, n, ii m ij > 0, for at least one i, j= Teorem 4 [8,section 7-22] A monoton matrix is nonsingular Since A 0 satisfies te conditions of Teorems 2 and 3, ten according Teorem 4, A 0 is nonsingular Considering te Eq 32, we can write; A 0 = 2 I + A A 0 M A A 0 33 Hence, by applying Teorem and Eqs 30, 3 and M = 0, we conclude: I + A A 0 M < b a 2 34 Having used te Eqs 33 and 34, we ave: In te following teorem we sow tat te inverse of exists Teorem 5 If Y = F < Proof Consider te Eq 36, ten ba 2 6b a 2 A 0 < 82 55b a 2 35 A = A 0 + λ 2 BF, 36 λ 2 ba 2 ten te inverse of A, defined by Eq 36, exists A = A 0 I + λ 2 A 0 BF 37 From Teorem 4 we know tat te A 0 exists Now, we need te existence of I + λ 2 A 0 BF According to Teorem is sufficient, we sow tat λ 2 A 0 BF < Having used Eq 35 and also B = 79, we obtain: λ 2 A 0 BF λ 2 A 0 B F < λ 2 b a b a 2Y 38 Considering assumption Y < ba 2, we ave: λ 2 ba 2 79 λ 2 A 0 BF < 39 Terefore, by using Teorem and Eqs 37 and 39 we conclude te existence of A We can also obtain a bound on te errors E = U U n in te maximum norm, were U = ut, ut 2,, ut n is te exact solution and U n = u, u 2,, u n is te approximate solution of Bratu s problem 2 From Teorem 5, we can derive a bound on E 240 IJNS omepage: ttp://wwwnonlinearscienceorguk/

6 8 International Journal of Nonlinear Science, Vol4202, No, pp 3-0 Teorem 6 Let T be te vector of local truncation error and AE = T, ten Proof By using Teorem 5 and AE = T, we can write: terefore, we get Having used Eq 39 and Teorem we obtain: Now, by applying Eqs 35, 38 and 43, we ave: E = O 6, wen α = 2, β = E = A T = A 0 + λ 2 BF T = I + λ 2 A 0 BF A 0 T, 4 E I + λ 2 A 0 BF A 0 T 42 I + λ 2 A 0 BF λ 2 A 0 BF 43 I + λ 2 A 0 BF 40K 40K 79Y λ 2 b a 2, 44 were K = b a 2 Considering te Eq 8 and α = 2, β = 5 2 M 8 = max a t b u8 t Terefore from Eq 4, we conclude tat: E, we obtain T M 8, were 240b a 2 8 M K 79Y λ 2 b a 2 = O Numerical illustrations In order to illustrate te performance of te parametric spline metod for te Bratu equation 2 and justify te accuracy and efficiency of te metod, we consider te following examples Te example ave been solved by presented metod wit different values of λ We take α = 2, β = 5 2 and n = 0 Te errors are reported on te set of uniform grid points S = {a = t 0,, t i,, t n = b}, t i = t 0 + i, Te absolute error on te uniform grid points S is i = 0,, 2,, n, = b a n 46 ut j u j, 0 j n, 47 were ut j is te exact solution of te given example, and u j is te computed solution by te parametric spline metod Te exact solution of te equation 2 is given in [-3] as: [ cos t 2 ut = 2 ln θ ] 2 cos, 48 θ 4 were θ satisfies θ = 2λ cos θ 49 4 Te Bratu problem as zero, one or two solutions wen λ > λ c, λ = λ c and λ < λ c respectively, were te critical value λ c satisfies te equation: IJNS for contribution: editor@nonlinearscienceorguk

7 M Zarebnia, Z Sarvari: Parametric Spline Metod for Solving Bratu s Problem 9 Table : λ =, α = 2, β = 5 2, n = 0 x P resent metod Laplace[20] Decomposition[2] B spline[9] Table 2: λ = 2, α = 2, β = 5 2, n = 0 x P resent metod Laplace[20] Decomposition[2] B spline[9] = 4 2λc sin θ c 50 4 It was evaluated in [2 4] tat te critical value λ c is given by λ c = Te maximum absolute errors in solutions of Bratu problems are compared wit metods in [9-2] for n=0 and tabulated in Tables -2 Te tables sow tat our results are more accurate Example Consider te following Bratu-type model wit boundary conditions u t + e ut = 0, 0 t, 5 u0 = 0, u = 0 In Eq 5, we ave λ = Applying te parametric spline metod, following approximations are obtained and te numerical results are tabulated in Table Example 2 Consider te following boundary value problem of te Bratu-type wen λ = 2: u t + 2e ut = 0, 0 t, u0 = u = 0 52 In Table 2, parametric spline solutions for te case λ = 2 are compared wit te numerical metods given in [9-2] IJNS omepage: ttp://wwwnonlinearscienceorguk/

8 0 International Journal of Nonlinear Science, Vol4202, No, pp Conclusion In tis paper, parametric quintic spline metod is applied for solving te Bratu equation Te parametric spline metod reduce te computation of te Bratu equation to some nonlinear algebraic equations Te analytical results are illustrated wit two numerical examples Te proposed sceme is simple and computationally attractive References [] U M Ascer and R Mateij and R D Russell Numerical solution of boundary value problems for ordinary differential equations SIAM, Piladelpia, PA 995 [2] J P Boyd Cebysev polynomial expansions for simultaneous approximation of two brances of a function wit application to te one-dimensional Bratu equation Matematics and Computation, : [3] R Buckmire Investigations of nonstandard Mickens-type finite-difference scemes for singular boundary value problems in cylindrical or sperical coordinates Numerical Metods for partial Differential equations, : [4] J Jacobson and K Scmitt Te Liouville-Bratu-Gelfand problem for radial operators Journal of Differential Equations, : [5] S Li and S J Liao An analytic approac to solve multiple solutions of a strongly nonlinear problem Applied Matematics and Computation, : [6] A M Wazwaz Adomian decomposition metod for a reliable treatment of te Bratu-type equations Applied Matematics and Computation, : [7] R Buckmire Application of Mickens finite-difference sceme to te cylindrical Bratu-Gelfand problem Numerical Metods for Partial Differential Equations, : [8] A S Mounim and B M de Dormale From te fitting tecniques to accurate scemes for te Liouville-Bratu- Gelfand problem Numerical Metods for Partial Differential Equations, : [9] I H A H Hassan and V S Erturk Applying differential transformation metod to te one-dimensional planar Bratu problem International Journal of Contemporary Matematical Sciences, 22007: [0] S A Kuri A new approac to Bratu s problem Applied Matematics and Computation, :3 36 [] Yiğit Aksoy and Memet Pakdemirli New perturbation-iteration solutions for Bratu-type equations Computers and Matematics wit Applications, 59200: [2] A Mosen, LF Sedeek and S A Moamed New smooter to enance multigrid-based metods for Bratu problem Applied Matematics and Computation, : [3] J P Boyd One-point pseudospectral collocation for te one-dimensional Bratu equation Applied Matematics and Computation, 2720: [4] R A Usmani and SA Wasrt Quintic spline solutions of boundary value problems Comput Mat wit Appl, 6980: [5] A Kan, I Kan and T Aziz A survey on parametric spline function approximation Applied Matematics and Computation, 72005: [6] A Kan Parametric cubic spline solution of two point boundary value problems Applied Matematics and Computation, :75 82 [7] B N Datta Numerical Linear Algebra and Applications Brooks/Cole Publications, Co, Pacific Grove, California 995 [8] P Henrici Discrete Variable Metods in Ordinary Differential Equations Wiley, New York 964 [9] H Caglar, N Caglar, M zer, Antonios Valaristos and Antonios N Anagnostopoulos B-spline metod for solving Bratus problem Int J Comput Mat, :885 89first publised [20] J S McGoug Numerical continuation and te Gelfand problem Appl Mat Comput, 89998: [2] S Liao and Y Tan A general approac to obtain series solutions of nonlinear differential equations Stud Appl Mat, 92007: IJNS for contribution: editor@nonlinearscienceorguk