adomian method

Surveys in Mathematics and its Applications ISSN 1842-6298 Volume 2(2007) , 11 – 20 ADOMIAN DECOMPOSITION METHOD FOR NONLINEAR STURM-LIOUVILLE PROBLEMS Sennur Somali and Guzin Gokmen Abstract . In this paper the Adomian decomposition method is applied to the nonlinear Sturm- Liouville problem −y/prime/prime+y(t)p=λy(t), y (t)>0, t∈I= (0 ,1), y(0) = y(1) = 0 , where p > 1 is a constant and λ > 0 is an eigenvalue parameter. Also, the eigenvalues and the behavior of eigenfuctions of the problem are demonstrated. 1 Introduction Recently a great deal of interest has been focused on the application of Adomian’s decomposition method for the solution of many different problems. For example in [6], [12], [15]-[20] boundary value problems, algebraic equations and partial dif- ferential equations are considered. The Adomian decomposition method, which accurately computes the series solution, is of great interest to applied sciences. The method provides the solution in a rapidly convergent series with components that are elegantly computed. The main advantage of the method is that it can be applied directly for all types of differential and integral equations, linear or nonlinear, homogeneous or inhomogeneous, with constant coefficients or with variable coefficients. Another important advantage is that the method is capable of greatly reducing the size of computation work while still maintaining high accuracy of the numerical solution. There has been great interest in nonlinear Sturm-Liouville problems. Theory and algorithms which compute a given number of eigenvalues of the radial symmetric p- Laplacian are presented in [ 13], [8] and [ 9]. Asymptotic expansion of the eigenvalues of a specific problem is studied in [ 14]. In this article we explore the possibilities of the decomposition method in the nonlinear Sturm-Liouville problem [ 14]. Let us consider a general functional equation y−N(y) =f, (1) 2000 Mathematics Subject Classification: 34L16. Keywords: Adomian decomposition method, nonlinear Sturm-Liouville problem. ****************************************************************************** http://www.utgjiu.ro/math/sma 12 S. Somali and G. Gokmen where Nis a nonlinear operator, fis a known function in which the solution y satisfying ( 1) is to be found. We assume that for every f, the problem ( 1) has a unique solution. The Adomian’s technique consists of approximating the solution of ( 1) as an infinite series y=∞/summationdisplay n=0yn (2) and decomposing the nonlinear operator Nas N(y) =∞/summationdisplay n=0An, (3) where Anare Adomain polynomials of y0, y1, ..., y n(see [ 3], [4], [5]) given by An=1 n!dn dµn/bracketleftBigg N/parenleftBigg∞/summationdisplay i=0µiyi/parenrightBigg/bracketrightBigg µ=0, n = 0,1,2, .... (4) Substituting ( 2) and ( 3) into ( 1) yields ∞/summationdisplay n=0yn−∞/summationdisplay n=0An=f. (5) Thus, we can identify y0=f, yn+1=An(y0, y1, . . . , y n), n = 0,1,2, . . . . (6) We then define the M-term approximant to the solution yby φM[y] =M/summationdisplay n=0yn with lim M→∞φM[y] =y. Convergence of the Adomian decomposition scheme was established by many authors by using fixed point theorems [ 1], [2], [5], [10]. ****************************************************************************** Surveys in Mathematics and its Applications 2(2007) , 11 – 20 http://www.utgjiu.ro/math/sma Adomian Decomposition Method for Nonlinear Sturm-Liouville Problems 13 2 Application to Sturm-Liouville Problems We consider the nonlinear Sturm-Liouville problem −y/prime/prime+y(t)p=λy(t), t ∈I= (0,1), y(t)>0, t ∈I, (7) y(0) = y(1) = 0 , where p > 1 is a constant and λ > 0 is an eigenvalue parameter. It is known by [7] and [ 11] that for each α >0,there exists a unique solution ( λ, y) = (λ(α), yα)∈ R+×C2(I) with /bardblyα/bardbl2=α. The purpose of this paper is to explore the eigenvalues and the structure of positive eigenfunctions of ( 7) as indicated in [ 14] by diagrams that are obtained approximately by using the Adomian Decomposition method. To begin with, ( 7) can be written in an operator form Ly =y(t)p−λy(t), (8) y(0) = y(1) = 0 , where L=d2 dt2is the differential operator. Operating on both sides of ( 8) with the inverse operator of L(namely L−1[.] =t/integraltext 0x/integraltext 0[.]dsdx ) and using the first boundary condition y(0) = 0 yields y(t, λ) =at+L−1(y(t)p−λy(t)), (9) where a=y/prime(0)/negationslash= 0 is not given but will be determined by using the other boundary condition. Substituting ( 2) and ( 3) into ( 9) gives ∞/summationdisplay n=0yn=at+L−1(∞/summationdisplay n=0An)−L−1(λ∞/summationdisplay n=0yn), (10) where Anare the Adomian polynomials. Identifying the zeroth component y0(t) byat,the remaining components yn(t), n≥1 can be determined by using the recurrence relation y0(t) = at, (11) yk+1=L−1(Ak−λyk), k ≥0, where Akare Adomian polynomials ( 4) involving the nonlinear term N(y) =ypand ****************************************************************************** Surveys in Mathematics and its Applications 2(2007) , 11 – 20 http://www.utgjiu.ro/math/sma 14 S. Somali and G. Gokmen given by A0=N(y0) =yp 0, A1=y1N/prime(y0) =pyp−1 0y1, A2=y2N/prime(y0) +1 2y2 1N/prime/prime(y0) =pyp−1 0y2+1 2p(p−1)yp−2 0y2 1, (12) A3=pyp−1 0y3+p(p−1)yp−2 0y1y2+1 6p(p−1)(p−2)yp−3 0y3 1, ... Combining ( 11) and ( 12) yields y0(t) = at, y1(t) =aptp+2 (p+ 1)( p+ 2)−λat3 3!, y2(t) =pa2p−1t2p+3 (p+ 1)( p+ 2)(2 p+ 2)(2 p+ 3), (13) −λaptp+4 (p+ 3)( p+ 4)(p 3!+1 (p+ 1)( p+ 2)) +λ2at5 5!, ... It is in principle, possible to calculate more components in the decomposition series to enhance the approximation. In view of ( 13), the solution y(t) is readily obtained in a series form by y(t;λ) =a√ λsin(√ λ)t (14) +aptp+2 (p+ 1)( p+ 2) +pa2p−1t2p+3 (p+ 1)( p+ 2)(2 p+ 2)(2 p+ 3) −λaptp+4 (p+ 3)( p+ 4)(p 3!+1 (p+ 1)( p+ 2)) +· · · The other boundary condition y(1, λ) = 0 (15) gives a nonlinear equation F(λ, a) = 0 ,from which it is possible to obtain the branching diagram of the problem ( 7). ****************************************************************************** Surveys in Mathematics and its Applications 2(2007) , 11 – 20 http://www.utgjiu.ro/math/sma Adomian Decomposition Method for Nonlinear Sturm-Liouville Problems 15 Since ( 7) is an autonomous and from Lemma 2.1 in [ 14] we know that y(t) satisfies for p >1 y(t) =y(1−t), 0≤t≤1. It follows that y/prime(0, λ) =y/prime(1, λ) (16) and then we have a nonlinear system of equations as F(λ, a) = 0 , G(λ, a) = 0 , (17) where G(λ, a) =y/prime(0, λ)−y/prime(1, λ).Solving the system ( 17) numerically, we can obtain the values of λanda. 2.1 The linear case: p= 1 The iterations are then determined in the following recursive way y0=at, yk+1=−L−1((λ−1)yk), k = 0,1,2, . . . . (18) or equivalently yk+1(t;λ) =−t/integraldisplay 0t1/integraldisplay 0(λ−1)yk(s;λ)dsdt 1. (19) It is clear that the convergence of the method depends on λand the size of the norm /bardblL−1/bardblfor the set {yn}.In the linear case, the decomposition method is equivalent to a classical iterative method, but the a posterior calculations of y(0) and y/prime(0), by imposing to eachM/summationtext n=0yn(t) to verify the boundary conditions, determines a set {yn} suitable for a good convergence. The recurrence relation ( 19) gives y0=at, yk(t;λ) = ( −1)kt/integraldisplay 0t2k−1/integraldisplay 0...t2/integraldisplay 0t1/integraldisplay 0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright 2k times(λ−1)ky0dsdt 1...dt 2k−2dt2k−1, (20) that is, yk(t;λ) =a(−1)k((λ−1)kt2k+1 (2k+ 1)!, k= 0,1, . . . . (21) ****************************************************************************** Surveys in Mathematics and its Applications 2(2007) , 11 – 20 http://www.utgjiu.ro/math/sma 16 S. Somali and G. Gokmen The solution in a series form is thus given by y(t;λ) = a∞/summationdisplay k=0(−1)k(λ−1)kt2k+1 (2k+ 1)!(22) =a√ λ−1∞/summationdisplay k=0(−1)k((√ λ−1)t)2k+1 (2k+ 1)!. It is clear that the infinite series of the solution converges to y(t;λ) =a√ λ−1sin(√ λ−1)t, a /negationslash= 0. (23) Using the boundary condition y(1) = 0, the eigenvalues are computed exactly λn= 1 + n2π2, n = 0,1, . . . . So, the effectiveness and the usefulness of the Adomain method are demonstrated by finding exact eigenvalues and eigenfunctions of linear Sturm-Liouville problem. 2.2 The nonlinear case: p= 2 Branching diagram for equation ( 7) is shown for M= 20 in Figure 1. Solution curves (eigenfunctions) for the linear and nonlinear cases are shown respectively in Figure 2 and Figure 3 corresponding to various eigenvalues. All computations are performed in Mathematica 4.0. The nonlinear problem behaves quite differently from the linear problem. It can be easily seen the smoothness of the solution curves in the linear case and the shape of the positive solutions associated with various eigenvalues λis almost ”box” in nonlinear case which agrees with [ 14]. In the following table, the approximate eigenvalues in [ π2,4π2) are given for M= 20 and the positive solutions corresponding to these eigenvalues are drawn as in Figure 3. ****************************************************************************** Surveys in Mathematics and its Applications 2(2007) , 11 – 20 http://www.utgjiu.ro/math/sma Adomian Decomposition Method for Nonlinear Sturm-Liouville Problems 17 a=y/prime(0) λ 2.2408×10−89.8696 0.04597 9.88203 0.20987 9.92628 0.70005 10.0584 1.7354 10.3364 3.58172 10.8287 6.55366 11.6117 11.035 12.7717 17.4844 14.4005 26.4008 16.5798 38.3541 19.4179 50.5994 21.6492 82.3224 24.2605 92.7438 25.225 0 50 100 150 200 Λ050100150200 /ScriptA Figure 1: ****************************************************************************** Surveys in Mathematics and its Applications 2(2007) , 11 – 20 http://www.utgjiu.ro/math/sma 18 S. Somali and G. Gokmen 0.2 0.4 0.6 0.8 1t -0.10.10.20.3y/LParen1t/RParen1 Figure 2: 0.2 0.4 0.6 0.8 1t2468101214y/LParen1t/RParen1 Figure 3: References [1]K. Abbaoui, Y. Cherruault, Convergence of Adomian’s method applied to non- linear equations , Math. Comput. Modelling 20(9) (1994) 60-73. MR1302630 . Zbl 822.65027 . ****************************************************************************** Surveys in Mathematics and its Applications 2(2007) , 11 – 20 http://www.utgjiu.ro/math/sma Adomian Decomposition Method for Nonlinear Sturm-Liouville Problems 19 [2]K. Abbaoui, Y. Cherruault, New ideas for proving convergence of decomposition methods , Comput. Math. Appl. 29 (7) (1995) 103-108. MR1321262 (95k:65057). Zbl 0832.47051 . [3]G.Adomian, Stochastic Systems, Academic Press , New York, 1983. MR0714710 (86d:93001). Zbl 0523.60056 . [4]G. Adomian, Nonlinear Stochastic Operator Equations , Academic Press, New York, 1986. MR0872695 (88j:60112). Zbl 0609.60072 . [5]G. 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Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type eqautions , Appl. Math. Comput. 166 (2005) 652-663. MR2151056 . Zbl 1073.65068 . Sennur Somali Guzin Gokmen Dokuz Eylul University Dokuz Eylul University Faculty of Arts and Sciences Faculty of Arts and Sciences Department of Mathematics, Department of Mathematics, Tinaztepe Kamp¨ us¨ u, 35160 Buca, Izmir, Tinaztepe Kamp¨ us¨ u, 35160 Buca, Izmir, Turkey. Turkey. e-mail: [email protected] e-mail:[email protected] ****************************************************************************** Surveys in Mathematics and its Applications 2(2007) , 11 – 20 http://www.utgjiu.ro/math/sma