A Modified Adomian Decomposition Method for Solving Higher-Order Singular Boundary Value Problems

Transcription

1 A Modified Adomian Decomposition Method for Solving Higher-Order Singular Boundary Value Problems Weonbae Kim a and Changbum Chun b a Department of Mathematics, Daejin University, Pocheon, Gyeonggi-do 487-7, Republic of Korea b Department of Mathematics, Sungkyunkwan University, Suwon , Republic of Korea Reprint requests to C C; Fax: , cbchun@skkuedu Z Naturforsch 65a, 9 ); received August 4, 9 / revised April 9, In this paper, we present a reliable modification of the Adomian decomposition method for solving higher-order singular boundary value problems He s polynomials are also used to overcome the complex and difficult calculation of Adomian polynomials occurring in the application of the Adomian decomposition method Numerical examples are given to illustrate the accuracy and efficiency of the presented method, revealing its reliability and applicability in handling the problems with singular nature Key words: Adomian Decomposition Method; Singular Boundary Value Problems; Higher-Order Ordinary Differential Equation; Modified Adomian Decomposition Method; He s polynomials; Homotopy Perturbation Method PACS number: 6Lj Introduction The Adomian decomposition method ADM) is one of the powerful and reliable methods for solving various kinds of problems arising in applied sciences, linear and nonlinear as well This method, which accurately computes the series solution, is of great interest to applied sciences, ] The method provides the solution in a rapidly convergent series solution, if the equation has a unique solution, and it has been successfully applied to a wide class of problems arising in applied sciences, see 6], and the references therein The convergence of the decomposition series have been investigated by several authors 7 9] In recent years, the Adomian method has been modified so as to solve boundary value problems with singular nature by several authors ] The difficulty of those singular problems is due to the singularity behaviour that occurs at the point x = In this paper we introduce a new reliable modification of the ADM to overcome the singularity difficulty for higher-order boundary value problems He s polynomials with the homotopy perturbation method HPM) are also used to overcome the complex and difficult calculation of Adomian polynomials occurring in the application of the Adomian decom- position method, demonstrating that He s polynomials can completely replace Adomian s polynomials The HPM, first proposed by He in 998, was developed and improved by He in 4 6] This method is a novel and effective method 7 9], and has been successfully applied to solve many types of problems For a comprehensive survey on the method, new interpretation, and its applications, the reader is referred to 7, 8] Several numerical examples will be considered to illustrate that the proposed framework is well suited to attain an accurate solution to the higher-order singular boundary value problems, revealing its reliability and applicability Modified Adomian Decomposition Method Consider the singular boundary value problem of n + )-order ordinary differential equation in the form y n+) + m x yn) + Ny = gx), ) y)=a,y )=a,,y r ) ) = a r,yb)=c,y b)=c,, y n r) b)=c n r, ) / / 9 $ 6 c Verlag der Zeitschrift für Naturforschung, Tübingen

2 94 W Kim and C Chun Modified ADM for Solving Higher-Order Singular Boundary Value Problems where N is a nonlinear differential operator of order less than n, gx) is a given function, a,a,,a r, c,c,,c n r,b are given constants, where m r n,r We first rewrite ) in the form dn x x n y +m n + )xy or equivalently, dn x n x n m d x +n m)n )y ] + Ny = gx), m n+ dy )] +n m)n )x dn y + Ny = gx) n Equation 4) can be written in the operator form ) 4) L L y = gx)+m n)n )L y Ny, 5) where the differential operator L employs the first two derivatives L = x n m d x m n+ d ), 6) dn L = x n 7) in order to overcome the singularity behaviour at x = In view of 6) and 7), the inverse operators L and L are the integral operators defined by L )= x n m x m n ), 8) b L )= x ) 9) }{{} n times By applying L on 5), we have such that L y = Ψ x)+l gx) L Ny, ) L Ψ x)= ) By applying L on ), we have yx)=ψ x)+l Ψ x)+l L gx) L L Ny, ) such that L Ψ x)= ) The Adomian decomposition method introduces the solution yx) and the nonlinear function Ny by infinite series and yx)= Ny = n= n= y n x) 4) A n y,y,,y n ), 5) where the Adomian polynomials A n are defined as )] d n A n = y i λ ) i 6) n! dλ n N n i= λ = The components y n x) of the solution yx) will be determined recurrently, and Adomian polynomials can be constructed for various classes of nonlinearity according to specific algorithms presented in 6] and ] Substituting 4) and 5) into ) gives n= y n = Ψ x)+l Ψ x)+l L gx) L L n= A n 7) Identifying y = Ψ x)+l Ψ x)+l L gx), the Adomian method admits the use of the recursive relation y = Ψ x)+l Ψ x)+l L gx), y n+ = L A n, n, which gives y = Ψx)+L gx), y = L A, y = L A, y = L A, 8) 9) This will lead to the complete determination of the components y n x) of yx) The series solution of yx)

3 W Kim and C Chun Modified ADM for Solving Higher-Order Singular Boundary Value Problems 95 defined by 4) follows immediately For numerical purposes, the n-term approximant n Ψ n = y k ) k= can be used to approximate the exact solution Numerical Examples Example Consider the nonlinear boundary value problem We define y x y = x y5, ) y)=, L = x d y )= 8 ) x d ), ) so that L )= x x ) 4) After multiplying ) by x, in an operator form, it becomes L y = x4 y5 5) Applying L on both sides of 5) we find yx)=y)+ y )x + L x 4 y5 6) Using the decomposition series for the linear function yx) and the polynomial series for the nonlinear term y 5, we obtain y n = y)+ n= y )x + L This gives the recursive relationship x 4 n= A n ) 7) y x)=y)+ y )x, 8) x y k+ x)=l 4 ) A k, k 9) The Adomian polynomials for the nonlinear term y 5 are computed as follows: A = y 5, A = 5y 4 y, A = 5y y 4 + y y, A = 5y y 4 + y y y + y y, Substituting ) into 9) gives the components y = 85x, y = 6787x x x x 8 586x x x 4 +, y = 6577x 4494x x 8 659x + 7x 4x 4 +, y = 668x + 446x x 8 596x + 864x 55x 4 +, ) ) The solution in a series form is thus approximately given by y = y + y + y + y = 66845x x 4 54x x 8 84x + 9x 668x 4 +, ) which is in good agreement with the Taylor series of the exact solution yx)= given by + x yx)= 66667x x 4 574x x 8 x + 94x 958x 4 + ) For the above and later computations, the Maple package was used and the digits environment variable was

4 96 W Kim and C Chun Modified ADM for Solving Higher-Order Singular Boundary Value Problems set to as the number of digits that Maple uses when calculating with software floating-point numbers The results rounded off to seven decimal places are presented for the sake of simplicity Example Consider the nonlinear boundary value problem y + ρ x y = gx), 4) y)=, y b)=b ρ ρ)cosb b ρ sinb, 5) where We define gx)= x ρ cosx ρ)x ρ sinx 6) L = x ρ d x ρ d ), 7) L = x, 8) so that L )= x ρ x ρ ), 9) b L )=x ) 4) In an operator form, 4) becomes We find that L L y = L L y = gx) 4) = x x x ρ x b x ρ x ρ d x ρ dy )] x ρ x ρ y b ρ y b) ] = yx) y) bρ ρ y b)x ρ 4) Applying L,L on both sides of 4), we find after imposing the given boundary conditions yx)=y)+ bρ ρ y b)x ρ + x ρ xcosx ρ)sinx] = b ρ)cosb bsinb]x ρ ρ +cosx cosb]x ρ + bsinb ρ x ρ = x ρ cosx, 4) which is the exact solution of the problem 4) and 5) Example Consider the nonlinear boundary value problem y + x y y = gx), 44) y)=, y )=, y)=e, 45) where gx)=4e x +6xe x +x e x +x e x x 9 e x We use the Taylor series of gx) with order, gx) g T = 4 + 6x + 6x + 5x + 4x x5 + x x7 + 6 x x x 46) We define so that L = x d x d ), 47) L = x d, 48) L )= x x ) 49) L x )= x ) 5) In an operator form, 44) becomes L L y = g T x)+l y + y 5) Applying L,L on both sides of 5) and then incorporating the given boundary conditions, we find yx)=l L g T x)+l y + L L y 5) Using the decomposition series for the linear function yx) and the polynomial series for the nonlinear term y, we obtain n= y n = L L g T x)+l + L L n=a n ) This gives the recursive relationship n=y n ) 5) y x)=l L g T x), 54) y k+ x)=l y k + L L A k, k 55)

5 W Kim and C Chun Modified ADM for Solving Higher-Order Singular Boundary Value Problems 97 The Adomian polynomials for the nonlinear term y are computed as follows: A = y, A = y y, A = y y + y y, A = y y + 6y y y + y, 56) Substituting 56) into 55) gives the components of the solution y = x + 975x x x x 7 + 8x x x + 46x 5759x 65x, y = x x 4 + 9x x x 7 + 8x x 9 + 9x + x +, y = 979x + 66x x 5 + 5x x 7 + x x x x +, y = 9x + 89x 4 + 7x x x x x x x, +, 57) The solution in a series form is thus approximately given by y = y + y + y + y = x x x x x 7 + 8x x x + 48x, 58) which is in good agreement with the Taylor series of the exact solution yx)=x e x given by yx)=x + x 4 + 5x x x 7 + 8x x x + 48x + 59) Example 4 Consider the nonlinear boundary value problem y 4) + x y) + y y = gx), 6) y)=, y )=, y)=e, y )=e, 6) where gx) =8x e x + e x + xe x + x e x + x 4 e x x 6 e x We use the terms of the Taylor series of gx), gx) 8 x x + x x x4 + 9 x5 + 8 x x x ) 4 x9 We define L = x d x d ), 6) d L = x, 64) so that L )= x x ), 65) L )= x ) 66) In an operator form, 6) then becomes L L y = gx)+y y 67) Applying L on both sides of 67) and then incorporating given boundary conditions, 6) becomes L y = L y + d x= L y) x x= + L gx)+l y y ) = y )x + L gx)+l y y ) = L gx)+l y y ) 68)

6 98 W Kim and C Chun Modified ADM for Solving Higher-Order Singular Boundary Value Problems Since L L y = yx) y) y )x, applying L on both sides of 68) with given boundary conditions yields yx)= e x + L L gx) 69) + L L y y ) Proceeding as before, we have the recursive relationship y x)= e x + L L gx), 7) y k+ x)=l L A k L L B k, k 7) The Adomian polynomials for the nonlinear terms y and y are computed as follows: A = y, A = y y, A = y y + y y, 7) A = y y + 6y y y + y, and B = y, B = y y, B = y y + y, B = y y + y y, 7) respectively Substituting 7) and 7) into 7) gives the components of the solution y = 99966x + x + 5x x x 6 + 8x x x x 5x 47x 897x, y = 7x 75x 8 446x 9 9x + 57x +, y = x x x x x +, y = x x x x x +, 74) We thus obtain an approximate solution given by y = y + y + y + y = 99866x + x + 5x x x 6 + 8x x x x x +, 75) which is in well agreement with Taylor series of the exact solution yx)=x e x given by yx)=x + x + 5x x x 6 + 8x x x x x + 4 Homotopy Perturbation Method with He s Polynomials 76) It should be noted that the calculation of Adomian s polynomials to deal with the nonlinear terms in Example, Example, and Example 4 is complex and difficult This aspect is the demerit of the application of Adomian s decomposition method, though this method has been proved to be more powerful and efficient than some existing methods ] This difficulty may be easily overcome without using Adomian s polynomials when the so-called He s polynomials are used to solve the problem We illustrate this in this section by revisiting and solving Example and Example by the homotopy perturbation method HPM) with He s polynomials 4] Revisited Example Consider the nonlinear boundary value problem y x y = x y5, 77) y)=, y )= 8 78) With equation 6), we construct the following homotopy: yx) y) y )x pl x 4 Ny)=, 79)

7 W Kim and C Chun Modified ADM for Solving Higher-Order Singular Boundary Value Problems 99 where Ny) =y 5, p,] is the embedding parameter, L ) is defined by 4) According to He s homotopy perturbation method 4, 5], we assume that the solution of 79) is y = y + py + p y + 8) When p, 8) becomes the approximate solution of 77) and 78), i e, lim y = y + y + y + 8) p is the exact solution of 77) and 78) The convergence of the series in 8) to the exact solution is discussed by He in 4, 5] The nonlinear term Ny) can be expressed in the form Ny)=Hy )+phy,y )+p Hy,y,y )+, 8) where Hy,,y n ) is called He s polynomial ] defined by Hy,,y n )= d n n ) n! dp n N p k y k, k= p= 8) n =,,, Substituting 8) and 8) into 79), and collecting the coefficients of like powers of p,wehave y y) ] y )x p + y L x 4 ] y5 p + y L x 4 ) ] 5y 4 y p + y L x 4 5y y 4 + y ) ] y p + y 4 L x 4 84) 5y y 4 + y y y + y y ) ] p 4 + =, from which we have simultaneously the approximate solution to 77) and 78) given in ) and He s polynomials Hy )=y 5, Hy,y )=5y 4 y, Hy,y,y )=5y y 4 + y y, Hy,y,y,y )=5y y 4 + y y y + y y, 85) Revisited Example boundary value problem Consider the nonlinear y + x y y = gx), 86) y)=, y )=, y)=e, 87) where gx)=4e x + 6xe x + x e x + x e x x 9 e x With equation 5), we construct the following homotopy: yx) L L g T x) L y pl L Ny)=, 88) where Ny)=y, p,] is the embedding parameter and, g T, L ) and L ) are defined by 46), 49), and 5), respectively We assume that the solution of 88) is y = y + py + p y + 89) The nonlinear term Ny) can be expressed in the form Ny)=Hy )+phy,y )+p Hy,y,y )+, 9) where Hy,,y n ) is the He polynomial defined by 8) Substituting 89) and 9) into 88), and collecting the coefficients of like powers of p, wehave y L L g ] T p + y L y L L y )] p + y L y L L y y ) ] p + y L y L L y y + y y )] p + y 4 L y L L y y + 6y y y + y )] p 4 + =, 9) from which we have simultaneously the approximate solution to 86) and 87) given in 58) and He s polynomials Hy )=y, Hy,y )=y y, Hy,y,y )=y y + y y, Hy,y,y,y )=y y + 6y y y + y, 9) He s polynomials were used in the above examples to overcome the complex and difficult calculation of the Adomian polynomials in the Adomian decomposition method We calculated He s polynomials and the solutions simultaneously, and observed that He s

8 W Kim and C Chun Modified ADM for Solving Higher-Order Singular Boundary Value Problems polynomials can completely replace Adomian s polynomials This result reveals the applicability and efficiency of the homotopy perturbation method with higher-order singular boundary value problems 5 Conclusion In this paper, we present a reliable modification of the Adomian decomposition method to solve linear and nonlinear problems with singular feature It is demonstrated that the presented approach can be well suited to attain an accurate solution to the higher-order singular boundary value problems, linear and nonlinear as well The difficulty of those singular problems, due to the existence of the singular point at x =, is overcome in this contribution He s polynomials are also used to overcome the complex and difficult calculation of Adomian s polynomials occurring in the application of Adomian s decomposition method, illustrating that He s polynomials can completely replace Adomian s polynomials The other types of differential equations with singular feature can also be similarly handled by the proposed approach Acknowledgements This work was supported by the Daejin University Research Grants in The authors would like to thank the referees for their useful comments and constructive suggestions which substantially improved the quality of this paper ] G Adomian, Math Comput Modelling, 7 99) ] G Adomian, Solving frontier problems of physics: The decomposition method, Kluwer, Boston, MA, 994 ] A M Wazwaz, Appl Math Comput 66, 68 5) 4] E Deeba, G Dibeh, and G Xie, Appl Math Comput 8, 8 ) 5] E Babolian and J Biazar, Appl Math Comput 9, 9 ) 6] A M Wazwaz, Partial differential equations and solitary waves theory, Springer, New York 9 7] K Abbaoui and Y Cherruault, Comput Math Appl 9, 995) 8] Y Cherruault, G Adomian, K Abbaoui, and R Rach, Bio-Medical Computing 8, ) 9] M M Hosseini and H Nasabzadeh, Appl Math Comput 8, 56 6) ] A M Wazwaz, Appl Math Comput 8, 45 ) ] A M Wazwaz, Appl Math Comput 6, 54 5) ] Y Q Hasan and L M Zhu, Commun Nonlinear Sci Numer Simul 4, 59 9) ] M M Hosseini and M Jafari, Commun Nonlinear Sci Numer Simul 4, 95 9) 4] J H He, Comput Methods Appl Mech Eng 78, ) 5] J H He, Int J Nonlinear Mech 5, 7 ) 6] J H He, Appl Math Comput 5, 7 ) 7] J H He, Int J Modern Phys B, 44 6) 8] J H He, Int J Modern Phys B, 56 6) 9] A Yildirim and D Agirseven, Int J Nonlinear Sci Numer Simul, 5 9) ] A M Wazwaz, Appl Math Comput, 5 ) ] H Bulut, Nonlinear Sci Lett A, 6 ) ] A Ghorbani and J S Nadjfi, Int J Nonlinear Sci Numer Simul 8, 9 7) ] A Ghorbani, Chaos, Solitons, and Fractals 9, 486 9) 4] S T Mohyun-Din, Math Probl Eng 9, )