Numerical solutions of fourth-order Volterra integro-differential equations by the Green s function and decomposition method

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1 Math Sci (16) DOI 117/s ORIGINAL RESEARCH Numerical solutions of fourth-order Volterra integro-differential equations by the Green s function and decomposition method Randhir Singh 1 Abdul-Majid Wazwaz Received June 15 / Accepted 16 August 16 / Published online 7 September 16 The Author(s) 16 This article is published with open access at Springerlinkcom Abstract We propose a reliable technique based on Adomian decomposition method (ADM) for the numerical solution of fourth-order boundary value problems for Volterra integro-differential equations We use Green s function technique to convert boundary value problem into the integral equation before establishing the recursive scheme for the solution components of a specific solution The advantage of the proposed technique over the standard ADM or modified ADM is that it provides not only better numerical results but also avoids solving a sequence of transcendental equations for unknown constant Approximations of the solutions are obtained in the form of series Convergence and error analysis is also discussed The accuracy and generality of the proposed scheme are demonstrated by solving some numerical examples Keywords Integro-differential equations Boundary value problems Adomian decomposition method Green s function Approximations Mathematics Subject Classification 34B15 34B7 34B5 65L1 65L & Randhir Singh randhirmath@gmailcom Abdul-Majid Wazwaz wazwaz@sxuedu 1 Department of Applied Mathematics, Birla Institute of Technology, Mesra, Ranchi 3515, India Department of Mathematics, Saint Xavier University, Chicago IL 6655, USA Introduction Consider the following class of fourth-order BVPs for Volterra IDEs [1 5] y ðivþ ðxþ ¼gðxÞþ Kðx; tþf ðyðtþþdt; x ½; b; ð1þ with the boundary conditions yðþ ¼a 1 ; y ðþ ¼a ; yðbþ ¼a 3 ; y ðbþ ¼a 4 ; ðþ where a i ; i ¼ 1; ; 3; 4 are any finite real constants, gðxþ C½; b, and Kðx; tþ Cð½; b½; bþ The IDEs are often involved in the mathematical formulation of physical and engineering phenomena [4 6] In general, the IDEs with given boundary conditions are difficult to solve analytically Therefore, these problems must be solved by various approximation and numerical methods The existence and uniqueness of solutions for such problems can be found in [1] There is considerable literature on the numerical-approximate treatment of the BVPs for IDEs, for example, the compact finite difference method [7], monotone iterative methods [7, ], spline collocation method [], the method of upper and lower solution [1], Haar wavelets [11], and pseudo-spectral method [1] Though, these numerical techniques have many advantages, a huge amount of computational work is involved that combines some root-finding techniques to obtain an accurate numerical solution especially for nonlinear problems Recently, some newly developed semi-numerical methods have also been applied to solve BVPs for IDEs such as, ADM [3], homotopy perturbation method (HPM) [4], and homotopy analysis method (HAM) [13] In [5], the 13

2 16 Math Sci (16) variational iteration method (VIM) was also used for solving the problem (1) () However, in [14] Wazwaz pointed out that VIM gives good approximations only when the problem is linear or nonlinear with the weak nonlinearity of the form (y n ; yy ; y n ; Þ, but the VIM suffers when the nonlinearity is of the form ðe y ; ln y; sin y; Þ (for details see [14]) It is well known that the ADM allows us to solve nonlinear BVPs without restrictive assumptions such as linearization, discretization and perturbation Many researchers [14 3] have shown interest to study the ADM for different scientific models According to the ADM, we rewrite the problem (1) in an operator form Ly ¼ g þ Ny; ð3þ where L ¼ d4 dx is a fourth-order linear differential operator, 4 g is a function of x and Ny ¼ R x Kðx; tþf ðyðtþþdt is a nonlinear term Inverse integral operator is usually defined as L 1 ½ ¼ ½dx dx dx dx ð4þ Operating with L 1 on both sides of (3) and using the conditions yðþ ¼a 1 and y ðþ ¼a, we obtain yðxþ ¼a 1 þ a x þ c 1 x þ c x 3 þ L 1 ½g þ Ny; ð5þ where c 1 ¼ y ðþ! and c ¼ y ðþ 3! are unknown constants to be determined The ADM relies on decomposing y by a series of components and nonlinear term f(y) by a series of Adomian polynomials as y ¼ X1 y j ðxþ and f ðyþ ¼ X1 A j ; ð6þ j¼ where A j are Adomian s polynomials [], which can be computed as "! A n ¼ 1 d n X # 1 n! dk n f y k k k ; n ¼ ; 1; ; ; ð7þ k¼ j¼ k¼ Several algorithms have also been given to generate the Adomian polynomial rapidly in [4 6] Substituting the series (6) in(5), we get X 1 j¼ y j ðxþ ¼a 1 þ a x þ c 1 x þ c x 3 þ L 1 ½g þ L 1 " # Kðx; tþ X1 A j dt ; ðþ On comparing both sides of equation (), the ADM is given by j¼ y ¼ a 1 þ a x þ c 1 x þ c x 3 þ L 1 ðgþ; y j ¼ L 1 Kðx; tþa j1 dt ; j ¼ 1; ðþ Wazwaz [7] suggested a modified ADM (MADM) which is given by y ¼ a 1 ; y 1 ¼ a x þ c 1 x þ c x 3 þ L 1 ½gþL 1 Kðx; tþa dt ; ; y j ¼ L 1 Kðx; tþa j1 dt ; j ¼ ; 3; ð1þ Hence, the n-term approximate series solution is obtained as / n ðx; c 1 ; c Þ¼ Xn j¼ y j ðx; c 1 ; c Þ ð11þ We note that the series solution / n ðx; c 1 ; c Þ depends on the unknown constants c 1 and c These unknown constants will be determined approximately by imposing the boundary condition at x ¼ b on / n ðx; c 1 ; c Þ, which leads a sequence of nonlinear system of equations as / n ðb;c 1 ;c Þ¼a 3 and / n ðb;c 1;c Þ¼a 4 ; n ¼ 1;; ð1þ To determine the unknown constants c 1 and c, we require root finding methods such as Newton Raphson s method which requires additional computational work But solving the nonlinear equation (1) for c 1 and c is a difficult task in general Moreover, in some cases the unknowns c 1 and c may not be uniquely determined This may be the main difficulty of the ADM In this work, we propose a new recursive scheme which does not involve any unknown constant to be determined In other words, we introduce a modification of the ADM to overcome the difficulties occurring in ADM or MADM for solving fourth-order BVPs for IDEs The decomposition method with Green s function Let us first consider homogeneous version of the problem (1) and () as ( u ðivþ ðxþ¼; x ½;b; ð13þ uðþ¼a 1 ; u ðþ¼a ; uðbþ¼a 3 ; u ðbþ¼a 4 Solving (13) analytically, we obtain 13

3 Math Sci (16) ð uðxþ ¼a 1 þ a x 3a 1 þ ba 3a 3 þ ba 4 Þ b x ð þ a 1 þ ba a 3 þ ba 4 Þ b 3 x 3 ð14þ We now construct Green s function of the following fourth-order boundary value problem ( y ðivþ ðxþ ¼hðxÞ; x ½; b; ð15þ yðþ ¼y ðþ ¼yðbÞ ¼y ðbþ ¼ The Green s function of (15) can be easily constructed and it is given by x þ n n3 b 3b Gðx; nþ ¼ 3 þ x n n > b þ n3 b ; x n; n þ x x3 b 3b 3 þ n x x > b þ x3 b ; n x b ð16þ Using (14) and (16), we transform BVPs for IDEs (1) and () into an integral equation as Z n yðxþ ¼uðxÞþ Gðx; nþ gðnþþ Kðn; tþf ðyðtþþdt ; dn ð17þ Substituting the series (6) in(17), we obtain X 1 Z n " # y j ðxþ ¼uðxÞþ Gðx; nþ gðnþþ Kðn; tþ X1 A j dt ; dn j¼ j¼ ð1þ Comparing both sides of (1), the decomposition with Green s function (DMGF) is given by the following recursive scheme as y j ¼ y ¼ uðxþþ Gðx; nþgðnþdn; Z n Gðx; nþ Kðn; tþa j1 dt dn; j ¼ 1; ; 3 ð1þ and the modified decomposition with Green s function (MDMGF) is given by the following recursive scheme as y ¼ u ; Z n y 1 ¼ u 1 þ Gðx; nþ gðnþþ Kðn;tÞA dt ; dn; ðþ Z n y j ¼ Gðx; nþ Kðn;tÞA j1 dt dn; j ¼ ;3 where u ¼ a 1 and u 1 ¼ a x 3a 1þba 3a 3 þba 4 b x þ ða 1 þba a 3 þba 4 Þ b x 3 Then-terms truncated series solution is 3 obtained as w n ¼ Xn j¼ y j ð Þ ð1þ Convergence and error estimate of the scheme (1) or () In this section, we shall show that sequence fw n g of the partial sums of series solution defined by (1) converges to the exact solution y of the problem (1), () Theorem 31 (Convergence theorem) Suppose that X ¼ C½; b is a Banach space with the norm kyk ¼max xi¼½;b jyðxþj; y X Assume that the function f(y) satisfies the Lipschitz condition such that jf ðyþ f ðy Þj ljy y j and denote kkk 1 ¼ max jkðn; tþj and kgk 1 ¼ max jgðx; nþj Further, we define d as d ¼ lkkk 1 kgk 1 b Then the sequence fw n g converges to the exact solution whenever d\1 and ky 1 k\1 Proof w n ¼ y þ Xn þ Xn From (1) or() and (1), we write j¼1 j¼1 4 ¼ uðxþþ y j ¼ uðxþþ Gðx; nþ Z n Gðx; nþ gðnþþ Gðx; nþgðnþdn 3 Kðn; tþa j1 dt ; dn 5 Z n " # Kðn; tþ Xn A j1 dt ; dn For all n; m N, with n [ m, consider Z n " # kw n w m k¼max Gðx; nþ Kðn; tþ Xn1 A j Xm1 A j dt xi ; dn j¼ j¼1 j¼ ðþ Using the relation P n j¼ A j f ðw n Þ (for details see, [, pp 44 45]) we have Z n kw n w m k max Gðx; nþ Kðn; tþ½f ðw n1 Þfðw m1 Þdt xi ; dn Z n max jgðx; nþj max jkðn; tþj ljw n1 w m1 jdt ; dn lkkk 1 kgk 1 kw n1 w m1 k max xi Z n dtdn lkkk 1 kgk 1 b kw n1 w m1 k¼dkw n1 w m1 k; 13

4 16 Math Sci (16) where d ¼ lkkk 1 kgk 1 b Setting n ¼ m þ 1 we obtain kw mþ1 w m kdkw m w m1 k Thus, we have kw mþ1 w m kdkw m w m1 k d kw m1 w m k d m kw 1 w k Using triangle inequality for any n; m N, with n [ m we have kw n w m k¼kðw n w n1 Þþðw n1 w n Þþþðw mþ1 w m Þk kw n w n1 kþkw n1 w n kþþkw mþ1 w m k ½d n1 þ d n þþd m kw 1 w k ¼ d m ½1 þ d þ d þþd nm1 kw 1 w k ¼ d m 1 d nm ky 1 k 1 d Thus, we obtain kw n w m k dm 1 d ky 1k; ð3þ which converges to zero, ie, kw n w m k!, as m!1 This implies that there exits an w such that lim n!1 w n ¼ w h In the next theorem, we give the error estimate of the series solution Theorem 3 (Error estimate) The maximum absolute truncation error of the series w m obtained by the scheme (1) or () to problem is given as ky w m k dmþ1 lð1 dþ ka k 1 where ka k 1 ¼ max xi ja j, A ¼ f ðy Þ ð4þ Proof Fixing m and letting n!1in the estimate (3) with n m, we obtain ky w m k dm 1 d ky 1k ð5þ From the scheme (1) we have y 1 ¼ R b Gðx; nþ R n K ðn; tþa dt dn, and following the steps of theorem (31), we have Z n ky 1 k¼max Gðx; nþ Kðn; tþa dt xi ; dn ð6þ kkk 1 kgk 1 ka k 1 b But we know that d ¼ lkkk 1 kgk 1 b Hence, the inequality (6) becomes ky 1 k d l ka k 1 ð7þ Combining the estimates (5) and (7), we get the desired result h Numerical results To demonstrate the efficiency and accuracy of the proposed recursive schemes, we consider four fourth-order BVPs for Volterra IDEs All symbolic and numerical computations are performed using Mathematica software package Example 41 Consider the following linear fourth-order BVP for Volterra IDE [, 3] y ðivþ ðxþ ¼gðxÞþ yðtþdt; x ½; 1 yðþ ¼y ðþ ¼1; yð1þ ¼1 þ e; y ; ðþ ð1þ ¼e; where gðxþ ¼x þ 5e x 1, and its exact solution is yðxþ ¼1 þ xe x Here, b ¼ 1, a 1 ¼ 1, a ¼ 1, a 3 ¼ 1 þ e, a 4 ¼ e, Kðx; tþ ¼1, f ðyþ ¼y, and gðnþ ¼nþ5e n 1 According to the MDMGF (), the problem () is transformed into the following recursive scheme y ¼ 1; ( ) y 1 ¼ x þðeþx þ x 3 þ R1 Gðx; nþ gðnþþ Rn Kðn; tþy dt dn; ( ) y j ¼ R1 R n Gðx; nþ Kðn; tþy j1 dt dn; j ¼ ; 3; >; Gðx; nþ is given by x þ n n3 þ x n > 3 Gðx; nþ ¼ n þ x x3 > þ n 3 n 1 þ n3 x x 1 þ x3 ðþ ; x n; ; n x 1 ð3þ Using () and (3), we compute the solution components y j as y ¼ 1; y 1 ¼5þ5e x 4x 156x 355x x 4 ; y ¼5þ5e x 5x 43x 414x 3 33x x 5 555x 6 þ To check the accuracy and efficiency of the proposed methods, the absolute error function is defined as E n ðxþ ¼jw n ðxþyðxþj; n ¼ 1; ; where y is the exact solution and w n is the nth-stage approximation obtained by the proposed (1) or() In Table 1, we list the numerical results of the absolute errors jw n yj (obtained by the proposed MDMGF ()[7]) 13

5 Math Sci (16) Table 1 The absolute error jw n yj and j/ n yj for n ¼ 1; ; 3 of Example 41 x MDMGF MADM [7] jw 1 yj jw yj jw 3 yj j/ 1 yj j/ yj j/ 3 yj 1 53E-5 354E- E-11 44E-3 454E-5 E E-4 3E-7 144E-1 41E-3 43E-5 474E E-4 363E-7 34E-1 66E-3 663E-5 337E E-4 E-7 1E-1 77E-3 6E-5 47E-7 1E-4 57E- 33E-11 E-3 77E-6 51E- y ¼ 1;y 1 ¼ xþðe5þx ðe3þx 3 Z 1 Z n þ Gðx; nþ gðnþþ Kðn;tÞA dt ; dn; y j ¼ Z 1 Gðx;nÞ Z n Kðn;tÞA j1 dt dn; j ¼ ;3; ð3þ where gðnþ¼1 and Gðx;nÞ is given by the Eq (3) The Adomian s polynomial f ðyþ¼y are obtained as A ¼ y ; A 1 ¼ y y 1 ; A ¼ y 1 þ y y ; ð33þ Using (3) and (33), we obtain the solution components y j as Fig 1 Plots of the exact and the approximate solution of Example 41 and j/ n yj [obtained by the existing MADM (1)] for n ¼ 1; ; 3 It is observed that the proposed MDMGF provides not only better numerical results but also avoids solving a sequence of transcendental equations for unknown constant In Fig 1, we plot the exact solution y and the approximate solution w 1 ¼ y þ y 1 We observe that only two-term approximations w 1 ¼ y þ y 1 coincide with the exact solution y Example 4 Consider the following nonlinear fourthorder BVPs for Volterra IDE [3] y iv ðxþ ¼gðxÞþ e t y ðtþdt; x ½; 1; ð31þ yðþ ¼y ðþ ¼1; yð1þ ¼y ; ð1þ ¼e; where gðxþ ¼1 The exact solution is yðxþ ¼e x Here, b ¼ 1, a 1 ¼ 1, a ¼ 1, a 3 ¼ e, a 4 ¼ e, Kðx; tþ ¼e t, and f ðyþ ¼y ðtþ In view of the MDMGF (), we transform the problem (31) into the following recursive scheme y ¼ 1; y 1 ¼ 1 e x þ 1415x þ 11413x 3 þ 33333x 4 ; y ¼ þ 65e x 1471x þ x 56665x 3 þ 3775x 4 þ In Table, we present the numerical results of the absolute errors jw n yj (obtained by the proposed MDMGF) and j/ n yj (obtained by MADM) for n ¼ 1; ; 3 It is observed that the proposed DMGF provides not only better numerical results but also avoids solving a sequence of transcendental equations for unknown constant In Fig, the exact solution y and the approximate solution w 1 are plotted From this figure, we observe that only two-term approximations w 1 coincide with the exact one Example 43 Consider the following nonlinear fourthorder BVPs for Volterra IDE y ðivþ ðxþ ¼gðxÞþ e yðtþ dt; x ½; 1; yðþ ¼lnð4Þ; y ðþ ¼ 1 4 ; yð1þ ¼lnð5Þ; y ð1þ ¼ 1 >; 5 ð34þ 13

6 164 Math Sci (16) Table The absolute error jw n yj and j/ n yj for n ¼ 1; ; 3 of Example 4 x MDMGF MADM [7] jw 1 yj jw yj jw 3 yj j/ 1 yj j/ yj j/ 3 yj 1 744E-5 143E-5 43E- 644E-4 4E-5 3E E-4 17E-5 75E-7 467E-3 16E-5 77E E-4 156E-4 454E-7 663E-3 366E-4 75E-5 7 6E-4 134E-4 37E-7 5E-3 454E-4 47E-5 13E-4 3E-5 761E- 33E-3 3E-5 761E-6 where gðnþ¼ n 6 4n and Gðx;nÞ is given by the ðnþ4þ 4 Eq (3) The Adomian s polynomials for f ðyþ¼e yðxþ are calculated as A ¼ e y ; A 1 ¼ e y y 1 ; A ¼ 1 ey y 1 þ ey y ðxþ; Using (35) and (36), we obtain the components y j as ð36þ Fig Plots of the exact and the approximate solution of Example 4 where gðxþ ¼ x 6 4x The exact solution is ðxþ4þ 4 yðxþ ¼lnð4 þ xþ Here, we have b ¼ 1, a 1 ¼ lnð4þ, a ¼ 1 4, a 3 ¼ lnð5þ, a 4 ¼ 1 5, Kðx; tþ ¼1, and f ðyþ ¼eyðtÞ According to the MDMGF (), the problem (34) is transformed into the following recursive scheme y ¼ lnð4þ; y 1 ¼ 5x 356x ( þ 371x ) 3 þ R1 Gðx;nÞ gðnþþ Rn Kðn;tÞA dt dn; ð35þ ( ) y j ¼ R1 R n Gðx;nÞ Kðn;tÞA j1 dt dn j ¼ ;3; >; y ¼ lnð4þ; y 1 ¼ 5x x þ 1763x 3 76x 4 þ 153x 5 þ; y ¼ þ x þ 371x 531x 3 þ; In Table 3, we present the numerical results of the absolute errors jw n yj (obtained by the proposed MDMGF) and j/ n yj (obtained by MADM) for n ¼ 1; ; 3 In this case, we also observe the same trend as was observed in last two examples that the proposed MDMGF gives better numerical results Moreover, the curves of the exact solution y and the approximate solution w 1 are plotted in Fig 3 We observe that only two-term approximations w 1 and the exact solution overlap each other Example 44 Consider the following nonlinear fourthorder BVPs for IDEs Table 3 The absolute error jw n yj and j/ n yj for n ¼ 1; ; 3 of Example 43 x MDMGF MADM [7] jw 1 yj jw yj jw 3 yj j/ 1 yj j/ yj j/ 3 yj 1 361E-5 171E-6 713E- 561E-3 571E-4 53E-6 3 6E-4 1E-5 464E-7 656E-3 51E-4 344E E-4 15E-5 E-7 76E-3 35E-4 41E-6 7 3E-4 157E-5 7E-7 54E-3 47E-4 4E-6 631E-5 34E-6 161E-7 431E-4 64E-4 571E-6 13

7 Math Sci (16) Fig 3 Plots of the exact and the approximate solution of Example 43 y ðivþ ðxþ ¼gðxÞþ e yðtþ dt; x ½; 1; yðþ ¼; y ðþ ¼1; yð1þ ¼lnðÞ; y ð1þ ¼ 1 where gðxþ ¼ x yðxþ ¼lnð1 þ xþ ð37þ x 6 ðxþ1þ 4 The exact solution is Here, b ¼ 1, a 1 ¼, a ¼ 1, a 3 ¼ lnðþ, a 4 ¼ 1, Kðx; tþ ¼1, and f ðyþ ¼e yðtþ According to MDMGF (), we transform the problem (37) into the following recursive scheme as y ¼ ; y 1 ¼ x 455x þ 11376x 3 þ R1 Gðx; nþ gðnþþ R1 Kðn; tþa dt dn; y j ¼ R1 Gðx; nþ R1 Kðn; tþa j1 dt dn; j ¼ ; 3; >; ð3þ where gðnþ ¼ n 6 n Using (36) and (3), we ðnþ1þ 4 obtain the solution components y j as Fig 4 Plots of the exact and the approximate solution of Example 44 y ¼ ; y 1 ¼41666x þ5555x 3 13x 6 þlnð1þxþ; y ¼ 5511 x 3364þ xþ x þ x 3 þ755x 4 þ66x 6 33x 7 þx 1 þ3364ð1þxþ 5 lnð1þxþ ; In Table 4, we present the numerical results of the absolute errors jw n yj (obtained by the proposed DMGF) and j/ n yj (obtained by MADM) for n ¼ 1; ; 3 We also plot the curves of the exact y and the approximate solution w 1 for n ¼ 1 in Fig 4 Like previous examples, it is observed that only two-term approximations w 1 coincide with the exact solution y Conclusions In this paper, we studied a reliable technique based on the decomposition method and Green s function for the numerical solution of the fourth-order BVPs for Volterra Table 4 The absolute error jw n yj and j/ n yj for n ¼ 1; ; 3 of Example 44 x MDMGF MADM [7] jw 1 yj jw yj jw 3 yj j/ 1 yj j/ yj j/ 3 yj 1 361E-5 53E-6 64E-7 561E-4 3E-5 1E-6 3 6E-4 345E-5 4E-6 46E-3 45E-4 75E E-4 53E-5 7E-6 36E-3 33E-4 E-5 7 3E-4 43E-5 63E-6 57E-3 63E-4 73E-5 631E-5 1E-5 147E-6 531E-4 3E-4 447E-5 13

8 166 Math Sci (16) IDEs The technique depends on constructing Green s function before establishing the recursive scheme for the solution components The proposed technique provides a direct recursive scheme for obtaining the approximations to the solutions of BVPs Unlike the existing ADM or the MADM, the proposed method DMGF or MDMGF avoids unnecessary evaluation of unknown constants and provides better numerical solutions Convergence and error analysis of the proposed technique have also been discussed The performance of the proposed recursive scheme have been examined by solving four numerical examples It has been shown that only two-term series solution is enough to obtain an accurate approximation to the solution Open Access This article is distributed under the terms of the Creative Commons Attribution 4 International License (http//crea tivecommonsorg/licenses/by/4/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made References 1 Agarwal, RP Boundary value problems for higher order integro-differential equations Nonlinear Anal Theory Methods Appl 7(3), 5 7 (13) Hashim, I Adomian decomposition method for solving BVPs for fourth-order integro-differential equations J Comput Appl Math 13(), (6) 3 Wazwaz, AM A reliable algorithm for solving boundary value problems for higher-order integro-differential equations Appl Math Comput 11(), (1) 4 Yıldırım, A Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method Comput Math Appl 56(1), () 5 Sweilam, N Fourth order integro-differential equations using variational iteration method Comput Math Appl 54(7), (7) 6 Wazwaz, AM A comparison study between the modified decomposition method and the traditional methods for solving nonlinear integral equations Appl Math Comput 11(), (6) 7 Zhao, J, Corless, RM Compact finite difference method for integro-differential equations Appl Math Comput 177(1), 71 (6) Al-Mdallal, QM Monotone iterative sequences for nonlinear integro-differential equations of second order Nonlinear Anal Real World Appl 1(6), (11) Brunner, H On the numerical solution of nonlinear Volterra integro-differential equations BIT Numer Math 13(4), 31 3 (173) 1 Liz, E, Nieto, JJ Boundary value problems for second order integro-differential equations of Fredholm type J Comput Appl Math 7(), 15 5 (16) 11 Lepik, Ü Haar wavelet method for nonlinear integro-differential equations Appl Math Comput 176(1), (6) 1 Sweilam, NH, Khader, MM, Kota, W Numerical and analytical study for fourth-order integro-differential equations using a pseudospectral method Math Probl Eng 13, 1 7 (13) doi11155/13/ Saeidy, M, Matinfar, M, Vahidi, J Analytical solution of BVPs for fourth-order integro-differential equations by using homotopy analysis method Int J Nonlinear Sci (4), (1) 14 Wazwaz, AM, Rach, R Comparison of the Adomian decomposition method and the variational iteration method for solving the Lane-Emden equations of the first and second kinds Kybernetes 4(/1), (11) 15 Singh, R, Kumar, J, Nelakanti, G Approximate series solution of singular boundary value problems with derivative dependence using Green s function technique Comput Appl Math 33(), (14) 16 Singh, R, Kumar, J The Adomian decomposition method with Green s function for solving nonlinear singular boundary value problems J Appl Math Comput 44(1), (14) 17 Al-Hayani, W Adomian decomposition method with Green s function for solving tenth-order boundary value problems Appl Math 5, (14) 1 Al-Hayani, W Adomian decomposition method with Green s function for solving twelfth-order boundary value problems Appl Math Sci (), (15) 1 Singh, R, Kumar, J, Nelakanti, G Numerical solution of singular boundary value problems using Green s function and improved decomposition method J Appl Math Comput 43(1), 4 45 (13) Wazwaz, AM The combined Laplace transform-adomian decomposition method for handling nonlinear Volterra integrodifferential equations Appl Math Comput 16(4), (1) 1 Wazwaz, AM A reliable study for extensions of the Bratu s problem with boundary conditions Math Methods Appl Sci 35(7), (1) Adomian, G, Rach, R Inversion of nonlinear stochastic operators J Math Anal Appl 1(1), 3 46 (13) 3 Adomian, G Solving Frontier Problems of Physics The Decomposition Method Kluwer Academic Publishers, Boston (14) 4 Duan, J An efficient algorithm for the multivariable Adomian polynomials Appl Math Comput 17(6), (1) 5 Rach, R A convenient computational form for the Adomian polynomials J Math Anal Appl 1(), (14) 6 Duan, J Convenient analytic recurrence algorithms for the Adomian polynomials Appl Math Comput 17(13), (11) 7 Wazwaz, AM A reliable modification of Adomian decomposition method Appl Math Comput 1(1), 77 6 (1) Rach, RC A new definition of the Adomian polynomials Kybernetes 37(7), 1 55 () 13