Solving Nth-order Integro-differential Equations Using the Combined Laplace Transform-adomian Decomposition Method

In this paper, the Combined Laplace Transform-Adomian Decomposition Method is used to solve nth-order integro-differential equations. The results show that the method is very simple and effective.


Introduction
In the recent literature there is a growing interest to solve integro-differential equations.The reader is referred to [1][2][3] for an overview of the recent work in this area.In the beginning of the 1980's, Adomian [4][5][6][7] proposed a new and fruitful method (so-called the Adomian decomposition method) for solving linear and nonlinear (algebraic, differential, partial differential, integral, etc.) equations.It has been shown that this method yields a rapid convergence of the solutions series to linear and nonlinear deterministic and stochastic equations.The main objective of this work is to use the Combined Laplace Transform-Adomian Decomposition Method (CLT-ADM) in solving the nth-order integro-differential equations.
Let us consider the general functional equation where is a nonlinear operator, N f is a known func- tion, and we are seeking the solution y satisfying (1.1).We assume that for every , f Equation (1.1) has one and only one solution.
The Adomian's technique consists of approximating the solution of (1.1) as an infinite series where n A are polynomials (called Adomian polynomials) of [4][5][6][7] given by 0 1 , , , n y y y  The proofs of the convergence of the series .
Thus all components of can be calculated once the y n A are given.We then define the n-terms approximant to the solution by with y

General nth-Order Integro-Differential Equations
Let us consider the general nth-order integro-differential equations of the type [1,2]: with initial conditions  are real constants, m and are integers and .In Equation (2.1) the functions and the kernel are given real-valued functions, and is the solution to be determined.We assume that Equation (2.1) has the unique solution.


To solve the general nth-order integro-differential Equation (2.1) using, the Laplace transform method, we recall that the Laplace transforms of the derivatives of are defined by Applying the Laplace transform to both sides of (2.1) and taking into account the fact that the convolution theorem for Laplace transform [13,14] gives: This can be reduced to The Adomian decomposition method presents the recursive relation Applying the inverse Laplace transform to both sides of the first part of (2.3) gives , and using the recursive relation (2.3) gives the components of We then define the -terms approximant to the solution n   y x by with . In this paper, the obtained series solution converges to the exact solution.

A Test of Convergence
The convergence of the method is established by Theorem 3.1 in [9].In fact, on each interval the inequality is a constant and is the maximum order of the approximant used in the computation.Of course, this is only a necessary condition for convergence, because it would be necessary to compute in order to conclude that the series is convergent.

Let
 be the successive approximations to the solution   ,

Applications
In this section, the CLT-ADM for solving nth-order inte- Copyright © 2013 SciRes.AM gro-differential equations is illustrated in the three examples given below.To show the high accuracy of the solution results from applying the present method to our problem (2.1) compared with the exact solution, the maximum error is defined as: where represents the number of iterations.Moreover, we give a comparison among the CLT-ADM, Homotopy perturbation method (HPM) [1] and the variational iteration method (VIM) [2].The computations associated with the examples were performed using Maple 13 package.
As mentioned above, taking Laplace transform of both sides of (3.1) gives where .Substituting the series as- sumption for   Y s as given above in (1.2), and using the recursive relation (2.3) we obtain Thus the series solution is given by lim e e 3! 30 x  .In Table 1, the maximum errors and the EOC are presented for Comparing it with the HPM and VIM results given in [1,2], we notice that the result obtained by the present method is very superior (lower error combined with less number of iterations) to that obtained by HPM and VIM.From Table 1, it can be deduced that, the error decreased monotically with the increment of the integer .
Example 2 Solve the third-order integro-differential equation by using the CLT-ADM [1,2]: As early mentioned, taking Laplace transform of both sides of (3.3) gives . Substituting the series assumption for   Y s as given above in (1.2), and using the recursive relation (2.3) we obtain  The series solution is therefore given by In Table 2, the maximum errors and the EOC are shown for Comparing it with the HPM and VIM results given in [1,2], we notice that the result obtained by the present method is very superior (lower error combined with less number of iterations) to that obtained by HPM and VIM.From Table 2, it can be concluded that, the error decreased monotically with the increment of the integer .
Example 3 Solve the eighth-order integro-differential equation by using the CLT-ADM [1,2]: x y x x y x x y t t y y y y y y y y As previously mentioned, taking Laplace transform of both sides of (3.5) gives and so on for other components.Consequently, the series solution is given by .Comparing it with the VIM results given in [2], we realize that the result obtained by the present method is very superior (lower error combined with less number of iterations) to that obtained by VIM.
From Table 3, it can be deduced that, the error decreased monotically with the increment of the integer .n


are given in [6,8-12].Substituting (1.2) and (1.3) into (1.1)yields 0 0 then we call the (estimated) Local Order of Convergence (EOC) at the point p i x .The constant K is called Convergence Factor at i x .
2)Taking the inverse Laplace transform of both sides of the first part of (3.2) gives Taking the inverse Laplace transform of both sides of the first part of (3.4) gives

Table 3 ,
the maximum errors and the EOC are given for