Abstract
We apply Adomian decomposition method (ADM) for obtaining approximate series solution of Urysohn integral equations. The ADM provides a direct recursive scheme for solving such problems approximately. The approximations of the solution are obtained in the form of series with easily calculable components. Furthermore, we also discuss the convergence and error analysis of the ADM. Moreover, three numerical examples are included to demonstrate the accuracy and applicability of the method.
1. Introduction
Nonlinear integral equations arise very frequently many branches of science and engineering such as biology, applied mathematics, physics, and many areas of analysis. We consider the following nonlinear Urysohn integral equation: Depending on or , (1) is named a nonlinear Volterra integral equation or a nonlinear Fredholm integral equation, respectively. We assume that the problem (1) has a unique solution. The following conditions are assumed:() is continuous;()the kernel is continuous on ;()let the kernel satisfy the Lipschitz condition where is Lipchitz constant.
Recently, a great deal of interest has been focused in the study of nonlinear Urysohn integral equation [1–8] and many of the references therein.
The objective of this paper is to apply the ADM for obtaining approximate series solution of the nonlinear Urysohn integral equations. Moreover, the convergence and error analysis of the ADM are discussed. Finally, we compare our numerical results with those obtained by [5].
2. Adomian Decomposition Method
In the recent past, a lot of researchers [9–16] have expressed their interest in the study of ADM for various scientific models. Adomian [12] asserted that the ADM provides an efficient and computationally worthy method for generating approximate series solution for a large class of differential as well as integral equations. In order to apply ADM to Urysohn integral equation, we rewrite (1) as where is a nonlinear operator.
We now decompose the solution by series as and the nonlinear functions is decomposed by series where , are Adomian's polynomials which can be obtained by using the formula given as in [13]: Recently, El-Kalla [16] proposed another programmable formula for Adomian's polynomials: where is the partial sum of the series solution .
Substituting the series (4) and (5) into (3), we obtain Comparing both sides of (8), we obtain the ADM scheme: Using the recursive scheme (9), the -term approximate series solution can be obtained as follows: Now we discuss the convergence analysis and error bounds for the recursive scheme (9). Let be the Banach space with the norm . Note that (3) can be written in the operator as where is given by We next discuss the existence of the unique solution of (11).
Theorem 1 (contraction principle). Let be the Banach space with the norm . Assume that the nonlinear kernel satisfies the Lipschitz condition such that Further, one defines , where is Lipschitz constant. If , then the solution of (11) exists and unique.
Proof. For any , and using Lipschitz continuity of as defined in (2), we have Thus we have where . If , then is contraction and hence, by the Banach contraction mapping theorem, (11) has a unique solution in .
Now we write scheme (9) in operator form as follows. Let be the sequence of partial sums of the series solution .
By using the recursive scheme (9) and (10), we have Using (7) into (16), it follows that which is equivalent to the following operator equation: In the next theorem, we discuss the convergence of the approximate series solution to the exact solution .
Theorem 2. Let be the nonlinear operator defined by (12) contractive; that is, If , then the sequence defined by (18) converges to the exact solution .
Proof. Using relation (18) and the fact that is contractive, we have Thus we have Now for all , , with , consider Since , so, , and , it follows that which converges to zero; that is, , as . This implies that there exists such that . Since, we have , that is, which is exact solution of (11).
In the following theorem we obtain the error bounds for the approximate series solution .
Theorem 3. Let be the exact solution of (11). Let be the sequence of approximate series solution given by (18). Then there holds
Proof. From estimate (23), for , , , we have Since , fixing and letting , we obtain Since and , we have Combining estimates (26) and (27), we obtain This completes the proof.
3. Numerical Results
In this section, we apply scheme (9) to solve three examples. All numerical results obtained by ADM are compared with known results and with those in [5].
Example 1. Consider the following nonlinear Fredholm integral equation Urysohn form [5]: with and . Its analytical solution is given by .
According to the ADM (9), the problem (29) is transformed into the following recursive scheme: The Adomian’s polynomials are computed for by using formula (6):
For quantitative comparison, we define the absolute error functions as where is analytical solution and is -term truncated series solutions obtained by ADM (9). Table 1 shows the comparison of the numerical results obtained by the present recursive (9) and Newton-Kantorovich-quadrature used in [5]. One can note that the present method gives much better numerical results compared to Newton-Kantorovich-quadrature method.
Example 2. We now consider the following nonlinear Urysohn Volterra-integral equation [5]: with and . The exact solution to this integral equation is .
According to the ADM (9), the problem (33) is transformed into the following recursive scheme: Using formula (6), we obtain the Adomian's polynomials for as follows:
Once again, we compare the numerical results obtained by the present recursive (9) and Newton-Kantorovich-quadrature used in [5] in Table 2. We again conclude that the present method gives much better numerical results compared to Newton-Kantorovich-quadrature method [5]. Furthermore, we also plot absolute error function , in Figures 1(a) and 1(b). From these figures we can clearly see that as the iterations increase the error decreases.
(a)
(b)
Example 3. Finally, we consider the following nonlinear Urysohn integral equation [8]: with and . The exact solution is given by , where is the solution of .
According to the ADM (9), the problem (31) is transformed into the following recursive scheme: Using formula (6), we obtain the Adomian's polynomials for as follows Like previous examples, Table 3 shows the numerical results obtained by present method (9). In addition, we also plot absolute error functions for in Figures 2(a) and 2(b), and it is shown that only a few terms are required to obtain acceptable approximation for the solution.
(a)
(b)
4. Conclusion
In this paper, we have applied the ADM to solve nonlinear Urysohn type integral equation. The accuracy of the numerical results indicates that the proposed method is well suited for the solution of such type of problems. The advantage of current approach is that it provides a direct scheme for obtaining approximations of the solutions. Moreover, the proposed method provides a reliable technique which requires less work compared to the traditional techniques such as finite difference method (FDM), cubic spline method (CSM), B-spline method (BSM), and Newton-Kantorovich-quadrature method. Unlike FDM, CSM, and any other discrete methods, the ADM does not require any linearization or discretization of variables. The numerical results show that only a few terms are required to obtain accurate solutions. By comparing the numerical results with other existing method used in [5], it has been shown that proposed method is a powerful method for solving nonlinear Urysohn integral equations. Finally, we have also discussed the convergence and error analysis of the ADM.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the anonymous referees for their useful comments and suggestions that led to improvement of the presentation and content of this paper. The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.