The variational Adomian decomposition method for solving nonlinear two- dimensional Volterra-Fredholm integro-differential equation

1 Introduction

In recent years, there has been a clear interest in the integro-differential equations which are a combination of differential and Volterra-Fredholm integral equations. Integro-differential equations play an important role in many branches of linear and non-linear functional analysis and their applications. The mentioned integro- differential equations are usually difficult to solve analytically, so approximation strategies are required to obtain the solution of the linear and nonlinear integro-differential equations (Huesin et al., 2008).

Several strategies are proposed to achieve this goal. The ADM is introduced by G. Adomian. It is based on the search for a solution in the form of a series and on decomposing the nonlinear operator into a series in which the terms are calculated recursively using the Adomian polynomials (Adomian, 1994). Many researchers studied and discussed the linear V-FIDE, Babolian et al. (2008) and Al-Jubory (2010) introduced some approximation method for solving Volterra-Fredholm integral and integro-differential equations. Dadkhah et al. (2010) utilized numerical solution of nonlinear V-FIDEs utilizing Legendre wavelets. Rabbani and Kiasodltani (2011) studied finding solution of nonlinear system of V-FIDE by utilizing discrete collocation method. Gherjalar and Mohammadikia, 2012 solved integral and integro-differential equations by utilizing B-splines function. The VIM (He, 2007, 1999) is a powerful device for solving various kinds of equations, linear and nonlinear. The technique has successfully been applied to many situations. For instance, He (2007) utilized the strategy to solve some integro-differential equations where he chose initial approximate solution in the form of exact solution with unknown constants.

In this work, the two-dimensional nonlinear V-FIDE is solved by the VADM. The two-dimensional nonlinear V-FIDE is given by

2 Adomian decomposition method

In what follows we display an outline for utilizing the ADM for solving the nonlinear V-FIDE. The Eq. (1) can be written as follows:

The ADM introduces the following expression

Substituting Eqs. (5) and (6) into Eq. (3) yields

To determine the components $u_0\text{,}{u}_1\text{,}{u}_2\text{,}\dots$ of the solution $u(x_1\text{,}{\text{ȷ}}_1)$ , we set the recurrence relation

The nonlinear terms given as follows: $$\gamma (u^l(y\text{,}\tau ))={\displaystyle \sum _{i=0}^{\infty }}{\mathrm{\Lambda }}_i\text{,}{D}^j(u(x_1\text{,}{\text{ȷ}}_1))={\displaystyle \sum _{i=0}^{\infty }}{L}_{i_j}\text{.}$$ where $\left(D^j=\frac{{\partial }^{j}u(x_1\text{,}{\text{ȷ}}_1)}{\partial x_1^j}\text{is derivative operator}\right)$ .

3 Variational iteration method

According to the VIM (He, 2007; He, 1999; He, 2000; He and Wang, 2007; Nadjafi and Tamamgar, 2008; Sadigh Behzadi, 2012) we can write the correction functional for Eq. (1) as

To find the optimal $\lambda$ , we proceed as follows $$\begin{array}{cc}\hfill \delta u_{i+1}(x_1\text{,}{\text{ȷ}}_1)=& \delta u_i(x_1\text{,}{\text{ȷ}}_1)+\delta {\int }_0^{{\text{ȷ}}_1}\lambda (\varsigma )[u_i^k(x_1\text{,}\varsigma )-\frac{\dot{f}(x_1\text{,}\varsigma )}{P_k(x_1\text{,}\varsigma )}\hfill \\ \hfill & -{\int }_a^{x_1}{\int }_{\mathrm{\Omega }}\frac{F(x_1\text{,}\varsigma \text{,}y\text{,}\tau )}{P_k(x_1\text{,}{\varsigma }_1)}\gamma (u^l(y\text{,}\tau ))\mathit{dyd}\tau +{\displaystyle \sum _{j=0}^{k-1}}\frac{P_j(x_1\text{,}\varsigma )}{P_k(x_1\text{,}\varsigma )}{u}^j(x_1\text{,}\varsigma )]d\varsigma \hfill \\ \hfill =& 0\hfill \end{array}$$

Then we apply the following stationary conditions $$\begin{array}{cc}\hfill 1-{\lambda }^{\prime }=& 0\text{,}\hfill \\ \hfill \lambda (\varsigma ={\text{ȷ}}_1)=& 0\text{,}\hfill \\ \hfill {\lambda }^{\prime \prime }=& 0\hfill \end{array}$$

This in turn gives $$\lambda =\varsigma -{\text{ȷ}}_1$$

The solution is given by

4 Variational Adomian decomposition method (VADM) for solving Eqs. (1) and (2)

This modified version of VADM which is obtained by the elegant coupling of VIM and ADM.

In VIM, from the correction functional for Eq. (1) we can write

Substituting from Eq. (6) in Eq. (12)we have

Then the recurrence relation is given by

The approximate solution is given by

To demonstrate the efficiency of the VADM we have considered the following examples.

5 Numerical examples

Example (5.1.) (Aghazadeh and Khajehnasiri, 2013; Poorfattah and Shaerlar, 2015; Darania et al., 2011): Consider the nonlinear integro - differential equation

This example is solved by the VADM formule (14), the results are shown in (Table 1).

Example (5.2) (Aghazadeh and Khajehnasiri, 2013; Poorfattah and Shaerlar, 2015; Darania et al., 2011): Consider the nonlinear integro-differential equation

This example is solved by the VADM formule (14), the results are written in (Table 2).

6 Conclusion

In this work, we applied the VADM for solving nonlinear mixed V-FIDE. The approximate solutions of V-FIDE are obtained by two powerful methods VIM and ADM in VADM. The given numerical examples showed the efficiency and accuracy of the introduced method, it reduces the size of computation without the restrictive assumption to handle nonlinear terms.