Abstract
Numerical solutions of linear and nonlinear integrodifferential-difference equations are presented using homotopy analysis method. The aim of the paper is to present an efficient numerical procedure for solving linear and nonlinear integrodifferential-difference equations. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system.
1. Introduction
Currently, considerable interest in mixed integrodifferential-difference equations (IDDEs) has been stimulated due to their numerous applications in the areas of engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory, and electrostatics [1–7]. In integrodifferential-difference equations, the unknown function appears to be under integration sign, and it may also include the derivatives and functional arguments of the unknown function. This type of equations can be grouped into Fredholm integrodifferential-difference equations and Volterra integrodifferential-difference equations. The upper bound of the integral part of Volterra type is variable, while it is a fixed number for that of Fredholm type [1]. Since IDDEs are usually difficult to solve in an analytical manner, or to obtain closed form solution, a numerical method is needed. Many numerical methods were applied such as Taylor matrix method [2], Chebyshev finite difference method [8], Legendre tau method [4], Bessel matrix method [5], and variational iteration method [9].
In this paper, we propose an efficient method, namely homotopy analysis method, to obtain the numerical solution of th-order linear Fredholm integrodifferential-difference equation with variable coefficients. Consider with mixed conditions where , and are functions defined on ; the real coefficients , and are appropriate constants.
Homotopy analysis method (HAM) was first introduced by Liao [10] to obtain series solutions of various linear and nonlinear problems. HAM is a promising method that gives us acceptable analytical results with convenient convergence [10]. In contrast to perturbation techniques, this approach is independent of any small parameters. In addition, different from all other analytic techniques, HAM provides us with a simple procedure to obtain the convergence of series of solutions, so that one can obtain accurate enough approximations by auxiliary convergence-controller parameter . In his papers, Liao used this method to solve many nonlinear problems. In [10], especially, he pointed out the basic ideas of the HAM and gave the details in theory and he has successfully applied it to many nonlinear problems [10–15]. Recently, this technique has successfully been applied to several nonlinear problems, such as the viscous flows of non-Newtonian fluids [16, 17], nonlinear heat transfer [18], nonlinear Fredholm integral equations [19], the KdV-type equations [20], differential-difference equations [21], time-dependent Emden-Fowler-type equations [22], Laplace equation with Dirichlet and Neumann conditions [23], and pantograph equations [24].
2. Homotopy Analysis Method
The purpose of the study is to apply the homotopy analysis method to the integrodifferential-difference equation. Now suppose that denotes an initial approximation guess of exact solution of . The auxiliary parameter , which is a convergence-controller parameter, the auxiliary function , and the auxiliary linear operator play important roles within the homotopy analysis method to adjust and control the convergence region of solution series. Liao constructs, using as an embedding parameter, the so-called zero-order deformation equation where is unknown function to be determined and is given by When , the zero-order deformation equation (3) becomes , and when , then the zero-order deformation equation (3) becomes . As increases from 0 to 1, the solution varies from the initial guess to the solution . So is exactly the solution of nonlinear equation (1). Expanding in Taylor series with respect to q, we have where If the auxiliary linear operator, the initial guess, the auxiliary parameter which will be determined, later and the auxiliary function are properly chosen, the power series of converges at , then the following series solution can be obtained: where the terms can be determined by the so-called high-order deformation equations. Now the vector is defined as , and differentiating (3) equation times with respect to embedding parameter q, then setting , and finally dividing by , we obtain the so-called mth-order deformation equation in the following form: where In order to obey both the first rule of solution expression and the rule of the coefficient ergodicity, the corresponding auxiliary function is determined uniquely by [10].
For any given nonlinear operator N and the term can be easily expressed by (10). So, we can obtain by means of solving the linear high-order deformation equation (8). The mth-order approximation of is given by The foregoing approximate solution consists of , which is a cornerstone of the HAM in determining convergence of series solution rapidly. We may adjust and control the convergence region and rate of the solution series (11) by means of the auxiliary parameter . To obtain valid region of , we first plot the so-called curves of , , and , where and so on. According to these curves, it is easy to discover the valid region of , which corresponds to the line segments nearly parallel to the horizontal axis.
Theorem 1 (convergence theorem). As long as the series (8) converges to , where is governed by the high-order deformation equation (9) under the definitions (10) and (11), it must be the exact solution of (3).
3. Numerical Examples and Error Analysis
In the present paper, the HAM technique is applied to the integrodifferential-difference equations, and how one can control the convergence of approximate solution and make the convergence fast is shown. The method is applied to the following problems.
Example 2. Firstly, let us consider the second-order integrodifferential-difference equation with variable coefficients [4] with the exact solution and initial conditions To solve (12) by means of homotopy analysis method, we choose as initial approximation which satisfies (13) and let us define as follows: Hence, the high-order deformation equation (8)–(10) is as follows: Therefore, by starting with , we successively obtain , by (15), In this way we can get by choosing proper convergence parameter with . To find the valid region of on the convergence of as goes to infinity, we plot the curves of and as shown in Figure 1 which clearly indicates that the valid region of is about −0,6 < < −0,15.
Some numerical results of , , and for = −0,368 are presented in Table 1, and the absolute errors of , , and for = −0,368 are given in Table 2. We also give the graphs of the exact solution and 20th order approximation for = −0,368 in Figure 2 and the absolute error of the 20th order approximation for = −0,368 in Figure 3.
Example 3. Now, consider the second-order integrodifferential-difference equation with variable coefficients [1] with exact solution and subjects to initial conditions In this case, we determine the high-order deformation equation (8)–(10) for (18) and (19) Choosing as initial approximation which satisfies (19), we successively obtain , by (20), and we can get The curves of and have been shown in Figure 4, and the valid region of is about −0,7 < < −0,2.
Finally, we have solutions of (15) and (18) for = −0,37 as shown in Table 3, and the absolute errors of , , and for = −0,37 are given in Table 4. We also give the graphs of the exact solution and 20th order approximation for = −0,37 in Figure 5 and the absolute error of the 20th order approximation for = −0,37 in Figure 6.
Example 4. Finally, consider the third-order nonlinear integrodifferential-difference equation [9], subject to initial conditions with exact solution . Now, the high-order deformation equation (8)–(10) for (22) and (23) is We choose as initial guess which satisfies (23) and we get In Figure 7, the curves of and have been plotted, and the valid region of is determined as −0,7 < < 0.
Then, we also have solutions of (22) for = −0,64 as shown in Table 5, and the absolute errors of , and , for = −0,64 are given in Table 6. We also give the graphs of the exact solution and 20th order approximation for = −0,64 in Figure 8 and the absolute error of the 20th order approximation for = −0,64 in Figure 9.
4. Conclusions
In this work, a reliable algorithm based on the HAM to solve various integrodifferential-difference equations with constant and variable coefficients is presented. Some examples are given to illustrate the validity and accuracy of this procedure. The series solutions of (12), (18), and (22) obtained by HAM contain the auxiliary parameter . In general, by means of the so-called curve, it is straightforward to choose a proper value of which ensures that the series solution is convergent. Figures 1, 4, and 7 show the curves obtained from the mth-order HAM approximation solutions. From these figures, the valid regions of correspond to the line segments nearly parallel to the horizontal axis. Finally, it can be concluded that the homotopy analysis method is a promising tool for both linear and nonlinear IDDEs.