Abstract
In this paper, recently developed perturbation iteration method is used to solve Fredholm and Volterra integral equations. The study shows that the new method can be applied to both types of integral equations. Some numerical examples are given, and results are compared with other studies to illustrate the efficiency of the method.
1. Introduction
As one of the most important subjects of mathematics, differential and integral equations are widely used to model a variety of physical problems. Perturbation methods have been used in search of approximate analytical solutions for over a century [1–3]. Algebraic equations, integral-differential equations, and difference equations could be solved by these techniques approximately.
However, a major difficulty in the implementation of perturbation methods is the requirement of a small parameter or inserting a small artificial parameter in the equation. Solutions obtained by these methods are therefore restricted by a validity range of physical parameters. To eliminate the small parameter assumption in regular perturbation analysis, iteration techniques are incorporated with perturbations. Many attempts in this issue appear in the literature recently [4–13].
Recently, a new perturbation-iteration algorithm has been developed by Pakdemirli and his coworkers [14–16]. A preliminary study of developing root finding algorithms systematically [17–19] finally led to generalization of the method to differential equations also [14–16]. An iterative scheme is constituted over the perturbation expansion in the new technique. The method has been successfully implemented to first-order equations [15] and Bratu-type second-order equations [14].
In this paper, this new technique is applied to integral equations for the first time. Fredholm and Volterra integral equations are considered, where , , and is the unknown function to be determined. Results are compared with some other studies.
2. Overview of the Method
In the present paper, the simplest perturbation-iteration algorithm is used by taking one correction term in the perturbation expansion and correction terms of only first derivatives in the Taylor series expansion, that is, , [14–16]. Consider the Volterra integral equation that has the form of where and is the artificially introduced perturbation parameter. In this method, we use only one correction term in the perturbation expansion:
Substituting (5) into (3) and expanding in a Taylor series with first-order derivatives only yield or All derivatives are evaluated at .
Starting with the initial condition , first has been calculated by the help of (7). Then we substitute into (5) to find . Iteration process is repeated using (7) and (5) until we obtain a satisfactory result.
3. Application
Example 1. Consider the Fredholm integral equation of the second kind
with exact solution
Equation (8) can be rewritten in the following form:
where is a small parameter. The terms in (7) are
Note that introducing the small parameter as a coefficient of the integral term simplifies (7) and makes it solvable. For this specific example (7) reads
When applying the iteration formula (5), we select an initial guess appropriate to the boundary condition and at each step we determine coefficients from the boundary condition. Starting with the initial function
and using the formula, the approximate solutions at each step are
Higher iterations are not given here for brevity. Using a symbolic manipulation software, iterations could be calculated up to any order. In Table 1, some of our iterations are compared with the exact solution and the error between the exact solution, and are given which are of order .
Example 2. Consider the following integral equation:
The exact solution of the problem is
Equation (15) can be rewritten in the following form:
where is a small artificial parameter. The terms in (7) are
Equation (7) reduces to
Choosing the initial guess
and using the formula, the approximate solutions at each step are
Higher iterations are not given for brevity. In Table 2, some of our iterations are compared with the exact solution, and the errors between the exact solution and are given which are of order .
Example 3. Consider the equation with the exact solution Equation (22) is rewritten in the following form: where is an artificially introduced small parameter. The terms in (7) are Equation (7) reduces to Choosing the initial guess and using the formula, the approximate solutions at each step are Higher iterations are not given for brevity. In Table 3, some of our iterations are compared with the exact solution, and the errors between the exact solution and are given which are of order .
Example 4. Consider the following integral equation:
The exact solution of the problem is
Equation (29) is rewritten in the following form:
and proceeding in a similar way yields the following iteration algorithm:
One may select the initial guess as . The successive approximations are
Higher iterations are not given for brevity. In Table 4, some of our iterations are compared with the exact solution, and the errors between the exact solution and are given which are of order .
4. Conclusion
In this paper, we have applied the newly developed Perturbation Iteration Algorithm to some Fredholm and Volterra type integral equations for the first time. Numerical results show that method is an effective perturbation-iteration technique producing successful analytical results for integral equations.