An expansion–iterative method for numerically solving Volterra integral equation of the first kind

Abstract

a b s t r a c tMost integral equations of the first kind are ill-posed, and obtaining their numerical solution often leads to solving a linear system of algebraic equations of a large condition number. So, solving this system is difficult or impossible. For numerically solving Volterra integral equation of the first kind an efficient expansion-iterative method based on the block-pulse functions is proposed. Using this method, solving the first kind integral equation reduces to solving a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to solve any linear system of algebraic equations. To show the convergence and stability of the method, some computable error bounds are obtained. Numerical examples are provided to illustrate that the method is practical and has good accuracy.

Bernstein Multiscaling polynomials and application by solving Volterra integral equations

“…However, our method is different from the methods, as presented in [32] and [33]. Consider Example 1, in Table 6, the expansion-iterative method and direct method are compared with our method.…”

Numerical expansion-iterative method for solving second kind Volterra and Fredholm integral equations using block-pulse functions

“…is an arbitrary vector norm [13,14]. Then, an approximate solution x(s) X T Φ(s) can be computed for Eq.…”

“…Two numerical direct methods based on vector forms of BPFs have been formulated in [11] and [12] to solve first kind Volterra and first or second kind Fredholm integral equations. BPFs are also used in [13] and [14] for constructing two numerical expansion-iterative methods for solving first kind Volterra and Fredholm integral equations. A part of the work done in the last two papers is based on numerical interpretations of the Neumann series concept.…”

“…With emphasis that some basic concepts required for formulating the method presented here are similar to those presented in [13] and [14], we organize this article as follows. A brief review on BPFs and their vector forms is provided in section 2.…”

A new approach to homotopy perturbation method for solving systems of Volterra integral equations of first kind

“…Volterra integral equations were also studied by several researchers with various numerical techniques. Masouri et al (2010) presented the numerical solution of Volterra integral equation of the first kind by an expansion-iterative method. Using Runge-Kutta method, Maleknejad and Shahrezaee (2004) solved the equations of type (1).…”