Modified method for determining an approximate solution of the Fredholm–Volterra integral equations by Taylor’s expansion

Abstract

A modified method for determining an approximate solution of the Fredholm–Volterra integral equations of the second kind is developed. Via Taylor’s expansion of the unknown function, the integral equation to be solved is approximately transformed into a system of linear equations for the unknown and its derivatives, which can be dealt with in an easy way. The obtained nth-order approximate solution is of high accuracy, and is exact for polynomials of degree n. In particular, an approximate solution with satisfactory accuracy of the weakly singular Volterra integral equation is also given. The efficiency of the method is illustrated by some numerical examples.

Keywords:

• Taylor’s expansion
• Fredholm integral equation
• Volterra integral equation
• Approximate solution
• Weakly singular kernel