Abstract
The present paper employs a numerical method based on the improved block–pulse basis functions (IBPFs). This was mainly performed to resolve the Volterra–Fredholm integral equations of the second kind. Those equations are often simplified into a linear system of algebraic equations through the use of IBPFs in addition to the operational matrix of integration. Typically, the classical alterations have enhanced the time taken by the computer program to solve the system of algebraic equations. The current modification works perfectly and has improved the efficiency over the regular block–pulse basis functions (BPF). Additionally, the paper handles the uniqueness plus the convergence theorems of the solution. Numerical examples have been presented to illustrate the efficiency as well as the accuracy of the method. Furthermore, tables and graphs are used to show and confirm how the method is highly efficient.
Highlights
In recent years, there has been a growing interest in the formulation of many engineering and physical problems in terms of integral equations
The non-linear Volterra–Fredholm integral equation given in Equation (2) will be solved using BPF
In order to solve the nonlinear Volterra–Fredholm integral equation given in Equation (2), the following approximations must be used: T
Summary
There has been a growing interest in the formulation of many engineering and physical problems in terms of integral equations. Many attempts have been made to solve integral equations by several researchers using numerical or perturbed methods. In 1999, He [8] tried to solve linear differential and integral equations by using a new method called the homotopy perturbation method (HPM). He managed to solve some nonlinear problems [9]. It can be deduced that the integration of the vector φm (t) defined in approximated by 2m This operational matrix will be modified to obtain better results than that used in [16].
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