Solving Systems of Volterra Integral Equations with Cardinal Splines

Abstract

This work is a continuation of the earlier article [1]. We establish new numerical methods for solving systems of Volterra integral equations with cardinal splines. The unknown functions are expressed as a linear combination of horizontal translations of certain cardinal spline functions with small compact supports. Then a simple system of equations on the coefficients is acquired for the system of integral equations. It is relatively straight forward to solve the system of unknowns and an approximation of the original solution with high accuracy is achieved. Several cardinal splines are applied in the paper to enhance the accuracy. The sufficient condition for the existence of the inverse matrix is examined and the convergence rate is investigated. We demonstrated the value of the methods using several examples.

The Cardinal Spline Methods for the Numerical Solution of Nonlinear Integral Equations

“…B n (x) are called one-dimensional B-splines, which are polynomial splines and have small supports (− ((n + 1)/2), (n + 1)/2),i.e., B n (x) � 0 for x > (n + 1)/2 or < − ((n + 1)/2) and excellent traits (cf [8]). In my previous papers [4,5], low-degree orthonormal spline and cardinal spline functions with small compact supports were applied in solving the second kind of Volterra integral equations. In this paper, we use the notation…”

“…Many differential equations with boundary values can be reformulated as integral equations; for example, for chemical integral equations with boundaries, Polyanin summarized different solutions of integral equations in [1][2][3], published in 2013 and 2016. In [4][5][6], we discussed numerical methods using cardinal splines in solving systems of linear integral equations. In this paper, we are going to explore the applications of cardinal splines in solving nonlinear integral equations.…”

The Numerical Solutions of Systems of Nonlinear Integral Equations with the Spline Functions

“…[8]). In my previous papers [4] and [5], low degree orthonormal spline and cardinal splines functions with small compact supports were applied in solving the second kind of Volterra integral equations. In this paper we use the notation ( )…”

“…Abundant papers have appeared on solving integral equations, for example, Polyanin summarized different solutions of integral equations in [1] and [2] [3] published in 2013 and 2016. In [4] [5] and [6], we discussed numerical methods using cardinal splines in solving systems of linear integral equations. In this paper we are going to explore the applications of cardinal splines in solving nonlinear systems of integral equations.…”