Numerical solution ofVolterra‐Fredholmintegral equation via hyperbolic basis functions

Abstract

In this paper, a new method using hyperbolic basis functions is presented to solve second kind linear Volterra‐Fredholm integral equation. In other words, our method approximates the solution of a Volterra‐Fredholm integral equation by the hyperbolic basis functions, which produce block‐pulse functions. Hence, the new method reduces the linear Volterra‐Fredholm integral equation to a system of algebraic equations. Some numerical examples are provided to illustrate the computational efficiency and accuracy of the new method.

An Algorithm for the Solution of Nonlinear Volterra–Fredholm Integral Equations with a Singular Kernel

“…Fixed point methods are some of the methods that have been used successfully, especially those based on rationalized Haar wavelets [23], fixed point methods in extended b-metric space [24], Schauder bases in an adequate Banach space [25], and Schauder bases [26]. Among the numerical methods that were also introduced using Hosoya polynomials [27], there are Haar wavelets [1], Bernstein polynomials [28], hyperbolic basis functions [29], and block-pulse functions [30]. Some of the methods that have been reported to be successful include a neural network approach [31], Tau methods [32], and parameter continuation methods [33].…”

An Algorithm for the Closed-Form Solution of Certain Classes of Volterra–Fredholm Integral Equations of Convolution Type

“…Fixed point methods based on Schauder bases [21,29,33,98], Schauder basis in an adequate Banach space [58], rationalized Haar wavelet [68], and fixed point methods in extended b-metric space [71] have been used successfully. Approximate methods using block-pulse functions [52,61,67], Bernstein polynomials [59], Haar wavelets [4], hyperbolic basis functions [84] and Hosoya polynomials [85] have also been presented. Other successful methods that have been reported are the reproducing kernel methods [14,78,88], Tau methods [50,55], modified hat functions method [56], optimal control method [28], scaling function interpolation wavelet method [39], hybrid contractive mapping and parameter continuation method [72], sinusoidal basis functions and a neural network approach [74].…”