Application of hybrid functions operational matrices in the numerical solution of two-dimensional nonlinear integral equations
type: Item
instance of: scholarly article
title: Application of hybrid functions operational matrices in the numerical solution of two-dimensional nonlinear integral equations
zbMATH Open document ID: 1433.65355
author: Y. Aharonov
published in: Applied Numerical Mathematics
publication date: 2018-12-19T00:00:00Z
review text: This paper is devoted to the two-dimensional nonlinear Volterra integral equations of the second kind \[ f(x,t)=g(x,t)+\int\limits_{0}^{t}\int\limits_{0}^{x}k(x,t,y,z)H(y,z,f(y,z))dydz, \; (x,t)\in\Omega=[0,1)\times [0,1) \] with respect to the unknown function $f(x,t)$ defined on $\Omega$. \par An effective numerical technique based on the two-dimensional hybrid of Block-pulse functions and Legendre polynomials is presented to approximate the solution of a class of two-dimensional nonlinear Volterra integral equations of the second kind. \par Two-dimensional hybrid of Block-pulse functions and Legendre polynomials on the $\Omega$ can be defined as follows \[ \psi_{i_1,j_1,i_2,j_2}(x,t)=\begin{cases} L_{j_1}(2Nx-2i_1+1)L_{j_2}(2Nt-2i_2+1), & (x,t)\in [\frac{i_1-1}{N},\frac{i_1}{N})\times [\frac{i_2-1}{N},\frac{i_2}{N}), \\ 0, & \text{otherwise,} \end{cases} \] where $i_1,i_2=\overline{1,N}$, $j_1,j_2=\overline{1,M-1}$, and $L_{j_i}, \; i=1,2,$ are the well-known Legendre polynomials of order $j_i$. \par The main idea of this work is to develop the operational matrices of hybrid functions into two-dimensional modes. These operational matrices are then applied to convert the two-dimensional nonlinear integral equations into a system of nonlinear algebraic equations that can be solved numerically by Newton's method. An approximation of the error bound for the proposed method is presented. Finally, some numerical examples are considered to confirm the applicability and the efficiency of the method, especially in cases where the solution is not sufficiently smooth.
reviewed by: Alexander N. Tynda
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45G15
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 65H10
zbMATH DE Number: 6995986
zbMATH Keywords: nonlinear Volterra integral equations
zbMATH Keywords: two-dimensional hybrid functions
zbMATH Keywords: product operational matrix
MaRDI profile type: Publication
full work available at URL: j.apnum.2018.09.014
OpenAlex ID: W2894195044
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DOI: 10.1016/J.APNUM.2018.09.014
description: scientific article; zbMATH DE number 6995986