Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method
type: Item
instance of: scholarly article
title: Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method
zbMATH Open document ID: 1176.65164
author: M. Y. Rahimi
author: F. Talati
author: Abolfazl Tari
author: Sedaghat Shahmorad
published in: Journal of Computational and Applied Mathematics
publication date: 2009-05-28T00:00:00Z
review text: The authors consider two-dimensional Volterra integral equations of the form \[ u(x, t)-\int_0^t \int_0^x K(x, t, y, z, u(y, z))\,dydz=f(x, t), \] where \(K\) takes the degenerate form \[ K(x, t, y, z)=\sum_{i=0}^p v_i(x, t)w_i(y, z, u(y, z)). \] In section 2 fundamental properties of the TDDT are summarised in a theorem. This is followed by the main theorem in the paper which states and derives differential transforms for \[ g(x, t)=\int_{t_0}^t \int_{x_0}^x u(y, z)v(y, z)\,dydz\text{ and }g(x, t)=h(x, t)\int_{t_0}^t \int_{x_0}^xu(y, z)\,dydz. \] The method is described in section 3. Section 4 includes illustrative examples (including a non-linear problem) to demonstrate the accuracy of the presented method. In each example a recurrence relation for the differential transform is obtained and results arising from the use of \textsc{Maple} are given. The authors anticipate that the method will be developed for solving two-dimensional Volterra integro-differential equations and their systems.
reviewed by: Pat Lumb
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 5559030
zbMATH Keywords: Volterra integral equations
zbMATH Keywords: two-dimensional differential transform method
zbMATH Keywords: numerical examples
zbMATH Keywords: degenerate kernel
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: j.cam.2008.08.038
OpenAlex ID: W1982754700
cites work: The Numerical Solution of Fredholm integral Equations of the Second Kind
cites work: Computational Methods for Integral Equations
cites work: Q4255464
cites work: Q5624183
cites work: Linear integral equations.
cites work: Solution of boundary value problems for integro-differential equations by using differential transform method
cites work: Applications of differential transform methods to differential-algebraic equations
cites work: Solutions of the system of differential equations by differential transform method
cites work: Solving partial differential equations by two-dimensional differential transform method
cites work: Extrapolation of Nyström solution for two dimensional nonlinear Fredholm integral equations
cites work: Richardson extrapolation of iterated discrete Galerkin solution for two-dimensional Fredholm integral equations
cites work: Two-dimensional differential transform for partial differential equations
cites work: Differential transform method for solving Volterra integral equation with separable kernels
DOI: 10.1016/J.CAM.2008.08.038
description: scientific article; zbMATH DE number 5559030