Abstract
In this article, an effective method is given to solve nonlinear two-dimensional Volterra integral equations of the second kind, which is arising from torsion problem for a long bar that consists of the nonlinear viscoelastic material type with a fixed elliptical cross section. First, the existence of a unique solution of this problem is discussed, and then, we find the solution of a nonlinear two-dimensional Volterra integral equation (NT-DVIE) using block-by-block method (B-by-BM) and degenerate kernel method (DKM). Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the method.
Highlights
The equations of the torsion problem were derived in detail with analytical solutions, by Muskhelishvili [1], Frank and Mises [2], Nowinski [3], and Sneddon and Berry [4]
In [5], boundary element method was developed for the nonuniform torsion of or multiply connected cylindrical bars of arbitrary cross section, where the bar is subjected to an arbitrary distributed twisting moment while its edges are restrained by the most general linear torsional
Fattahzadeh, in [11], solved two-dimensional linear and nonlinear Fredholm integral equations of the first kind based on Haar wavelet
Summary
The equations of the torsion problem were derived in detail with analytical solutions, by Muskhelishvili [1], Frank and Mises [2], Nowinski [3], and Sneddon and Berry [4]. In [6], nonlinear inelastic uniform torsion of bars by BEM was studied. Sapountzakis and Tsipiras in [7] used the boundary element method solution to the nonlinear inelastic uniform torsion problem of composite bars. El-Kalla and AL-Bugami in [8] discussed the nonlinear VolterraFredholm integral equation and torsion problems. Sheshtawy and Ghaleb in [9], discussed approximate solution to the problem of torsion by a boundary integral method. Fattahzadeh, in [11], solved two-dimensional linear and nonlinear Fredholm integral equations of the first kind based on Haar wavelet. In [13], solved two-dimensional Fredholm integral equations of the first kind using regularization-homotopy method. Effective numerical methods are proposed to obtain the solution of nonlinear two-dimensional Volterra integral equations of the second kind and study the values of absolute errors
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