Abstract
The goal of this paper is study the mixed integral equation with singular kernel in two-dimensional adding to the time in the Volterra integral term numerically. We established the problem from the plane strain problem for the bounded layer medium composed of different materials that contains a crack on one of the interface. Also, the existence of a unique solution of the equation proved. Therefore, a numerical method is used to translate our problem to a system of two-dimensional Fredholm integral equations (STDFIEs). Then, Toeplitz matrix (TMM) and the Nystrom product methods (NPM) are used to solve the STDFIEs with Cauchy kernel. Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the methods. The codes were written in Maple.
Highlights
Many problems of engineering, mathematical physical, and contact problems in the theory of elasticity lead to singular integral equations
In [4, 5], used different methods to obtain the solution of F-VIE of the first and second kinds in which the Fredholm integral term is considered in position while the Volterra integral term is considered in time
EL-Borai et al, in [6], studied the numerical solution for the T-DFIE with weak singular kernel, but they have studied the problem on a rectangular path of the parties only
Summary
Mathematical physical, and contact problems in the theory of elasticity lead to singular integral equations. In [4, 5], used different methods to obtain the solution of F-VIE of the first and second kinds in which the Fredholm integral term is considered in position while the Volterra integral term is considered in time. ALBugami in [8] studied and discussed the solution of the two-dimensional singular Fredholm integral equation (TDFIE) with time. In [10], the authors studied the linear two-dimensional Volterra integral equation with continuous kernel numerically. Formula (1) is called the MIE with singular kernel in Advances in Mathematical Physics two-dimensional of the second kind with Cauchy kernel in ðL2ð−1, 1Þ × L2ð−1, 1ÞÞ × Cð0 ; YÞ ; Y < 1, where the FI term is considered in position with singular kernel, and the VI term is considered in time with a positive and continuous kernelζðt, τÞ. The numerical coefficient λ is called the parameter of the IE
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