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In this paper, we propose the difference splitting scheme for a mixed problem posed for n - dimensional symmetric t-hyperbolic systems. We construct the difference splitting scheme for the numerical calculation of stable solutions for this system. To build a difference scheme, a multidimensional problem is split into one-dimensional ones and solved for each direction. A discrete analogue of the Lyapunov's function is constructed for the numerical verification of stability solutions for the considered problem. A priori estimate is obtained for the discrete analogue of the Lyapunov's function. This estimate allows us to assert the exponential stability of the numerical solution. A theorem on the exponential stability solution of the boundary value problem for linear hyperbolic system was proved. These stability theorems give us the opportunity to prove the convergence of the numerical solution. [ABSTRACT FROM AUTHOR]

In this note, we consider a hypersingular integral equations (HSIEs) of the first kind on the interval [-1, 1] with the assumption that kernel of the hypersingular integral is constant on the diagonal of the domain D =[1, -1] ×[-1, 1]. Projection method together with Chebyshev polynomials of the first and second kinds are used to find bounded and unbounded solutions of HSIEs respectively. Exact calculations of hypersingular and singular integrals for Chebyshev polynomials allows us to obtain high accurate approximate solution. Gauss-Chebyshev quadrature with Gauss-Lobotto nodes are presented as the high accurate computation of regular kernel integrals. Six examples are provided to verify the validity and accuracy of the proposed method. Comparisons with other methods are also given. Numerical examples reveals that approximate solutions are exact if solution of HSIEs is of the polynomial forms with corresponding weights. It is worth to note that proposed method works well for large value of node points n and errors are drastically decreases. Comparisons of SPU times are also shown to demonstrate effectiveness of the method and less complexity computations. [ABSTRACT FROM AUTHOR]

We consider an n×n system of nonlinear integral equations of Volterra type (nonlinear VIEs) arising from an economic model. By applying the Newton-Kantorovich method to the nonlinear VIEs we linearize them into linear Volterra type integral equations (linear VIEs). Uniqueness of the solution of the system is shown. An idea has been proposed to find the approximate solution by transforming the system of linear VIEs into a system of linear Fredholm integral equations by using sub-collocation points. Then the backward Newton interpolation formula is used to find the approximate solution at the collocation points. Each iteration is solved by the Nystrom type Gauss-Legendre quadrature formula (QF). It is found that by increasing the number of collocation points of QF with fewer iterations, a high accurate approximate solution can be obtained. Finally, an illustrative example is demonstrated to validate the accuracy of the method. [ABSTRACT FROM AUTHOR]

A flat crack, Ω, is lying in a three-dimensional homogenous isotropic elastic solid subjected to shear loading. A mathematical formulation is developed based on the mixed boundary values for Ω such that the problem of finding the resulting force can be written in the form of hypersingular integral equation. Employing conformal mapping, the integral equation is transformed to a similar equation over a circular region, D. By making a suitable representation of hypersingular integral equation, the problem is reduced to solve a system of linear equations. Numerical solution for the shear stress intensity factors, maximum stress intensity, and strain energy release rate is obtained. Our results give an excellent agreement to the existing asymptotic solutions. [ABSTRACT FROM AUTHOR]

This paper presents an efficient approximate method to obtain a numerical solution, which is bounded at the end point x = -1, for Cauchy type singular integral equations of the first kind on the interval [-1,1]. The solution is derived by approximating the unknown density function using the weighted Chebyshev polynomials of the third kind, and then computing the Cauchy singular integral which is obtained analytically. The known force function is interpolated using the Chebyshev polynomials of the fourth kind. The exactness of this approximate method is shown for characteristic equation when the force function is a cubic. Particular result is also given to show the exactness of this method. [ABSTRACT FROM AUTHOR]