Numerical Solution of Direct and Inverse Problems for Time-Dependent Volterra Integro-Differential Equation Using Finite Integration Method with Shifted Chebyshev Polynomials

Abstract

In this article, the direct and inverse problems for the one-dimensional time-dependent Volterra integro-differential equation involving two integration terms of the unknown function (i.e., with respect to time and space) are considered. In order to acquire accurate numerical results, we apply the finite integration method based on shifted Chebyshev polynomials (FIM-SCP) to handle the spatial variable. These shifted Chebyshev polynomials are symmetric (either with respect to the point x = L 2 or the vertical line x = L 2 depending on their degree) over [ 0 , L ] , and their zeros in the interval are distributed symmetrically. We use these zeros to construct the main tool of FIM-SCP: the Chebyshev integration matrix. The forward difference quotient is used to deal with the temporal variable. Then, we obtain efficient numerical algorithms for solving both the direct and inverse problems. However, the ill-posedness of the inverse problem causes instability in the solution and, so, the Tikhonov regularization method is utilized to stabilize the solution. Furthermore, several direct and inverse numerical experiments are illustrated. Evidently, our proposed algorithms for both the direct and inverse problems give a highly accurate result with low computational cost, due to the small number of iterations and discretization.

The mechanism of decrease in bearing capacity and performance evaluation of the yielding bolt in coal mine

“…The element's strain could be expressed as follows 25,28 : where the geometric matrix B is . Where B i is:…”

“…The element's shape function matrix is as follows 25,28 :…”

Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials

“…This modified FIM also provides much higher accuracy than the FDM and those original FIMs with small computational nodes. Recently, the modified FIM was widely utilized to apply with many applications, see [20][21][22][23]. Also, it was demonstrated that results obtained by the modified FIM achieve significant improvement in terms of accuracy more than several existing methods.…”

“…The major aim of this paper is to develop the modified FIM [19] by using the shifted Chebyshev polynomial which thereafter will be referred to as FIM-SCP and also constructs the operational matrix of fractional integration in order to devise two numerical procedures for solving numerically the systems of both FIDEs and CIDEs of the Volterra type. Actually, the technique of FIM-SCP has been proposed in [20]. It is used to find numerical solutions of direct and inverse problems for the time-dependent Volterra integro-differential equation (VIDE).…”

“…It is used to find numerical solutions of direct and inverse problems for the time-dependent Volterra integro-differential equation (VIDE). Hence, in this paper, we continue our study from [20] by extending the VIDEs to be a kind of system that involves the fractional-order differential operator. The problem mainly considered in this article is called the system of FIDEs which is studied in the form presented in [24], i.e.,…”