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 Burgers’ equation is one of the nonlinear partial differential equations that has been studied by many researchers, especially, in terms of the fractional derivatives. In this article, the numerical algorithms are invented to obtain the approximate solutions of time-fractional Burgers’ equations both in one and two dimensions as well as time-fractional coupled Burgers’ equations which their fractional derivatives are described in the Caputo sense. These proposed algorithms are constructed by applying the finite integration method combined with the shifted Chebyshev polynomials to deal the spatial discretizations and further using the forward difference quotient to handle the temporal discretizations. Moreover, numerical examples demonstrate the ability of the proposed method to produce the decent approximate solutions in terms of accuracy. The rate of convergence and computational cost for each example are also presented.

Abstract. Let q, a, b, and T be real numbers with q ≥ 0, a > 0, 0 < b < 1, and T > 0. This article studies the following degenerate semilinear parabolic first initial-boundary value problem,

In this paper, the finite integration method and the operational matrix of fractional integration are implemented based on the shifted Chebyshev polynomial. They are utilized to devise two numerical procedures for solving the systems of fractional and classical integro-differential equations. The fractional derivatives are described in the Caputo sense. The devised procedure can be successfully applied to solve the stiff system of ODEs. To demonstrate the efficiency, accuracy and numerical convergence order of these procedures, several experimental examples are given. As a consequence, the numerical computations illustrate that our presented procedures achieve significant improvement in terms of accuracy with less computational cost.

A closed knight’s tour of a normal two-dimensional chessboard by using legal moves of the knight has been generalized in several ways. One way is to consider a closed knight’s tour on a ringboard of width [Formula: see text], which is the [Formula: see text] chessboard with the middle part missing and the rim contains [Formula: see text] rows and [Formula: see text] columns. Another way is to stack [Formula: see text] copies of the [Formula: see text] chessboard to construct an [Formula: see text] rectangular chessboard and the closed knight’s tour can be on the surface or within the [Formula: see text] rectangular chessboard. This paper combines these two ideas by stacking [Formula: see text] copies of [Formula: see text] ringboard of width [Formula: see text], which we call the [Formula: see text]-rectangular tube. We explore the existence and the nonexistence of closed knight’s tours for [Formula: see text]-tube and [Formula: see text]-tube.