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.
Highlights
An integro-differential equation (IDE) is an equation which contains both derivatives and integrals of an unknown function
Many of these problems cannot be directly solved, because we may not know all necessary information or an incomplete system may be provided. This has led to the study of both direct and inverse problems for a certain type of one-dimensional IDE involving time, which is called the one-dimensional time-dependent Volterra
We summarize the aforementioned procedures for finding approximate solutions to the direct (8) and inverse (19) TVIDE problems in Sections 3.1 and 3.2, respectively, as the numerical Algorithms 1 and 2, which are in the form of pseudocode
Summary
An integro-differential equation (IDE) is an equation which contains both derivatives and integrals of an unknown function. We concentrate on providing a decent numerical procedure to find approximate solutions for both the direct and inverse problems of the proposed TVIDE (1) It is well-known that the classification of problems involving differential equations was defined by Hadamard [13] in 1902. There exist many schemes which are generally used to solve both direct and inverse problems of Volterra IDEs, such as the above-mentioned methods Those methods utilize the process of approximating differentiation. Their technique [17] has not yet been utilized to overcome the direct and inverse problems of TVIDE, which are the major focuses of this work.
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