**Ratinan Boonklurb \* , Ampol Duangpan and Phansphitcha Gugaew**

Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand; ty\_math@hotmail.com (A.D.); poohdd28@hotmail.com (P.G.) **\*** Correspondence: ratinan.b@chula.ac.th

Received: 3 February 2020; Accepted: 5 March 2020; Published: 30 March 2020

**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* = *<sup>L</sup>* <sup>2</sup> or the vertical line *<sup>x</sup>* <sup>=</sup> *<sup>L</sup>* <sup>2</sup> 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.

**Keywords:** finite integration method; shifted Chebyshev polynomial; direct and inverse problems; Volterra integro-differential equation; Tikhonov regularization method

**MSC:** 65R20; 65R32
