**1. Introduction**

An integro-differential equation (IDE) is an equation which contains both derivatives and integrals of an unknown function. Several situations in the branches of science and engineering can be demonstrated by developing mathematical models which are often in the form of IDEs, such as in RLC circuit analysis, the activity of interacting inhibitory and excitatory neurons, the Wilson–Cowan model, and so on; see Reference [1] for more applications. In fact, 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 IDE (TVIDE). Hence, in this study, we investigate the TVIDE of the following form

$$v\_l(\mathbf{x},t) + \mathcal{L}v(\mathbf{x},t) = \int\_0^t \kappa\_1(\mathbf{x},\eta)v(\mathbf{x},\eta)d\eta + \int\_0^\mathbf{x} \kappa\_2(\xi,t)v(\xi,t)d\xi + F(\mathbf{x},t),\tag{1}$$

for all (*x*, *t*) ∈ (0, *L*) × (0, *T*], where *x* and *t* represent space and time variables, respectively; L is the spatial linear differential operator of order *n*; *κ*1(*x*, *t*) and *κ*2(*x*, *t*) are the given continuously integrable kernel functions; and *v*(*x*, *t*) is an unknown function, which is to be determined subject to prescribed initial and boundary conditions. We remark that, if a forcing term *F*(*x*, *t*) of (1) is given, then this problem has only one unknown *<sup>v</sup>*(*x*, *<sup>t</sup>*) ∈ *<sup>C</sup>n*,1([0, *<sup>L</sup>*] × [0, *<sup>T</sup>*]) to be solved, and it is called a direct problem. In contrast, if the forcing term *F*(*x*, *t*) is missing, then this problem has two unknowns *<sup>F</sup>*(*x*, *<sup>t</sup>*) ∈ *<sup>C</sup>*([0, *<sup>L</sup>*] × [0, *<sup>T</sup>*]) and *<sup>v</sup>*(*x*, *<sup>t</sup>*) ∈ *<sup>C</sup>n*,1([0, *<sup>L</sup>*] × [0, *<sup>T</sup>*]) to be solved, and it is called an inverse problem. However, for the inverse problem in this paper, we specifically define the forcing term *F*(*x*, *t*) := *β*(*t*)*f*(*x*, *t*), where *β*(*t*) is a missing source function to be retrieved and *f*(*x*, *t*) is the given function. We note that (1) has both *<sup>t</sup>* <sup>0</sup> *<sup>κ</sup>*1(*x*, *<sup>η</sup>*)*v*(*x*, *<sup>η</sup>*)*d<sup>η</sup>* and *<sup>x</sup>* <sup>0</sup> *κ*2(*ξ*, *t*)*v*(*ξ*, *t*)*dξ*, while several studies in the literature have considered similar problems containing only one of these two terms.

The Volterra IDE containing only an integration term with respect to time arises in many applications, including the compression of poro-viscoelastic media, blow-up problems, analysis of space–time-dependent nuclear reactor dynamics, and so on; see Reference [2]. The existence, uniqueness, and asymptotic behavior of its solution have been discussed in Reference [3]. There are many authors who have studied the numerical solution of this type of problem by using techniques such as the finite element method [2], finite difference method (FDM) [4], collocation methods in polynomial spline [5], the implicit Runge–Kutta–Nyström method [6], the Legendre collocation method [7], and so on.

On the other hand, the Volterra IDE containing only an integration term with respect to space has also been studied in various areas, such as for the one-dimensional viscoelastic problem and one-dimensional heat flow in materials with memory [8], modeling heat/mass diffusion processes, biological species coexisting together with increasing and decreasing rates of growth, electromagnetism, and ocean circulation, among others [9]. Moreover, the existence and uniqueness for this type of Volterra IDE were shown in Reference [8]. Consequently, abundant numerical methods have appeared for finding solutions to this type of Volterra IDE using, for example, spline collocation method [10], collocation method with implicit Runge–Kutta method [11], decomposition method [12], and so on.

However, our problem deals with a Volterra IDE involving both temporal and spatial integrations. There have been no results in the literature regarding the existence and uniqueness of solutions to this type of problem. In this paper, we concentrate on providing a decent numerical procedure to find approximate solutions for both the direct and inverse problems of the proposed TVIDE (1).

Generally, it is well-known that the classification of problems involving differential equations was defined by Hadamard [13] in 1902. Mathematical problems involving differential equations are well-posed if the following conditions hold: existence, uniqueness, and stability. Otherwise, the problem is called ill-posed; this normally occurs in the inverse problem. Even though the initial and boundary conditions are prescribed, it is not sufficient to guarantee that our inverse problem (1) has unique solutions *β*(*t*) and *v*(*x*, *t*). Hence, additional conditions (e.g., the observation or measurement of data) need to be involved. In practice, there are many kinds of additional conditions; for example, a fixed point of the system, an average time of the system, or an integral of the system. After the additional conditions has been added as an auxiliary condition in our inverse problem (1), we can obtain the existence and uniqueness of *β*(*t*) and *v*(*x*, *t*). However, the additional condition may contains measurement or observation errors, which may cause the instability in the solutions; namely, a small perturbation in the input data can produce a considerable error, especially for *β*(*t*). Thus, some regularization techniques are required to overcome the ill-posedness and stabilize the solution.

There exist many schemes which are generally used to solve both direct and inverse problems of Volterra IDEs, such as the above-mentioned methods. However, those methods utilize the process of approximating differentiation. It is well-known that numerical differentiation is very sensitive to rounding errors, as its manipulation task involves division by a small step-size. On the other hand, the process of numerical integration involves multiplication by a small step-size and, so, it is very insensitive to rounding errors. In recent years, the finite integration method (FIM) has been developed to find approximate solutions of linear boundary value problems for partial differential equations (PDEs). The concept of FIM is to transform a given PDE into an equivalent integral equation, following which a numerical integration method, such as the trapezoid, Simpson, or Newton–Cotes methods (see References [14–16]), are applied. In 2018, Boonklurb et al. [17] modified the traditional FIM by using Chebyshev polynomials to solve one- and two-dimensional linear PDEs and obtained a more accurate result compared to the traditional FIMs and FDM. However, 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.

In this paper, we formulate numerical algorithms for solving the direct and inverse problems of TVIDE (1). We manipulate the idea of FIM in Reference [17] by using shifted Chebyshev polynomials, which we call the FIM with shifted Chebyshev polynomials (FIM-SCP), to deal with the spatial variable and use the forward difference quotient to estimate the time derivative. We further apply the Tikhonov regularization method to stabilize our ill-posed problem (1). The rest of the paper is organized as follows. In Section 2, the definition and some basic properties concerning the shifted Chebyshev polynomial are given to construct the shifted Chebyshev integration matrices. The Tikhonov regularization method is also presented in Section 2. In Section 3, we use the FIM-SCP and the forward difference quotient to devise efficient numerical algorithms to find approximate solutions to the direct and inverse problems of (1). Then, we implement our proposed algorithms through several examples, in order to demonstrate their efficiency compared with their analytical solutions. Furthermore, we also display the time convergence rate and CPU time (s) in Section 4. Finally, the conclusion and some directions for future work are given in Section 5.
