**5. Conclusions and Discussion**

In this paper, we utilized FIM-SCP combined with the forward difference quotient to create efficient and accurate numerical algorithms for solving the considered direct and inverse TVIDE problems. According to the numerical examples in Section 4, we have demonstrated the performance of our proposed Algorithm 1 for seeking the approximate solutions of direct TVIDE problems in Examples 1 and 2. We can see that, for Example 1—which involved a second-order derivative with constant coefficients—Algorithm 1 provided an accurate result. Furthermore, for a problem involving a higher-order derivative with variable coefficients, it still provided high accuracy, in terms of solutions, as demonstrated in Example 2. Moreover, we handled inverse TVIDE problems using Algorithm 2, the effectiveness of which was illustrated in Examples 3 and 4. We used the Tikhonov regularization method to deal with the instability of the inverse problem; it can be seen that, in the examples, the regularization parameter *λ* was close to zero. Algorithm 2 could handle both continuous and piecewise-defined forcing terms with high accuracy, as demonstrated in Examples 3 and 4. Furthermore, when we perturbed the problems by adding noisy values, our Algorithm 2 still overcame the noise and provided approximate results that approached the analytical solutions. We further notice that our presented methods provide high accuracy, even when using only a small number of nodal points. Evidently, when we decrease the time step, they will furnish more accurate results. The rates of convergence with respect to time (based on the -<sup>∞</sup> norm) of our methods were observed to be linear. Finally, we also depicted the computational times for each example. However, we realize that there exist no theoretical error analysis results for the proposed numerical algorithms. Thus, our future research will study the error analysis, in order to find theories for order of accuracy and rate of convergence for our method. Another interesting direction for our future work is to extend our techniques to solve other types of IDEs and non-linear IDEs.

**Author Contributions:** Conceptualization, R.B., A.D., and P.G.; methodology, R.B. and A.D.; software, A.D. and P.G.; validation, R.B., A.D., and P.G.; formal analysis, R.B.; investigation, A.D. and P.G.; writing—original draft preparation, A.D. and P.G.; writing—review and editing, R.B.; visualization, A.D. and P.G.; supervision, R.B.; project administration, R.B.; funding acquisition, R.B. All authors have read and agreed to the published version of the manuscript.

**Acknowledgments:** The authors would like to thank the reviewers for their thoughtful comments and efforts towards improving our manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.
