Numerical Solution of Direct and Inverse Problems for Time-Dependent Volterra Integro-Differential Equation Using Finite Integration Method with Shifted Chebyshev Polynomials
Round 1
Reviewer 1 Report
The manuscript proposes a numerical method for the solution of direct and inverse problems for time-dependent integro-differential equations. The method is based on forward finite different in time and Chebyshev polynomial approximation in space. For handling space derivatives, finite integration methods are used and combined with quadrature formulas based on Chebyshev polynomials. Tikhonov regularization is used for ill-posed inverse problems. The efficiency and accuracy of the method is shown with some numerical examples.
I think the ideas of the method are explained clearly and in detail. I enjoyed the structure of the manuscript and the proposed examples. I only have a few minor comments that I report below. The language, however, is sometimes poor with several grammar mistakes. I recommend a revision with the help of a native English speaker
I recommend the paper for acceptance in Symmetry, after minor revisions and a thorough revision of the English language.
Minor comments:
- Abstract line 10, “less computational cost”: please explain less than what? Or otherwise change “less” into “low”
- When you refer to the pseudocodes (last line of section 3.1 and section 3.2), I recommend to give an explicit number to the pseudocode, like it is done for figures and tables, and refer to it in the text. Indeed, in the current versions the pseudocodes are moved away from the text where they are referred to.
- Equation (22): since you have defined g^\epsilon already, you can delete the intermediate term “g(t)+\epsilon” from the equation (only leave the second equality)
- Table 2,4,6: please write explicitly how is the convergence rate calculated
Author Response
Please see our response in the file attached.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Here are also some remarks concerning this manuscript:
\begin{enumerate}
\item The question of existence and uniqueness of solution for inverse problem should be highlighted more explicitly;
\item The smoothness properties of solution and the kernel functions of considered TVIDE with two integration terms should also be pointed (if available in literature).
\item There are no theoretical convergence results for suggested numerical algorithms. This question should be briefly discussed at least.
\end{enumerate}
Comments for author File: Comments.pdf
Author Response
Please see our response in the file attached.
Author Response File: Author Response.pdf
Reviewer 3 Report
The manuscript deals with a numerical method for solving the direct and inverse problem of Eq. (1) by employing Shifted Chebyshev polynomials (SCPs) for the spatial variable and forward difference schemes for the time derivative.
In section 2 some results regarding the SCPs and Tikhonov's Regularization Method are recalled. The proposed numerical techniques for the direct problem and the inverse problem are explained in Section 3. Four selected examples are solved in Section 4. Lastly, some conclusions are drawn in Section 5.
The paper is well organized in sections and subsections. The title and the abstract reflect the content and the results.
The equations are clearly presented and they can be easily followed. However, some of them have to be put in more rigorous mathematical and accurate form (for example, Eq. (8)).
An error analysis is missing and should be added.
An extensive editing of English language is necessary.
Author Response
Please see our response in the file attached.
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
The manuscript has been revised according to the comments of the reviewer.
The error analysis requested will be included in a sequel paper.
