A Tau Approach for Solving Time-Fractional Heat Equation Based on the Shifted Sixth-Kind Chebyshev Polynomials

Round 1

Reviewer 1 Report

Title: Infallible Tau Approach for Solving Time-Fractional Heat Equation Based on the Shifted Sixth-Kind Chebyshev Polynomials

 

Manuscript ID: symmetry-2231322

 

Dear Editor,

 

In this article, the Authors present the spectral tau method based on the sixth-kind Chebyshev polynomials to solve the time-fractional heat equation governed by nonlocal conditions.  Firstly, the approximate solution accepts a product of the Chebyshev polynomials, and the tau method can obtain an algebraic system. Moreover, an error analysis is added.

Section 2,presents some basic definitions and properties about fractional calculus, fractional differential equations and 6^th Chebyshev polynomials. Then, in Section 3, the Tau method is applied to the time-fractional heat equation with nonlocal conditions. 

 

Further, this article is well-written, clear and well-organized.  Therefore, I advise to publish this article in Symmetry after some minor refinement. I think those refinements are required:

 

 

1-       I think that “Infallible”  worlds can be deleted. That is; title of the article can be  “A Tau approach ……”

 

2-      We know that Chebyshev polynomials arise many more branches in applied mathematics.  So, Chebyshev polynomials are used to find numerical solutions more and more problems. Authors put some studies but it needs to be revised. Therefore, the following articles should be added in the Reference list. Moreover, some studies can be added for the Tau method in the References list.

 

 K. Maleknejad, S. Sohrabi, H. Derili, A new computational method for solution of non-linear Volterra–Fredholm integro-differential equations, International Journal of Computer Mathematics, 87(2) (2010) 327-338.

 

      A spectral collocation matrixmethod for solving linear Fredholm integro-differential–difference equations, Computational and Applied Mathematics, 10, 218.

 

 

J. C. Mason, D. C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC, New York (2003).

 

Operational Tau approximation for a general class fractional integro-differential equations, Comp. Appl. Math. 30(3) (2011) 655-674.

 

An operational matrix method for solving Lane-Emden equations arising in astrophysics, Mathematical Methods in Applied Sciences 37 (2014) 2227-2235.

 

Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Appl. Math. Comp. 188 (2007) 417-425.

 

 An operational matrix method to solve linear Fredholm–Volterra integro‑differential equations with piecewise intervals, Mathematical Sciences, 15, 189-197.

 

Solution for the system of Lane-Emden type equations using Chebyshev polynomials, Mathematics, 6(10), 181.

 

3-      “Tau method” and “PDEs” should be added in Keywords.

4-      In Figs., the errors could be clearer. A Contour Graph of absolute errors should be added for examples.

5-      The reason (advantage) of using the 6-th kind of Chebyshev polynomials is explained in the Conclusion section.

 

 

Best regards.

 

Author Response

Please see the attached file: Response 1.pdf

Author Response File: Author Response.pdf

Reviewer 2 Report

Report on the paper

Infallible Tau Approach for Solving Time-Fractional Heat Equation Based on the Shifted Sixth-Kind Chebyshev Polynomials

by Esraa Magdy Abdelghany, Waleed Mohamed Abd-Elhameed, Galal Mahrous Moatimid, Youssri Hassan Youssri, and Ahmed Gamal Atta

The authors' aim is 'to treat numerically the time-fractional differential equation using the shifted sixth-kind Chebyshev polynomials'. 

The basic idea is 'to transform the problem governed by the underlying conditions into a set of linear algebraic equations that can be solved by means of an appropriate numerical scheme'.

The calculations seem correct to me and the results sound well. Some examples are considered, improving someone obtained using different methods.

The content is valuable and I recommend the publication of the work.

 

Author Response

All authors are thanking the reviewer for the time exerted while revising the manuscript.