Dear Mohammed K A Kaabar,

HAM and HPM coupled with the integral transformations like Laplace, Fourier, etc. Especially if you apply them to linear/nonlinear problems by considering the fractional operators without a singular kernel, I hope you will obtain great results.

Thank you Luis - Felipe Velázquez - León very much for your answers and suggestions!! I greatly appreciate that!!

HAM and HPM you can not have better nowadays in mathematics. Please learn it. Its the future of mathematics.

Thank you Ndolane Sene very much for your suggestions!!

Dear Mohammed K A Kaabar,

HAM and HPM coupled with the integral transformations like Laplace, Fourier, etc. Especially if you apply them to linear/nonlinear problems by considering the fractional operators without a singular kernel, I hope you will obtain great results.

Thank you very much Mehmet Yavuz for your suggestion!! I greatly appreciate it!

Reduce differential transform method (RDTM) is better

Thank you very much Hassan Kamil Jassim for your suggestion!!

Adams Bashforth Moulton method provides reliable results for system of ordinary differential equations of fractional order.

Thank you Akinyemi Samuel for your suggestion!!

Dear Mohammed,

I recommend you to read my publications on this subject:

Katsikadelis J.T. (2009). Numerical Solution of Multi-term Fractional Differential Equations, ZAMM, Zeitschrift für Angewandte Mathematik und Mechanik, 89, (7), 593 – 608 (2009) / DOI 10.1002/zamm.200900252.

1. Katsikadelis, J.T. (2011). The BEM for Numerical Solution of Partial Fractional Differential Equations, Computers and Mathematics with Applications, 62, pp. 891–901,doi:10.1016/j.camwa.2011.04.001.

They might be usefully.

With kind regards

John Katsikadelis

You might be interested in

Conference Paper Time-domain solutions for vibration systems with fading memory

Thank you Nils Wagner very much for sharing your conference paper here!! I greatly appreciate that!

Thank you Dr. John T. Katsikadelis very much for your suggested publications!! I greatly appreciate that!!

I can only advise you to read this thesis.

Thank you Mansouri Aouatef very much for your suggested publication!!

Dear Mohammed K A Kaabar

I recommend you to read my publication on this subject:

https://www.researchgate.net/publication/334388517_On_Psi-Laplace_transform_method_and_its_applications_to_Psi-fractional_differential_equations

Thank you very much Mr. Hafiz Muhammad Fahad for sharing your research with me!!

Dear all professors and researchers,

I would like to share with you my new research article (preprint) in the area of applied mathematics, fractional differential equations, and numerical methods. This is a link to my new research article (preprint) titled "Novel Methods for solving the Conformable Wave Equation": https://hal.archives-ouvertes.fr/hal-02267015

Please recommend my research article and cite it if it is possible. I greatly appreciate all your brilliant efforts!!

Thank you very much all your helpful comments and answers, Best Regards, Mohammed K A Kaabar

Thank you very much Dr. Maged Gumaan Bin-Saad for your answer!! I will read about this method!!

I suggest you to read the article

A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions

Rahimkhani, Parisa, Ordokhani, YadollahJournal:Numerical Methods for Partial Differential Equations

Thank you, Dr. Maged Gumaan Bin-Saad for suggesting this article! that's a great help to my research!

SPSS

HAM

HPM

ADM

Adam's Predictor-Corrector Method

Thank you Ramashis Banerjee for your answer!! I greatly appreciate it!

I suggest the methods contained in the following books

[1] I. Podlubny, Fractional Differential Equations

[2] A. A. Kilbas, H. M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations.

Moreover, if the orders are rational or the equations can be rewritten as sequential differential equations of fractional order, I suggest to read Chap. 7 of [2] with particular attention. I used this approach to solve a set of coupled fractional differential equations and to find some useful analytical solution of noisy oscillator with rational order element in my paper Article Statistical correlation of fractional oscillator response by...

Thank you Dr. Francesco Paolo Pinnola for your answer! I greatly appreciate it! That's very helpful for my research!!

Dear Mohammed,

I know that my answer might be a bit too late, but I hope you would be still benefited from it. I hope also I am not repeating somebody's else answers.

For powerful and accurate analytical and numerical methods for solving FDEs I would suggest the articles of Prof. Garrapa from Italy and Profs. Diethelm and Luchko from Germany. Here are some examples:

Garrappa R.: Numerical Solution of Fractional Differential Equations: a Survey and a Software Tutorial, Mathematics 2018, 6(2)

Kai Diethelm. The analysis of fractional differential equations, volume

2004 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2010.

Kai Diethelm, Neville J. Ford, and Alan D. Freed. Detailed error analysis

for a fractional Adams method. Numer. Algorithms, 36(1):31{52,2004.

Kai Diethelm and Yuri Luchko. Numerical solution of linear multiterm

initial value problems of fractional order. J. Comput. Anal. Appl.,

6(3):243{263, 2004.

Prof. Garrappa has also developed Matlab routines for that, here is an example:

https://de.mathworks.com/matlabcentral/fileexchange/66603-solving-multiterm-fractional-differential-equations-fde

best wishes and regards

Tareq