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Numerical Solutions of Fractional Variable Order Differential Equations via Using Shifted Legendre Polynomials

Kamal Shah1,2, Hafsa Naz2, Thabet Abdeljawad1,3,*, Aziz Khan1, Manar A. Alqudah4

1 Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia
2 Department of Mathematics, University of Malakand, Chakdara Dir (L), Khyber Pakhtunkhwa, 18000, Pakistan
3 Department of Medical Research, China Medical University, Taichung, 40402, Taiwan
4 Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah bint Abdurahman University, Riyadh, 11671, Saudi Arabia

* Corresponding Author: Thabet Abdeljawad. Email:

(This article belongs to this Special Issue: Advanced Numerical Methods for Fractional Differential Equations)

Computer Modeling in Engineering & Sciences 2023, 134(2), 941-955.


In this manuscript, an algorithm for the computation of numerical solutions to some variable order fractional differential equations (FDEs) subject to the boundary and initial conditions is developed. We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices. Further, operational matrices are constructed using variable order differentiation and integration. We are finding the operational matrices of variable order differentiation and integration by omitting the discretization of data. With the help of aforesaid matrices, considered FDEs are converted to algebraic equations of Sylvester type. Finally, the algebraic equations we get are solved with the help of mathematical software like Matlab or Mathematica to compute numerical solutions. Some examples are given to check the proposed method’s accuracy and graphical representations. Exact and numerical solutions are also compared in the paper for some examples. The efficiency of the method can be enhanced further by increasing the scale level.


Cite This Article

Shah, K., Naz, H., Abdeljawad, T., Khan, A., Alqudah, M. A. (2023). Numerical Solutions of Fractional Variable Order Differential Equations via Using Shifted Legendre Polynomials. CMES-Computer Modeling in Engineering & Sciences, 134(2), 941–955.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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