TY  - EJOU
AU  - Shah, Kamal 
AU  - Naz, Hafsa 
AU  - Abdeljawad, Thabet 
AU  - Khan, Aziz 
AU  - Alqudah, Manar A. 

TI  - Numerical Solutions of Fractional Variable Order Differential Equations via Using Shifted Legendre Polynomials
T2  - Computer Modeling in Engineering \& Sciences

PY  - 2023
VL  - 134
IS  - 2
SN  - 1526-1506

AB  - 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.
KW  - Operational matrices; shifted legendre polynomials; FDEs; variable order

DO  - 10.32604/cmes.2022.021483