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

• Published in: Computer modeling in engineering & sciences Vol. 134; no. 2; pp. 941 - 955
• Main Author: Shah, Kamal
• Format: Journal Article
• Language: English
• Published: Henderson Tech Science Press 2023
• Subjects: Algebra; Algorithms; Approximation; Calculus; Differential equations; Differentiation; Fractional calculus; Graphical representations; Initial conditions; Mathematical analysis; Methods; Numerical analysis; Polynomials; Researchers
• ISSN: 1526-1506; 1526-1492; 1526-1506
• DOI: 10.32604/cmes.2022.021483

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.