Operational Matrix for Solving Variable Order Generalized Caputo Fractional Derivatives

Abstract

In this study, new generalized derivative and integral operators are introduced, stemming from the newly developed new generalized Caputo variable order fractional derivatives (NGCVFDs). Utilizing these operators, a numerical method is devised to address variable order fractional differential equations (VOFDEs). The solutions of VOFDEs are approximated using shifted Legendre polynomials (SLPs) as basis vectors, and the derivative operational matrix of SLPs is extended to a generalized derivative operational matrix in the context of NGCVFDs. The efficiency of the numerical method is assessed through various test examples. Additionally, the outcomes of the proposed method are compared with existing methodologies in the literature. The variable-order fractional differential operators of the generalized Caputo is categorized into three types in this paper:

 (i) Different values in ρ and Fractional variable order  parameters, (ii) Different values in fractional parameter whilst ρ parameters and Fractional variables orders are constant, and (iii) Different values in Fractional variables orders parameters controlled fraction and ρ parameters. The example of  numerical methods show theoretical interpretation and prove effectiveness of suggested technique.