SOLVING FRACTIONAL VARIABLE-ORDER DIFFERENTIAL EQUATIONS FOR GENERALIZED CAPUTO-TYPE WITH TIME-VARYING DELAY BY USING THE OPERATION MATRIX METHOD

Abstract

This study introduces novel derivative and integral operators derived from newly formulated generalized Caputo fractional derivatives (GCFDs). A numerical approach isdeveloped utilizing these operators to solve variable-order fractional differential equationswith time-varying delay (VOFDETVD). The approach entails approximating the solutionsof VOFDETVD by employing shifted Legendre polynomials (SLPs) as basis vectors. Toassess the efficiency of the numerical method, tests are conducted on various examples, withdierent values of ϱ specified for the fractional differential operator of the new generalized Caputo. Moreover, the present method is comprehensively evaluated for robustness andeectiveness through comparisons with existing approaches.