A New Numerical Approach for the Analysis of Variable Fractal and Fractional Order Differential Equations

Abstract

The variable fractional dimensions differential and integral operator overrides the phenomenon of the constant fractional order. This leads to exploring some new ideas in the proposed direction due to its varied applications in the recent era of science and engineering. The present papers deal with the replacement of the constant fractional order by variable fractional order in various fractal-fractional differential equations. An advanced numerical scheme is developed with the help of Lagrange three-point interpolation and further, it is employed for the solution of the proposed differential equations. However, the properties of these new operators are presented in detail. Finally, the error analysis is also conducted for the numerical scheme deployed. The results are validated by the suitable choice of applications to real-life problems. The well- known multi-step-Adams–Bashforth numerical scheme for classical differential equations is recovered when the non-integer order is one.

Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operators

“…In [26], the authors studied the analytic and numerical solution of diferent fractal-fractional differential equations. Jena et al [27] studied the numerical solution of variable fractal-fractional order diferential equations. Te authors of [28] studied the numerical solution of fractal-fractional equations using the Laplace transform method.…”