On a Boundary-Initial Value Problem for Fractional Differential Equation with Sequential Caputo derivatives

TL;DR

This paper solves a fractional differential equation with sequential Caputo derivatives by deriving an exact solution using bivariate Mittag-Leffler functions and developing a numerical scheme with finite elements.

Contribution

It introduces an exact analytic solution for the problem and a numerical method based on a sequential reformulation and L1-finite element approach.

Findings

Exact solution expressed via bivariate Mittag-Leffler functions

Properties of the bivariate Mittag-Leffler function are established

A numerical scheme using L1-finite element method is developed

Abstract

In this paper, we investigate a fractional differential equation involving sequential Caputo derivatives, motivated by recent research on fractional models with multiple memory effects. Using techniques inspired by earlier works on sequential fractional operators, we derive the exact analytic solution of the problem in terms of the bivariate Mittag-Leffler function. Additionally, several useful properties of the bivariate Mittag-Leffler function are formulated to support the solution construction. Furthermore, we develop a numerical scheme using a sequential reformulation and the L1-finite element method.