Solutions of Linear Fractional non-Homogeneous Differential Equations with Jumarie Fractional Derivative and Evaluation of Particular Integrals

TL;DR

This paper presents a method for solving linear non-homogeneous fractional differential equations with Jumarie derivatives, providing solutions in terms of Mittag-Leffler functions and extending classical methods to fractional systems.

Contribution

It introduces a novel approach to find particular integrals of non-homogeneous FDEs with Jumarie derivatives, extending previous homogeneous solutions and demonstrating applicability to physical systems.

Findings

Solutions expressed via Mittag-Leffler functions, fractional sine and cosine functions.

Method successfully applied to fractional damped forced differential equations.

Extended classical solution techniques to fractional differential equations.

Abstract

In this paper we describe a method to solve the linear non-homogeneous fractional differential equations (FDE), composed with Jumarie type Fractional Derivative, and describe this method developed by us, to find out Particular Integrals, for several types of forcing functions. The solutions are obtained in terms of Mittag-Leffler functions, fractional sine and cosine functions. We have used our earlier developed method of finding solution to homogeneous FDE composed via Jumarie fractional derivative, and extended this to non-homogeneous FDE. We have demonstrated these developed methods with few examples of FDE, and also applied in fractional damped forced differential equation. This method proposed by us is useful as it is having conjugation with the classical methods of solving non-homogeneous linear differential equations, and also useful in understanding physical systems described by FDE.