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

Uttam Ghosh^1, , Susmita Sarkar² and Shantanu Das^{3, 4}

¹Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India

²Department of Applied Mathematics, University of Calcutta, Kolkata, India

³Reactor Control System Design Section Bhabha Atomic Research Centre, Mumbai, India

⁴Department of Physics, Jadavpur University Kolkata, West Bengal, India

Cite this paper:
Uttam Ghosh, Susmita Sarkar and Shantanu Das. Solutions of Linear Fractional non-Homogeneous Differential Equations with Jumarie Fractional Derivative and Evaluation of Particular Integrals. American Journal of Mathematical Analysis. 2015; 3(3):54-64. doi: 10.12691/ajma-3-3-1

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. The short cut rules, that are developed here in this paper to replace the operator D^a or operator D^2a as were used in classical calculus, gives ease in evaluating particular integrals. Therefore this method proposed by us is useful and advantageous 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.

Keywords:
Mittag-Leffler functions non-homogeneous fractional differential equations modified riemann-liouville definition

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