Computable solution of fractional kinetic equations using Mathieu-type series

Dumitru Baleanu,Owais Khan,Nabiullah Khan,Kottakkaran Sooppy Nisar

doi:10.1186/s13662-019-2167-4

Abstract

The Mathieu series appeared in the study of elasticity of solid bodies in the work of Émile Leonard Mathieu. Since then numerous authors have studied various problems arising from the Mathieu series in several diverse ways. In this line, our aim is to study the solution of fractional kinetic equations involving generalized Mathieu-type series. The generality of this series will help us to deduce results related to a fractional kinetic equation involving another form of Mathieu series. To obtain the solution, we use the Laplace transform technique. Besides, a graphical representation is given to observe the behavior of the obtained solutions.

Highlights

Khan et al Advances in Difference EquationsIn [14], Haubold and Mathai explored the kinetic equation describing the rate of change of detraction, production, and reaction, which is given as dN dt = p(Nt) – d(Nt),
Introduction and preliminariesFractional calculus (FC) is a useful mathematical tool to study fractional order integrals and derivatives
The special functions and their applications appear in the solutions of fractional integral and differential equations and are related to comprehensive problems in the several areas of mathematics and mathematical physics

Summary

Khan et al Advances in Difference Equations

In [14], Haubold and Mathai explored the kinetic equation describing the rate of change of detraction, production, and reaction, which is given as dN dt = p(Nt) – d(Nt),. A fractional generalization of (3) was given by Haubold and Mathai [14] in the following form: N(t) – N0 = –cν 0D–t ν N(t),. Saxena et al [26] investigated solutions of three generalized forms of (4) in terms of the following generalized Mittag-Leffler function (see, e.g., [40]):. Sexena et al [28] investigated the following generalized fractional kinetic equation:. Saxena and Kalla [25] investigated the following fractional kinetic equation: N(t) – N0f (t) = –cν 0D–t νN(t) ν, c ∈ R+ ,. The Laplace convolution of functions f (t) and g(t) is given by the following integral:.

We find

Laplace transform and applying the relation

Conclusion