Analysis of Fractional Order Differential Equation Using Laplace Transform

Exploring the functioning of analytical solutions of differential equations defined by the fractional order operators is one of the hottest topics in the field of research in stability problems. This article analyze the stability of a certain type of fractional order differential equation. Employing Banach contraction mapping principle, existence and uniqueness of solutions are obtained. A sufficient condition to assure the reliability of solving the fractional order differential equation by Laplace Transform method and Generalized Laplace Transform are presented. Exponential stability results for the solution is discussed using Gronwall inequality. Application of the generalized Gronwall inequality to fractional order differential equation under investigation yields new sufficient condition for stability of fractional order differential equation. Applicability of the theoretical results on stability are demonstrated with an example. In the analysis of fractional order differential equation, Laplace transform is proved to be a valid tool under certain conditions.