Symmetry Analysis of Nonlinear Fractional Partial Differential Equations

Abstract

Fractional calculus is a branch of mathematics that deals with real number or complex number powers of the differential operator and integral operator. Although the idea of fractional calculus was born more than 300 years ago, serious efforts have been dedicated to its study recently. Fractional differential equations (FDEs) are gen- eralization of the differential equations of integer order, studied through the theory of fractional calculus. Lie symmetry method is a powerful technique for solving integer order differential equations. In this thesis, its various extensions are proposed for the symmetry analysis of nonlinear systems of FDEs. The aim of this thesis is to extend the symmetry approach in order to apply them to a wider class of FDEs including time fractional nonlinear systems, space-time fractional nonlinear systems, higher dimensional nonlinear systems, and variable coefficient nonlinear systems. The thesis consists of six chapters comprising various novel extensions and appli- cations of the symmetry method. Chapter 1 provides the history of fractional calculus, basic definitions, and properties of the Riemann-Liouville fractional operators used in this study. The main features, background and methodology of the Lie classical method by Sophus Lie are also discussed in the introductory chapter. Chapter 2 deals with the extension of Lie symmetry method for studying i ii time fractional systems of partial differential equations (PDEs). The prolongation for- mulae given in a recent paper [86] for symmetry analysis of time fractional systems are proved incomplete and the correct formulae are suggested in this chapter. The prolon- gation operators are derived for time fractional systems having two independent and an arbitrary number of dependent variables. Also, the technique to investigate nonlinear self-adjointness and conservation laws is extended for time fractional systems of PDEs.