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. The proposed methods are applied for the symmetry analysis and derivation of conserved vectors of five time fractional nonlinear systems of PDEs including Ito system, Burg- ers system, coupled KdV system, Hirota-Satsuma coupled KdV system, coupled Hirota equations. As a result, these systems are reduced into fractional nonlinear systems of ordinary differential equations (ODEs). Chapter 3 is devoted to extending the Lie group method and Noether operators for computing Lie symmetries and conserved vectors of space-time fractional PDEs. The complete Lie group classification is performed and concept of nonlinear self-adjointness is extended for space-time fractional PDEs. Two space-time fractional nonlinear PDEs namely Gilson-Pickering equation and generalized KdV equation are studied for their Lie symmetries resulting in their reductions into fractional nonlinear ODEs in the Erd ́ e lyi- Kober operators. In addition, the conservation laws for both the fractional partial differ- ential equations (FPDEs) are obtained successfully. Chapter 4 is concerned with the investigation of space-time fractional nonlinear systems of PDEs for their Lie symmetry analysis. For this purpose, the symmetry method is proposed for space-time fractional systems of PDEs by derivation of the required pro- longations. Using the extended prolongation operators, the group infinitesimals for five space-time fractional nonlinear systems are successfully calculated. The resulting group iii invariant solutions are used to obtain their symmetry reductions into nonlinear systems of fractional ordinary differential equations (FODEs). The discussed fractional nonlinear systems of PDEs are as follows: space-time fractional Ito system, space-time fractional coupled Burgers equations, space-time fractional coupled KdV system, space-time frac- tional Hirota-Satsuma coupled KdV system, space-time fractional coupled Hirota equa- tions. In chapter 5 , a generalized symmetry approach is proposed for systems of FDEs having an arbitrary number of independent as well as dependent variables. The derivation of the prolongation operators is discussed for generalized fractional order sys- tems. The symmetry analysis of higher dimensional systems can be discussed using the suggested approach. The efficiency of the presented symmetry method is proved by its application to five higher dimensional nonlinear systems namely (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov (ANNV) system, (3+1)-dimensional Burgers sys- tem, (3+1)-dimensional Navier-Stokes system, (3+1)-dimensional fractional incompress- ible non-hydrostatic Boussinesq system, fractional (3+1)-dimensional incompressible non- hydrostatic Boussinesq system with viscosity. Their symmetries and symmetry reductions into lower dimensional nonlinear fractional order systems are deduced in terms of ex- tended Erd ́ e lyi-Kober fractional operators systematically. In chapter 6 , certain variable coefficient fractional nonlinear PDEs are inves- tigated using the Lie group analysis. The complete group classification of fractional nonlinear variable coefficient PDEs is demonstrated for some single time fractional PDEs as well as systems of time fractional PDEs with variable coefficients. The studied variable coefficient fractional nonlinear PDEs include KdV-Burger-Kuramoto equation and gen- eralized seventh order KdV equation. The considered time fractional nonlinear systems of PDEs with variable coefficients are coupled Boussinesq system, coupled KdV system and Hirota-Satsuma coupled KdV system. After computing their group infinitesimals, the optimal sets of inequivalent one dimensional subalgebras are calculated. Finally, for each component of the optimal set, the similarity reductions of the considered variable coefficient fractional PDEs are obtained successfully.