Some Methods For Solving Fractional Partial Differential Equations

In the field of mathematical physics,the study of the fractional partial differential equations can enlarge the research area and provide some mathematical basis and new methods for fractional calculus.At the same time,the theory and methods of the fractional partial differential equations can provide many effective tool to solve the problems which appear in the quantum mechanics,biological tissue,chaos and turbulence,viscoelastic mechanics and non-Newtonian mechanics,finance,chemistry,mechanics and so on.Therefore,in recent years,the seeking the solutions of fractional partial differential equations and the practical applications of these solutions have become a frontier research area of mathematical physics.In this paper,based on the modified Riemann-Liouville derivative and its properties,the exact solutions of fractional modified Camassa-Holm equation,KdV-mKdV equation,mBBM equation,Klein-Gordon equation,Vakhnenko-Parkes equation and Sawada-Kotera equation are constructed by using the quasi-method,the auxiliary equation method and the F-expansion method.The first chapter is the introduction,which mainly introduces the significance of the research of fractional partial differential equations.The backgrounds,developments and trends to study of fractional partial differential equation at home and abroad are also introduced.Finally introduces the main work of this thesis.In Chapter 2,we first introduce the basic steps of ansatz-method,and apply it to find the exact solutions of the fractional modified Camassa-Holm equation and fractional KdV-mKdV equation.The obtained exact solutions include the bright soliton solutions and singular soliton solutions,etc.In the third chapter,the main procedures of the auxiliary equation method are introduced.And then,the auxiliary equation method is applied to solve the fractional order mBBM equation and the fractional order Klein-Gordon equation.The hyperbolic,triangular and elliptic function type exact solutions of the equations are obtained.In Chapter 4,the fractional Vakhnenko-Parkes equation and the fifth-order time fractional Sawada-Kotera equation are solved by the F-expansion method,and many Jacobi elliptic function solutions of these equations are presented.The fifth chapter is the conclusions.