Cauchy Problems For Some Classes Of Fractional Differential Equations

Fractional differential equations(FDEs)have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics,dynamical systems,physics,chemistry and engineering systems.Thus,great attention has been given to finding numerical/exact/approximate solutions of FDEs.Our concern in this paper is on analytical solutions and approximate analytical solutions for FDEs.In this paper,we study the operator method and integral transform method for solving fractional differential equations,and apply the DGJ method to deal with some non-linear fractional differential equations.This paper is divided into five chapters:In chapter 1,we mainly introduce the historical development and application of fractional calculus,simultaneously review a number of Chinese and foreign scholars'study result.In chapter 2,we give several operator solutions for fractional partial differential equations and use the operator method to solve the problems as follows:Where,DLt represents Laguerre derivative,WDt? represents Weyl derivative,CDt? rep-resents Caputo derivative,0<?? 1.In chapter 3,we use the integral transform method to solve a class of fractional reaction diffusion equation:Where(?)?1,?2 is a constant that not all equals to 0;1<?1?2,0<?2?1,0???1;?>0,p>0 is a known constant;g(x,t),fi??(R)(i=1,2,3)is a known equation.In chapter 4,the basic principles of DGJ methods is inroduced comprehensively,meanwhile we use it to solve several kinds of nonlinear fractional differential equations:1.Where N(u)is the nonlinear term,f(x,t)is a given function,1<??2.2.Where r>0 is a constant,N(u)is the nonlinear function of u,f(x,t)is a given function.3.CD?u(t)= u1/3(t),0<t?0.5.(0.0.20)Where 0<??1.Finally,concluding remarks are given in chapter 5.