DGJ Method For Some Classes Of Fractional Differential Equations

Fractional calculus is a mathematical theory which studys function with arbitrary order derivative and integral.It is the generalization of integer order calculus.In recent years,fractional calculus theory,especially the theory of fractional differential equations has been increasingly used to simulate appeared many complex phenomena in the natural science and social sciences,especially in nonlinear phenomena.However,the fractional differential equation models summarized from practical problems are often nonlinear,with variable coefficient.Usually,these problems could not find the exact analytical solution,only to get the numerical solution,so,it must be solved by an approximate method to get the approximate analytical solution.These methods have their advantages and disadvantages,currently used include Adomian decomposition method,the variational iteration method(VIM),Operator method,multiple integral transform method,the homotopy perturbation method(HPM),the homotopy analysis method(HAM),etc.By using these methods,we can obtain a sufficiently accurate approximate analytical solution,which brings a lot of help for theoretical analysis and practical application.This article is divided into seven chapters.Chapter 1 of this paper,mainly introduces the background of the paper and the results obtained.In Chapter 2,we present some of the basic knowledge of fractional order calculus and the basic principle of DGJ method.In Chapter 3,we use DGJ method to study the following questions:,where,D?tu denotes Caputo fractional derivative of ?.The research of this study generalized the Hemeda's result.In Chapter 4,we use DGJ method to study the following questions:,here,the initial condition is u(x,y,0)= h1(x,y),v(x,y,0)= h2(x,y),w(x,y,0)=h3(x,y),D?is Caputo fractional derivative.In Chapter 5,we combine DGJ method and E transformation,propose a new method for solving the following problem:,In Chapter 6,we use DGJ method to solve a class of nonlinear integral equation.Theoretically,the convergence is studied in detail,and make a numerical experiments to verify its accuracy.In Chapter 7,as the last section of paper,we make a summary and make a prospect for the future development.