Approximate Analytical Solutions Of Several Fractional Differential Equations

In the past several decades,the fractional derivative has gained considerable attention of mathematicians and physicists.There are many kinds of definitions on the fractional derivatives.The most used ones are: Riemann-Liouville derivative,Caputo derivative,Jumare's derivative and Conformable derivative,and so on.With the development of fractional derivatives,many kinds of interdisciplinary problems can be modeled in fractional differential equations with the help of fractional derivatives in many fields of physical and engineering.Such as heat and optical system,materials science and mechanics and material processing and signal system,system identification,cybernetics and intelligent robot,etc.However,it is rather difficult for us to find the exact solutions of fractional differential equations therefore approximate techniques have to be used.Many powerful techniques have been used to solve linear and nonlinear fractional differential equations.These powerful techniques include the Homotopy Analysis Method(HAM),Homotopy Perturbation Method(HPM),Variational Iteration Method(VIM)and Adomian Decomposition Method(ADM),Finite element method,Finite difference method,Spectral method,Liner Multistep method and Wavelets method,etc.All of these methods have their own advantages and disadvantages.In this dissertation,we establish several new and powerful methods to solve the fractional differential equations based on fractional Sumudu transform and fractional Elzaki transform.These new approximate methods are successfully used to find the approximate solutions of different types of fractional differential equations.We compare the obtained solutions with ones derived from the other classical methods.The results clearly show that our methods are very efficient and highly accurate for fractional differential equations.In this dissertation,we establish four new methods to solve the fractional equations.1.Fractional homotopy analysis transform method(FHATM).Its advantage of the Fractional homotopy analysis transform method lies in that the approximate solution is controlled by the auxiliary parameter h.In Fractional homotopy analysis transform method,we successfully use fractional Elzaki transform so that the calculation much simpler.We successfully used Fractional homotopy analysis transform method to solve fractional Fornberg-Whitham equation,two-dimensional time fractional diffusion equation,two-dimensional time fractional wave equation and three-dimensional time fractional diffusion equation.2.New iterative transform method(NITM).The advantages of New iterative transform method(NITM)are simple and efficient.The method need less preparative knowledge.The solutions obtained with New iterative transform method(NITM)have higher accuracy than the existing methods.We successfully used New iterative transform method(NITM)to solve fractional Fornberg-Whitham equation,homogeneous and non-homogeneous spacefractional telegraph equations,time fractional Cauchy diffusion equation,nonlinear reaction diffusion equation,two-dimensional time fractional wave equation and three-dimensional time fractional diffusion equation.3.Fractional reduced differential transform method(FRDTM).The advantages of Fractional reduced differential transform method(FRDTM)are simple and efficient.In this dissertation,we obtain the approximate solutions of the fractional Navier-Stokes equations by using Fractional reduced differential transform method(FRDTM).4.The method of solving local fractional differential equation.We combined fractional complex transform(FCT)and classical methods to find the approximate solution of local fractional differential equation.Local fractional heat transfer equation,Local fractional porous media equation and local fractional Bratu-type equation are researched by our proposed method.