The Methods For Solving Differential Equations Of Fractional Order

This text is a summarized article. We introduce several kinds of methods for solving differential equations of fractional order.There are eight parts in this text.The first part is an introduction.The second part,consider the fractional Riccati differential equationthe initial conditionswhere A(t),B(t),C(t) are given functions,c_j(j= 0,1,…, m-1) are arbitrary constants,αis a parameter describing the order of the fractional derivative. In the case ofα= 1 ,the fractional Riccati differential equation reduces to the classical Riccati differential equation, the importanceof this equation usually arises in the optimal control problems. The valueα=(?) is especially popular.This is because in classical fractional calculus, many of the model equationsdeveloped used these particular orders of derivatives[11]. In modern applications much more general values of the orderαappear in the equations and therefore one needs to considernumerical and analytical methods to solve differential equations of arbitrary order. The Adomian decomposition method is very powerful and efficient in finding analytical as well as numerical solutions for wide classes of linear and nonlinear fractional differential equations.It provides more realistic series solutions that converge very rapidly in real physical problems.The solution takes the form of a convergent series with easily computable components. The third part,consider the linear fractional differential equation of the formthe initial conditionswhere C_j, j = 0,1,…, m-1 ,are arbitrary constants and u(t) is assumed to be a causal function of time. The fractional derivatives is considered in the Caputo sense. We refer to the equation as to the composite fractional relaxation and to the composite fractional oscillation equation in the cases {0<α≤1,m=1} and (1 <α≤2,m=2),respectively. The composite fractional relaxation equation corresponds to the Basset Problem[14].It represents a classical problem in fluid dynamics where the unsteady motion of a particle accelerates in a viscous fluid due to the gravity force. The composite fractional oscillation equation describes the motion of a rigid plate immersed in a Newtonian fluid[15]. Apply the variational iteration method,obtain the following iteration formula:for m = 1,for m = 2,The variational iteration method and the Adomian decomposition method can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations.They provide more realistic series solutions that converge very rapidly in real physical problems.The fourth part,the homotopy analysis method is a generic method,it is valid to solve many kinds of differential equations. Applying the homotopy analysis method to solve fractionalpartial differential equations is similar with solving integral differential equtions. Construct deformation equation and obtain the series solutions. The homotopy analysis method logically contains the Adomian decomposition method. The fifth part,the generalized differential transform method is based on the two-dimensional differential transform method,generalized Taylor's formula and Caputo fractionalderivative.It is very effective and convenient for solving linear partial differential equationsof fractional order. Consider a function of two variables u(x,y) ,and suppose that it can be represented as a product of two single variable functions: u(x,y) = f(x)g(y).On the basis of the properties of generalized two-dimensional differential transform [18,19],the function u(x,y) can be represented as:where 0<α,β≤1,U_α,β(k,h) = F_α(k)G_β(h) is called the spectrum of u(x,y).The generalized two-dimensional differential transform of the function u(x, y) is as follows:where (?) represents the original function,U_α,β(k, h) stands for the rtansformed function. And there are three important theorems.The sixth part,the space-time fractional advection dispersion equation is obtained from the standard advection dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of orderα∈(0,1], and the first-order and second-order space derivatives by the Riemann-Liouville fractional derivatives of orderβ∈(0,1] and of orderγ∈(1,2],respectively. We prove that the IDM is unconditionally stable and convergent ,nevertheless,the EDM is conditionally stable and convergent.The techniques can be applied to solve fractional partial differential equations.The seventh part,linear multistep method is an important method for solving differential equations of integer order. Applying linear multistep method to solve differential equations of fractional order,by using an expansion of the local truncation error it is possible to unify,in a some sense,the way in which "fractional" and "classical" methods.The eighth part,the modified homotopy perturbation method does not require a small parameterin an equation. It may eliminate localization of traditional perturbation method,and it is very effective and convenient for solving nonlinear differential equations of fractional order.A homotopy with an imbedding parameter is constructed and series solutions are obtained.