Numerical Approximation Of The Fractional Reaction-diffusion Equation

In recent years, fractional calculation plays a more and more importantrole in various fields of science, especially in engineering, physics, finance, andhydrology. Differential equations with fractional order have recently proved tobe valuable tools to modeling of many physical phenomena. Compared withinteger-order models, the most significant advantage of the fractional ordermodels is based on important fundamental practical background and physicalinterpretation. The fractional diffusion-reaction equation has been recentlytreated by many authors. However, the numerical methods and analysis ofthe stability and convergence for the fractional reaction-diffusion equation arequite limited. Especially it is difficult to deal with the case in which thereaction term is nonlinear.In this paper, we discuss numerical approximation of the fractional reaction-diffusion equation in the following three aspects: (1) An implicit differencescheme for the time-fractional reaction-diffusion equation ;(2) An explicit dif-ference scheme and an implicit difference scheme for the space-time fractionalreaction-diffusion equation; (3) Numerical approximation for the linear andnonlinear space-time fractional reaction-diffusion equations by using Adomiandecomposition method. Firstly, the time-fractional order reaction-diffusionequation is considered, in which the time fractional derivativeα(0<α≤1) isdefined as the Caputo fractional derivative. In order to solve this equation, weuse effective difference scheme to approximate the time fractional derivativeand have an effective implicit difference scheme. The detailed analysis of thestability and convergence of implicit difference schemes are derived by applyingmathematical induction and characteristics of fractional dispersed coefficients. Secondly, the space-time fractional reaction-diffusion equation is considered,in which the time fractional derivativeα(0<α≤1) is defined as the Caputofractional derivative and the space fractional derivativeβ(1<β≤2) is definedas the Riemman-Liouville fractional derivative. In order to solve this equation,we use an effective difference scheme to approximate the time fractional deriv-ative and shift Griinwald formula to approximate the space fractional deriva-tive. The Effective explicit difference scheme and implicit difference scheme aregiven. The detailed analysis of the stability and convergence of both differenceschemes are derived by applying mathematical induction and characteristics offractional dispersed coefficients. Finally, the linear and nonlinear space-timefractional reaction-diffusion equations are considered, in which the time frac-tional derivativeα(0<α≤1) and the space fractional derivativeβ(1<β≤2)are defined as the Caputo fractional derivatives. The Adomian decompositionmethod can perfectly deal with nonlinear reaction term, and provide highlyaccurate numerical solutions without discretization for the problem. The over-all errors can be reduced to a much smaller extent by adding new terms of thedecomposition series. We can obtain approximate solutions of the linear andnonlinear space-time fractional reaction-diffusion equations by using Adomiandecomposition method. Numerical examples are presented in each chapter,which verify the efficiency of the above numerical methods. The techniquescan also be applied to deal with other types of fractional order differentialequations.