Numerical Solving For The Non-linear Fractional Reaction-subdiffusion Equations

Fractional integrals(FIs) and fractional derivatives(FDs) has drawn much attentiondue to its wide application in many science and engineering fields. They arevery useful mathematic tools in the memory of many kinds of materials, anomalousdiffusion, signal processing, controllers theory, vibration control of viscoelastic systemand pliable structure objects, fractional biological neurons, advection-diffusionin porous or fractured medium, chaotic, etc. Comparing to the classical differentialintegrals model, it is more adequate to simulate the memory and inheritabilityof the materials. For inter-order differential equations, numerical arithmetics aremature relatively, but the investigation of numerical methods for fractional differentialequations is underway. Particularly, the theoretical results of numericalmethods for fractional differential equations are quite limited.In this paper, we considered the numerical method for solving fractional reactionsubdiffusion equation (NFR-SubDE). We recall some basic definitions andproperties of fractional calculus. An implicit difference scheme was given for fractionalreaction subdiffusion equation with initial and boundary conditions. Usingthe new energy method, the stability and convergence of this method are proved.From the theoretical results can be seen that the accuracy of our numerical methodis higher than which is constructed by Zhuang and Liu (P. Zhuang, F. Liu, etal. Stability and convergence of an implicit numerical method for the nonlinearfractional reaction-subdiffusion process. IMA J. Applied Mathematics, 2009, 74:645-667). Some numerical examples are presented to show the the effectiveness ofthis method. This technique can also be applied to solve other types of fractionalpartial differential equations and higher dimensional problems.