Finite Element Approximations For Time-fractional Reaction-diffusion

Time fractional reaction-diffusion equation is derived from the classical reaction-diffusion equation derivative, which use theαderivative replaced by the first timederivative.In this paper, a numerical approximation for a Caputo’s time-fractional diffusionequation with initial and boundary conditions is discussed. The variational method andfinite element method is applied to the fractional partial equations. A different method isproposed for approximating the Caputo’s fractional derivative.In Chapter1, surveys of the history of the theory of fractional calculus are introduced.Furthermore some related knowledge about fractional derivatives are presented.In Chapter2, a system equivalent to the original problem is obtained by using arelationship between Riemann-Liouville fractional derivative and Caputo’s. Then semi-discretization is executed at the time direction. And with the help of Grüwald-Letnikovderivative, a variation equation is deduced by approximating the differential operator withliner operator. And we get the error estimate of theα-norm sense.In Chapter3, there is a isosceles triangle subdivision in the given rectangle region,and we get the high precision of integral identities, which was used as a tool to obtain highprecision error analysis and get the equation to achieve the super approximation on thefinite element space. Interpolated postprocessing operator, then we will get the globalsuperconvergence from the superapproximation.