Finite Element Approximations For Time-Fractional Partial Differential Equations

A numerical approximation for a Caputo's time-fractional diffusion equation withinitial and boundary conditions is discussed in the paper. Fractional differential equa-tions are a type of differential equations in which the definition of classical integer orderderivative is replaced by that of fractional derivative. It is better to model some natu-ral phenomenon and physical processes by fractional differential equations than integerorder equations. The finite element method is applied to fractional partial equations. Adifferent method is proposed for approximating the Caputo's fractional derivative,andthe stability of iterative scheme is proved.First,recent researches of fractional partial differential equations ,as well as histo-ry background,and the problem to be solved are stated.Secondly,a system equivalentto the original problem is obtained by using a relationship between Riemann-Liouvillefractional derivative and Caputo's.After that,semiffdiscretization is executed at the timedirection.With the help of Gru¨wald-Letnikov derivative,a variational equation is de-duced by approximating the differential operator with difference operator.The error isestimated in theα-norm sense.Then,the full-discrete equation is derived,and both errorestimation in theα-norm and 0-norm sense are obtained. The finite element equationform is presented for the practical computation.Then, the corresponding stiffness matrixand algebraic equations are derived.Also the stability about initial values of iterationis proved.Last,two numerical examples and some diagrams are given to illustrate thefeasibility of this method.It is indicated that results coincide with theoretical analysis.