Difference Schemes For Fractional Partial Differential Equations With Variable Coefficients

Recently, more and more researchers are finding that a variety of impor-tant dynamical exhibit fractional order or diffusion coefficients behavior that may vary with time or space,Such as heat conduction or liquid flow process on porous medium, propagation of seismic waves, etc. We focus on investigat-ing difference schemes for fractional partial differential equations with variable coefficients in this paper,which can be sorted as three part because of the different equations.In the first part,we consider a finite difference scheme for the fractional d-iffusion equation with a variable coefficient. because of the variable coefficient a(x),we can not use the common central difference scheme and the compact difference scheme. Here we consider the half integer point first, then use direct difference method, we can get a second order accuracy approximation for the space derivative. We use the Caputo fractional derivative for the time frac-tional part. The unique solvability of the difference solution is discussed. The stability and convergence of the finite difference scheme are proved using the maximum method.We also derive a ADI finite difference scheme for the two dimension fractional diffusion equation with variable coefficients.In the second part,we consider a finite difference scheme for the variable order fractional diffusion equation with a variable coefficient. For the space derivative we use the same method used in part one and use the Coimbra fractional derivative for the time fractional part. The convergence order of the difference scheme is O(τ+h2).The unique solvability of the difference solution is discussed. The stability and convergence of the finite difference scheme are proved using the maximum method.We also derive a finite difference scheme for the two dimension variable order fractional diffusion equation with variable coefficients.In the third part,we consider a finite difference scheme for the variable order fractional advection diffusion equation with variable coefficients, we use the same method used in part one for the second order space derivative, up-wind difference scheme for the advection term and use the Coimbra fractional derivative for the time fractional part. The convergence order of the difference scheme is O(τ+h).The unique solvability of the difference solution is discussed. The stability and convergence of the finite difference scheme are proved using the maximum method.We also provide numerical examples for each equation to show the effec-tiveness and accuracy of the method.