Numerical Solving For Two Classes Of Variable-order Fractional Order Diffusion Equations

Fractional order differential equation is a generalization of the integer order differentialequation, it can very effectively describe memory and genetic properties of material. Inrecent decades, the fractional order differential equations are widely used in many scientificand engineering fields, such as physics, finance, and hydrology, etc. Many scholars haveconstructed lots of the analytical solution of the fractional order differential equation throughusing special function. But many special functions are very complex and hard to calculate.At this time, the numerical solving for fractional order differential equation has importantpractical significance.The main contents of this paper are represented1) In order to solve two-dimensional variable coefficient spatial variable-order advection-dispersion equation as followswe present the explicit finite difference schemes and the implicit finite difference schemes,respectively. Then we prove that the former is conditionally stable and convergent with onedegree accuracy，while the latter is uncondition stable and converges with first order.2) In order to solve variable time fractional diffusion equation with a nonlinear sourceterma new implicit finite difference scheme is given. The stability and convergence of thisscheme are considered. We prove that this finite difference scheme is unconditional sta-ble and converges with first order temporal accuracy.Some numerical experiments are given and the results further verify the effectivenessof the algorithm and the correctness of the theoretical analysis.