ABSTRACT
This chapter investigates a solution to the differential equations with spatial and temporal fractional derivatives by using two novel finite difference schemes. A Crank-Nicolson difference method used for the temporal fractional derivatives in (0,1). Also, a compact difference scheme used for the temporal fractional derivatives in (1,2) of the fractional oscillation motion equation with viscoelastic damping are constructed when the spatial fractional derivatives in (1,2) for finding the numerical solutions. This model is used to describe the mechanical oscillation mechanism in a viscoelastic medium. Analysis of the stability and convergence are rigorously discussed for the two schemes. Finally, numerical experiments are presented to show that the suggested methods are very effective.
