Alternating Direction Implicit Orthogonal Spline Collocation Method For The Two-dimensional Fractional Diffusion Equation

Fractional calculus as a research direction developed in recent years, has been used influid mechanics, viscoelasticity, biology, physics and engineering due to its moreaccurately describe the actual phenomenon. The study of fractional differential equationsnumerical analysis has attracted widespread interest in recent years. Generally, getting theanalytical solutions of fractional differential equations is vary difficult. It is valuable tofind the numerical solution of these equations.In this paper, numerical methods for two dimensional fractional diffusion equationswith initial and boundary condition is considered. The fractional derivative is described inthe Caputo sense. First, we give a brief introduction to the background of this topic andthe overview of the research related to this topic at home and abroad. In order to study thistopic, some prior knowledge of the fractional calculus is given. Next, we consider timefractional diffusion equation with initial and boundary conditions (the time fractionalorder a such that0<aï¿¡1). For this equation, we use the L1approximation formula toapproximate the time fractional derivative, and orthogonal spline collocation method (forshort OSC) is used in spatial to attain the numerical solution since orthogonal splinecollocation method has the advantages of super convergence, less computation and easyimplementation. In order to get the alternating direction implicit orthogonal splinecollocation method for the equation, we add a term, so that we can solve atwo-dimensional problem by solving two one-dimensional problems. Then we prove thescheme is unconditional stable by combining the characteristic of the coefficients in thescheme and mathematical induction. Finally, we further consider the higher order methodof alternating direction implicit orthogonal spline collocation for fractional diffusionequation. We add two terms with truncation error of and, separately,then two kinds of ADI OSC scheme are given for the equation. For the threecomputational schemes, numerical examples are given to verify the effective of themethods which are proposed in this paper, and we make a comparison of them.