Numerical solution of fractional partial differential equations

Abstract

Partial differential equations are at the heart of many, if not most, computer analyses or simulations of continuous physical systems, such as fluids, electromagnetic fields, the human body. Fractional calculus is an extension of derivatives and integrals to noninteger orders, and a partial differential equation involving the fractional calculus operators is called the fractional PDE. They have many applications in science and engineering. However not only the analytical solution existed for a limited number of cases, but also the numerical methods are very complicated and difficult. In this chapter, the numerical methods for fractional partial differential equations will be reviewed, especially the approach based on the operational matrices of the orthogonal functions. It transforms the problem to a simple Lyapunov matrix equation solving. Advantages of the operational method include (1) the computation is simple and computer oriented, (2) it can solve the partial differential equations numerically, even the ones with fractional order, (3) the scope of application is wide and (4) the step size used could be large and the result obtained is still satisfactory. The numerically unstable problem does not occur in the operational method.