ABSTRACT
The principal-value integral can be approximated by numerical methods to yield a discrete representation that provides us with a basis for numerical computation. This chapter discusses the simplest numerical approximation scheme based on the mid-point integration rule. The numerical discretization is interesting from a physical standpoint in that it leads us back to the physical concepts that motivated the introduction of the fractional Laplacian. Having established the fractional Laplacian differentiation matrix, D(α), the chapter proceeds to develop a numerical method for solving the fractional Poisson equation for a function, f(x), in a finite or truncated solution domain. The use of an implicit time integration scheme, such as the first-order integration scheme, is mandated by issues of numerical stability. Explicit time integration schemes require a prohibitively small time step to prevent numerical instability.
