Matrix approach to discrete fractional calculus II: Partial fractional differential equations

Abstract

a b s t r a c tA new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) . Four examples of numerical solution of fractional diffusion equation with various combinations of time-/space-fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

“…Based on the discussion in [72], the Caputo derivative is frequently used for the derivative with respect to time. The left-sided Riemann-Liouville derivative of real order α with n − 1 < α < n, is defined by…”

“…We will follow the discretize-then-optimize approach for PDE-constrained optimization problems [11,35] (details on the numerical treatment of FDEs can be found in [50,51,69,71,72]). We consider Ω = [0, 1]d and assume that Ω u and Ωȳ are also cubes.…”

“…As in [72] we approximate the spatial derivative of order 1 ≤ β ≤ 2 using the symmetric Riesz derivative taken as the half sum of the left-and right-sided Riemann-Liouville fractional derivatives. The resulting differentiation matrix is given by…”

“…Historically, the finite difference-based discretization techniques are arguably the most popular [50,51,71,72,73]. Adomian decomposition should be mentioned as a popular semi-analytical approach [1], although its use is limited.…”

“…Our work here is motivated by some recent results in [72] where the discretization via finite differences is considered in a purely algebraic framework.…”