Fractional pseudo-spectral methods for distributed-order fractional PDEs

We develop a pseudo-spectral method of Petrov-Galerkin sense, employing nodal expansions in the weak formulation of distributed-order fractional partial differential equations. We define the underlying distributed Sobolev spaces and the associated norms. Then, we formulate the scheme, using the fractional Lagrange interpolants of first and second kind as the nodal bases and test functions, respectively. We construct the corresponding weak distributed differentiation matrices for the operators with one-/two-sided fractional derivatives, leading to an improved conditioning of the resulting linear system. Subsequently, we study the effect of distribution function and interpolation points on the condition number, and further design several distributed pre-conditioners. Furthermore, we investigate efficiency of the proposed scheme by considering several linear/nonlinear numerical examples, including initial value problem, (1+1)-D space distributed-order Burgers' equation, and (1+2)-D two-sided space distributed-order nonlinear reaction-diffusion equation.