A unified Petrov–Galerkin spectral method for fractional PDEs

A unified Petrov–Galerkin spectral method for fractional PDEs

Existing numerical methods for fractional PDEs suffer from low accuracy and inefficiency in dealing with three-dimensional problems or with long-time integrations. We develop a unified and spectrally accurate Petrov–Galerkin (PG) spectral method for a weak formulation of the general linear Fractional Partial Differential Equations (FPDEs) of the form 0D t u + d j=1 c j [a jD 2μ j x j u ] + γ u...

The Unified Transform Method for Linear PDEs

There exist certain nonlinear evolution PDEs which can be mapped to linear evolution PDEs. The prototypical such linearizable PDE is the so-called Burger's equation u t − u xx = 2uu x , (1.1) which can be written in the conservation form ∂ t u = ∂ x (u x + u 2). Employing the so-called Cole-Hopf transformation u = q x q (1.2) and using the identity ∂ t q x q = ∂ t ∂ x ln q = ∂ x ∂ t ln q = ∂ x ...

A Short Course on: Spectral Methods for Fractional ODEs/PDEs

The spectral iterative method for Solving Fractional-Order Logistic ‎Equation

In this paper, a new spectral-iterative method is employed to give approximate solutions of fractional logistic differential equation. This approach is based on combination of two different methods, i.e. the iterative method cite{35} and the spectral method. The method reduces the differential equation to systems of linear algebraic equations and then the resulting systems are solved by a numer...

Hermite Spectral Methods for Fractional PDEs in Unbounded Domains

Numerical approximations of fractional PDEs in unbounded domains are considered in this paper. Since their solutions decay slowly with power laws at infinity, a domain truncation approach is not effective as no transparent boundary condition is available. We develop efficient Hermite-collocation and Hermite–Galerkin methods for solving a class of fractional PDEs in unbounded domains directly, a...