A Petrov-Galerkin spectral method for fractional convection–diffusion equations with two-sided fractional derivative

Abstract

In this paper, we study a boundary value problem of fractional convection–diffusion equation with general two-sided fractional derivative. The well-posedness of the variation formulation is investigated under some properly assumptions. A Petrov-Galerkin method is developed, which employs the Jacobi poly-fractonomials for the trial and test space. The new approach allows the derivation of optimal error estimates in properly weighted Sobolev space and the matrix of the leading term is diagonal. We show that even for smooth data, we only obtain algebraic convergence due to the regularity of the solution. Some numerical examples are presented to demonstrate the validity of our theoretical results.

Keywords:

• Petrov-Galerkin spectral method
• fractional convection–diffusion equations
• error estimates
• two-sided fractional derivative
• boundary value problem

2010 Mathematics Subject Classifications:

• 65N12
• 65N35

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work are supported by the State Key Program of National Natural Science Foundation of China (11931003) and National Natural Science Foundation of China (41974133, 11671157, 11971410).