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In this paper we investigate the variable coefficient two-sided fractionaldiffusion, advection, reaction equations on a bounded interval. It is knownthat the fractional diffusion operator may lose coercivity due to the variablecoefficient, which makes both the mathematical and numerical analysischallenging. To resolve this issue, we design appropriate test and trialfunctions to prove the inf-sup condition of the variable coefficient fractionaldiffusion, advection, reaction operators in suitable function spaces. Based onthis property, we prove the well-posedness and regularity of the solutions, aswell as analyze the Petrov-Galerkin approximation scheme for the proposedmodel. Numerical experiments are presented to substantiate the theoreticalfindings and to compare the behaviors of different models.

Analysis and Petrov-Galerkin numerical approximation for variable coefficient two-sided fractional diffusion, advection, reaction equations