Regularity and Numerical Analysis of Multi‐Term General Fractional Advection‐Diffusion Equations With Scale and Weight Dependencies

ABSTRACT This article investigates a multi‐term general fractional advection‐diffusion equation, where the time derivative is a general Caputo fractional operator incorporating a weight and a strictly increasing scale function. First, the analytical solution is derived using the weighted Laplace transform and the weighted Fourier sine transform. A thorough regularity analysis of the solution is then conducted, revealing that it exhibits a weak singularity at the initial time , characterized by a term of the form . This initial layer necessitates specialized temporal discretizations. To handle this singularity effectively, we develop and analyze a numerical scheme. The temporal discretization is implemented on uniform, graded, and adaptive meshes, while a high‐order compact difference scheme is employed for the spatial domain. The adaptive mesh is generated through the equidistribution of a positive monitor function. A rigorous stability and convergence analysis is performed for the uniform and graded mesh schemes. We rigorously prove that these schemes achieve an optimal convergence order of in time and a fourth‐order accuracy in space, where is the leading fractional order and is the grading parameter. The performance of the proposed schemes is evaluated through three numerical examples with non‐smooth solutions. The adaptive mesh consistently outperforms both graded and uniform meshes in these tests. Furthermore, the results clearly illustrate the significant impact of the weight and scale functions on the solution's behavior.