The Unstructured Mesh Finite Element Method for the Two-Dimensional Multi-term Time–Space Fractional Diffusion-Wave Equation on an Irregular Convex Domain



In this paper, the two-dimensional multi-term time-space fractional diffusion-wave equation on an irregular convex domain is considered as a much more general case for wider applications in fluid mechanics. A novel unstructured mesh finite element method is proposed for the considered equation. In most existing works, the finite element method is applied on regular domains using uniform meshes. The case of irregular convex domains, which would require subdivision using unstructured meshes, is mostly still open. Furthermore, the orders of the multi-term time-fractional derivatives have been considered to belong to (0, 1] or (1, 2] separately in existing models. In this paper, we consider two-dimensional multi-term time-space fractional diffusion-wave equations with the time fractional orders belonging to the whole interval (0, 2) on an irregular convex domain. We propose to use a mixed difference scheme in time and an unstructured mesh finite element method in space. Detailed implementation and the stability and convergence analyses of the proposed numerical scheme are given. Numerical examples are conducted to evaluate the theoretical analysis.


Multi-term time-space fractional diffusion-wave equation Irregular convex domain Unstructured mesh Stability and convergence analysis 

Mathematics Subject Classification

26A33 65M12 65N30 



We would like to express sincere thanks to the referees for their many constructive comments and suggestions to improve the paper.


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Authors and Affiliations

  1. 1.School of ScienceJiangnan UniversityWuxiPeople’s Republic of China
  2. 2.School of MathematicsShandong UniversityJinanPeople’s Republic of China
  3. 3.School of Mathematical SciencesQueensland University of TechnologyBrisbaneAustralia

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