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A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain

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Published/Copyright: April 28, 2017
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 20 Issue 2

Abstract

Most existing research on applying the finite element method to discretize space fractional operators is studied on regular domains using either uniform structured triangular meshes, or quadrilateral meshes. Since many practical problems involve irregular convex domains, such as the human brain or heart, which are difficult to partition well with a structured mesh, the existing finite element method using the structured mesh is less efficient. Research on the finite element method using a completely unstructured mesh on an irregular domain is of great significance. In this paper, a novel unstructured mesh finite element method is developed for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. The novel unstructured mesh Galerkin finite element method is used to discretize in space and the Crank-Nicholson scheme is used to discretize the Caputo time fractional derivative. The implementation of the unstructured mesh Crank-Nicholson Galerkin method (CNGM) is detailed and the stability and convergence of the numerical scheme are analyzed. Numerical examples are presented to verify the theoretical analysis. To highlight the ability of the proposed unstructured mesh Galerkin finite element method, a comparison of the unstructured mesh with the structured mesh in the implementation of the numerical scheme is conducted. The proposed numerical method using an unstructured mesh is shown to be more effective and feasible for practical applications involving irregular convex domains.

MSC 2010: Primary 26A33; Secondary 65M06; 65M12; 65M15; 35R11
Key Words and Phrases: two-dimensional time-space fractional wave equation; finite element method; unstructured mesh; irregular domain

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants 11472161 and 91130017), the Independent Innovation Foundation of Shandong University (Grant 2013ZRYQ002), and the Natural Science Foundation of Shandong Province (Grant ZR2014AQ015), the Australian Research Council Grant DP150103675, and the State Scholarship Fund from China Scholarship Council. The authors would like to express their sincere thanks to the anonymous referees for their constructive comments and suggestions to improve the quality of the paper.

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Received: 2016-9-8
Published Online: 2017-4-28
Published in Print: 2017-4-25

© 2017 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 20–2–2017)
  4. Survey Paper
  5. The Chronicles of Fractional Calculus
  6. Research Paper
  7. A Numerical Study of the Homogeneous Elliptic Equation with Fractional Boundary Conditions
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  9. A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain
  10. Research Paper
  11. Ulam stability for Hilfer type fractional differential inclusions via the weakly Picard operators theory
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  19. Invariant subspace method: A tool for solving fractional partial differential equations
  20. Research Paper
  21. Cauchy formula for the time-varying linear systems with caputo derivative
  22. Research Paper
  23. Overconvergence of series in generalized mittag-leffler functions
  24. Research Paper
  25. On a new class of constitutive equations for linear viscoelastic body
  26. Research Paper
  27. Observability for fractional diffusion equations by interior control
  28. Research Paper
  29. Null controllability of fractional dynamical systems with constrained control
https://doi.org/10.1515/fca-2017-0019
Keywords for this article
two-dimensional time-space fractional wave equation; finite element method; unstructured mesh; irregular domain

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 20–2–2017)
  4. Survey Paper
  5. The Chronicles of Fractional Calculus
  6. Research Paper
  7. A Numerical Study of the Homogeneous Elliptic Equation with Fractional Boundary Conditions
  8. Research Paper
  9. A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain
  10. Research Paper
  11. Ulam stability for Hilfer type fractional differential inclusions via the weakly Picard operators theory
  12. Research Paper
  13. Determination of two unknown thermal coefficients through an inverse one-phase fractional Stefan problem
  14. Research Paper
  15. Fractional integral operators characterized by some new hypergeometric summation formulas
  16. Research Paper
  17. A high-order predictor-corrector method for solving nonlinear differential equations of fractional order
  18. Research Paper
  19. Invariant subspace method: A tool for solving fractional partial differential equations
  20. Research Paper
  21. Cauchy formula for the time-varying linear systems with caputo derivative
  22. Research Paper
  23. Overconvergence of series in generalized mittag-leffler functions
  24. Research Paper
  25. On a new class of constitutive equations for linear viscoelastic body
  26. Research Paper
  27. Observability for fractional diffusion equations by interior control
  28. Research Paper
  29. Null controllability of fractional dynamical systems with constrained control
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