Numerical Algorithms

, Volume 73, Issue 2, pp 445–476 | Cite as

Analysis of a meshless method for the time fractional diffusion-wave equation

Original Paper


In this paper a numerical technique is proposed for solving the time fractional diffusion-wave equation. We obtain a time discrete scheme based on finite difference formula. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method and the convergence order of the time discrete scheme is \(\mathcal {O}(\tau ^{3-\alpha })\). Firstly, we change the main problem based on Dirichlet boundary condition to a new problem based on Robin boundary condition and then, we consider a semi-discrete scheme with Robin boundary condition and show when \(\beta \rightarrow +\infty \) solution of the main semi-discrete problem with Dirichlet boundary condition is convergent to the solution of the new semi-discrete problem with Robin boundary condition. We consider the new semi-discrete problem with Robin boundary condition and use the meshless Galerkin method to approximate the spatial derivatives. Finally, we obtain an error bound for the new problem. We prove that convergence order of the numerical scheme based on Galekin meshless is \(\mathcal {O}(h)\). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. The main aim of the current paper is to obtain an error estimate for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme.


Time fractional diffusion-wave equation Fractional derivative Convergence analysis Error estimate Caputo derivative Meshless Galerkin method Radial basis functions 

Mathematics Subject Classification (2010)

65M70 34A34 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Mehdi Dehghan
    • 1
  • Mostafa Abbaszadeh
    • 2
  • Akbar Mohebbi
    • 1
  1. 1.Department of Applied Mathematics, Faculty of Mathematics and Computer SciencesAmirkabir University of TechnologyTehranIran
  2. 2.Department of Applied Mathematics, Faculty of Mathematical ScienceUniversity of KashanKashanIran

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