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Numerical Solution of Nonstationary Problems for a Space-Fractional Diffusion Equation

  • Petr N. Vabishchevich EMAIL logo
Published/Copyright: March 9, 2016
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 19 Issue 1

Abstract

An unsteady problem is considered for a space-fractional diffusion equation in abounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary conditions of Robin type. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the fractional power of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The results of numerical experiments are presented for a model two-dimensional problem.

MSC: Primary 26A33; Secondary 35R11; 65F60; 65M06
Key Words and Phrases: fractional partial differential equations; elliptic operator; fractional power of an operator; two-level difference scheme

Acknowledgements

This work was supported by the Russian Foundation for Basic Research (Projects 14-01-00785, 15-01-00026).

References

[1] D. Baleanu, Fractional Calculus: Models and Numerical Methods. World Scientific, New York (2012).10.1142/8180Search in Google Scholar

[2] Å. Björck, Numerical Methods in Matrix Computations. Springer, Berlin (2015).10.1007/978-3-319-05089-8Search in Google Scholar

[3] A. Bonito, and J. E. Pasciak, Numerical approximation of fractional powers of elliptic operators. Mathematics of Computation64, No 295 (2015), 2083–2110.10.1090/S0025-5718-2015-02937-8Search in Google Scholar

[4] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods. Springer, New York (2008).10.1007/978-0-387-75934-0Search in Google Scholar

[5] A. Bueno-Orovio, D. Kay, and K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numerical Mathematics54, No 4 (2014), 937–954.10.1007/s10543-014-0484-2Search in Google Scholar

[6] K. Burrage, N. Hale, and D. Kay, An efficient implicit fem scheme for fractional-in-space reaction-diffusion equations. SIAM J. On Scientific Computing34, No 4 (2012), A2145–A2172.10.1137/110847007Search in Google Scholar

[7] S. Chen, F. Liu, I. Turner, and V. Anh, An implicit numerical method for the two-dimensional fractional percolation equation. Applied Mathematics and Computation219, No 9 (2013), 4322-4331.10.1016/j.amc.2012.10.003Search in Google Scholar

[8] A. G. Churbanov, and P. N. Vabishchevich, Numerical investigation of a space-fractional model of turbulent fluid flow in rectangular ducts. arXiv Preprint, arXiv: 1505.01519 (2015).10.1016/j.jcp.2016.06.009Search in Google Scholar

[9] A. C. Eringen, Nonlocal Continuum Field Theories. Springer, New York (2002).Search in Google Scholar

[10] L. Fang and H. K. Du, Young's inequality for positive operators. J. of Mathematical Research & Exposition31, No 5 (2011), 915–922.Search in Google Scholar

[11] G. H. Golub and C. F. Van Loan, Matrix Computations. JHU Press, Baltimore (2012).10.56021/9781421407944Search in Google Scholar

[12] N. J. Higham, Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008).10.1137/1.9780898717778Search in Google Scholar

[13] M. Ilic, F. Liu, I. Turner, and V. Anh, Numerical approximation of a fractional-in-space diffusion equation I. Fract. Calc. Appl. Anal. 8, No 3 (2005), 323-341; at http://www.math.bas.bg/∼fcaa.Search in Google Scholar

[14] M. Ilic, F. Liu, I. Turner, and V. Anh, Numerical approximation of a fractional-in-space diffusion equation (II) – with nonhomogeneous boundary conditions. Fract. Calc. Appl. Anal. 9, No 4 (2006), 333–349; at http://www.math.bas.bg/∼fcaa.Search in Google Scholar

[15] M. Ilić, F. Liu, I. Turner, and V. Anh, Numerical solution using an adaptively preconditioned lanczos method for a class of linear systems related with the fractional poisson equation. International J. of Stochastic Analysis (2008), Article ID 104525, 26 pages.10.1155/2008/104525Search in Google Scholar

[16] B. Jin, R. Lazarov, J. Pasciak, and Z. Zhou, Error analysis of finite element methods for space-fractional parabolic equations. SIAM J. Numer. Anal. 52, No 5 (2014), 2272-2294.10.1137/13093933XSearch in Google Scholar

[17] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Math. Studies, Elsevier, Amsterdam (2006).Search in Google Scholar

[18] P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer Verlag, New York (2003).Search in Google Scholar

[19] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin (1994).10.1007/978-3-540-85268-1Search in Google Scholar

[20] J. P. Roop, Computational aspects of fem approximation of fractional advection dispersion equations on bounded domains in R2. J. of Computational and Applied Mathematics193, No 1 (2006), 243–268.10.1016/j.cam.2005.06.005Search in Google Scholar

[21] A. A. Samarskii, The Theory of Difference Schemes. Marcel Dekker, New York (2001).10.1201/9780203908518Search in Google Scholar

[22] A. A. Samarskii, P. P. Matus, and P. N. Vabishchevich, Difference Schemes with Operator Factors. Kluwer, Boston (2002).10.1007/978-94-015-9874-3Search in Google Scholar

[23] R. Stern, F. Effenberger, H. Fichtner, and T. Schäfer, The space-fractional diffusion-advection equation: Analytical solutions and critical assessment of numerical solutions. Fract. Calc. Appl. Anal. 17, No 1 (2014), 171–190; DOI: 10.2478/sl3540-0l4-0l6l-9; http//www.degruyter.com/view/j/fca.2014.17.issue-1/issue-files/fca.2014.17.issue-l.xml.10.2478/sl3540-0l4-0l6l-9Search in Google Scholar

[24] C. Tadjeran and M. M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. of Computational Physics220, No 2 (2007), 813–823.10.1016/j.jcp.2006.05.030Search in Google Scholar

[25] C. Tadjeran, M. M. Meerschaert, and .H-P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation. J. of Computational Physics213, No 1 (2006), 205–213.10.1016/j.jcp.2005.08.008Search in Google Scholar

[26] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer Verlag, Berlin (2006).Search in Google Scholar

[27] P. N. Vabishchevich, Additive Operator-Difference Schemes: Splitting Schemes. De Gruyter, Berlin (2014).10.1515/9783110321463Search in Google Scholar

[28] P. N. Vabishchevich, Numerical solving the boundary value problem for fractional powers of elliptic operators. J. of Computational Physics282, No 1 (2015), 219–302.10.1016/j.jcp.2014.11.022Search in Google Scholar

[29] J. L. Väzquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. arXiv Preprint, arXiv: 1401.3640 (2014). 10.3934/dcdss.2014.7.857Search in Google Scholar

[30] A. Yagi, Abstract Parabolic Evolution Equations and Their Applications. Springer, Berlin (2009).10.1007/978-3-642-04631-5Search in Google Scholar

[31] Q. Yang, F. Liu, and I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Applied Mathematical Modelling34, No 1 (2010), 200–218.10.1016/j.apm.2009.04.006Search in Google Scholar

  1. Please cite to this paper as published in:

    Fract. Calc. Appl. Anal., Vol. 19, No 1 (2016), pp. 116–139, DOI: 10.1515/fca-2016-0007

Received: 2014-12-5
Revised: 2015-7-9
Published Online: 2016-3-9
Published in Print: 2016-2-1

© 2016 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA Related News, Events and Books (FCAA–Volume 19–1–2016)
  4. Research Paper
  5. Smallest Eigenvalues for a Right Focal Boundary Value Problem
  6. Research Paper
  7. High-Order Algorithms for Riesz Derivative and their Applications (III)
  8. Research Paper
  9. Existence and Uniqueness for a Class of Stochastic Time Fractional Space Pseudo-Differential Equations
  10. Research Paper
  11. Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data
  12. Research Paper
  13. Bogolyubov-Type Theorem with Constraints Generated by a Fractional Control System
  14. Research Paper
  15. Numerical Solution of Nonstationary Problems for a Space-Fractional Diffusion Equation
  16. Research Paper
  17. Solving 3D Time-Fractional Diffusion Equations by High-Performance Parallel Computing
  18. Discussion Survey
  19. Physical and Geometrical Interpretation of Grünwald-Letnikov Differintegrals: Measurement of Path and Acceleration
  20. Discussion Survey
  21. Some Applications of Fractional Velocities
  22. Research Paper
  23. Maximum Principles for Multi-Term Space-Time Variable-Order Fractional Diffusion Equations and their Applications
  24. Survey Paper
  25. Atypical Case of the Dielectric Relaxation Responses and its Fractional Kinetic Equation
  26. Research Paper
  27. Operator Method for Construction of Solutions of Linear Fractional Differential Equations with Constant Coefficients
  28. Research Article
  29. On the Existence and Multiplicity of Solutions for Dirichlet’s problem for Fractional Differential equations
  30. Research Paper
  31. Approximate controllability for semilinear composite fractional relaxation equations
https://doi.org/10.1515/fca-2016-0007
Keywords for this article
fractional partial differential equations; elliptic operator; fractional power of an operator; two-level difference scheme

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA Related News, Events and Books (FCAA–Volume 19–1–2016)
  4. Research Paper
  5. Smallest Eigenvalues for a Right Focal Boundary Value Problem
  6. Research Paper
  7. High-Order Algorithms for Riesz Derivative and their Applications (III)
  8. Research Paper
  9. Existence and Uniqueness for a Class of Stochastic Time Fractional Space Pseudo-Differential Equations
  10. Research Paper
  11. Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data
  12. Research Paper
  13. Bogolyubov-Type Theorem with Constraints Generated by a Fractional Control System
  14. Research Paper
  15. Numerical Solution of Nonstationary Problems for a Space-Fractional Diffusion Equation
  16. Research Paper
  17. Solving 3D Time-Fractional Diffusion Equations by High-Performance Parallel Computing
  18. Discussion Survey
  19. Physical and Geometrical Interpretation of Grünwald-Letnikov Differintegrals: Measurement of Path and Acceleration
  20. Discussion Survey
  21. Some Applications of Fractional Velocities
  22. Research Paper
  23. Maximum Principles for Multi-Term Space-Time Variable-Order Fractional Diffusion Equations and their Applications
  24. Survey Paper
  25. Atypical Case of the Dielectric Relaxation Responses and its Fractional Kinetic Equation
  26. Research Paper
  27. Operator Method for Construction of Solutions of Linear Fractional Differential Equations with Constant Coefficients
  28. Research Article
  29. On the Existence and Multiplicity of Solutions for Dirichlet’s problem for Fractional Differential equations
  30. Research Paper
  31. Approximate controllability for semilinear composite fractional relaxation equations
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