Implicit finite difference method for the space fractional heat conduction equation with the mixed boundary condition

• Autorzy: Brociek R. , Słota D.
• Języki publikacji: EN

Abstrakty

EN

This paper presents the numerical solution of the space fractional heat conduction equation with Neumann and Robin boundary conditions. In described equation the Riemann-Liouville fractional derivative is used. Considered model is solved by using the implicit finite difference method. The paper also presents the numerical examples to illustrate the accuracy and stability of described method.

Słowa kluczowe

EN

heat conduction equation; implicit finite difference method; space fractional heat conduction equation

PL

równanie przewodnictwa ciepła; schemat niejawny metody różnic skończonych; równanie przewodnictwa ciepła z pochodną ułamkową względem przestrzeni

• Wydawca: Wydawnictwo Politechniki Śląskiej
• Czasopismo: Silesian Journal of Pure and Applied Mathematics
• Rocznik: 2016
• Tom: Vol. 6, iss. 1
• Strony: 125--136
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Brociek R.

• Institute of Mathematics, Silesian University of Technology, Gliwice, Poland

autor

Słota D.

• Institute of Mathematics, Silesian University of Technology, Gliwice, Poland

Bibliografia

• 1. Brociek R.: Implicite finite difference method for time fractional heat equation with mixed boundary conditions. Zesz. Nauk. PŚl., Mat. Stosow. 4 (2014), 73– 87.
• 2. Carpinteri A., Mainardi F: Fractal and Fractional Calculus in Continuum Mechanics. Springer, New York 1997.
• 3. Chen W., Ye L., Sun H.: Fractional diffusion equations by the kansa method. Comput. Math. Appl. 59 (2010), 1614–1620.
• 4. Diethelm K.: The Analysis of Fractional Differential Equations. Springer, Berlin 2010.
• 5. Feng L., Zhuang P., Liu F., Turner I, Yang Q.: Second-order approximation for the space fractional diffusion equation with variable coefficient. Progr. Fract. Differ. Appl. 1 (2015), 23–35.
• 6. Guo B., Pu X., Huang F.: Fractional Partial Differential Equations and Their Numerical Solution. World Scientific, Singapore 2015.
• 7. Klafter J., Lim S.C., Metzler R.: Fractional dynamics. Resent advances. World Scientific, New Jersey 2012.
• 8. Liu F., Zhuang P., Turner I., Burrage K., Anh V.: A new fractional finite volume method for solving the fractional diffusion equation. Appl. Math. Modelling 38 (2014), 3871–3878.
• 9. Meerschaert M.M., Benson D.A., Scheffler H.P., Becker-Kern P.: Governing equations and solutions of anomalous random walk limits. Phys. Rev. E 66 (2002), 102R–105R.
• 10. Meerschaert M.M., Scheffler H.P., Tadjeran Ch.: Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211 (2006), 249–261.
• 11. Meerschaert M.M., Tadjeran Ch.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172 (2006), 65–77.
• 12. Meerschaert M.M., Tadjeran Ch.: Finite difference approximations for twosided space-fractional partial differential equations. Appl. Numer. Math. 56 (2006), 80–90.
• 13. Metzler R., Klafter J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. Rev. A 37 (2004), R161–R208.
• 14. Miller K., Ross B.: An Introduction to the Fractional Calculus and Fractional Differential. Wiley, New York 1993.
• 15. Mitkowski W., Kacprzyk J., Baranowski J.: Advances in the Theory and Applications of Non-integer Order Systems, Springer Inter. Publ., Cham 2013.
• 16. Murio D.A.: Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 56 (2008), 1138–1145.
• 17. Podlubny I.: Fractional Differential Equations. Academic Press, San Diego 1999.
• 18. Rabsztyn Sz., Słota D., Wituła R.: Functions gamma and beta, vol. 1 and 2. Wyd. Pol. Śl., Gliwice 2012 (in Polish).
• 19. Sabatier J., Agrawal O.P., Tenreiro Machado J.A.: Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht 2007.
• 20. Sudha Priya G, Prakash P., Nieto J.J., Kayar Z.: Higher-order numerical scheme for the fractional heat equation with Dirichlet and Neumann boundary conditions. Numer. Heat Transfer B 63 (2013), 540–559.
• 21. Sweilam N.H., Nagy A.M., El-Sayed A.A.: On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polynomials of the third kind. J. of King Saud Univesity – Science 28 (2016), 41–47.
• 22. Tadjeran Ch., Meerschaert M.M., Scheffler H.P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213 (2006), 205–213.