Numerical treatment of fractional heat equations

Applied Numerical Mathematics

Volumn 58, Issue 8, 2008, Pages 1212-1223

  Scherer, R ^a   Kalla, S L ^b   Boyadjiev, L ^c   Al Saqabi, B ^b  

^a UNIVERSITY OF KARLSRUHE   (Germany)

^b KUWAIT UNIVERSITY   (Kuwait)

^c TECHNICAL UNIVERSITY OF SOFIA   (Bulgaria)

Author keywords

CFL condition; Difference methods; Error estimates; Fractional derivatives; Fractional heat equation; Gr nwald Letnikov's approach; Stability

Indexed keywords

APPROXIMATION THEORY; ERROR ANALYSIS; FINITE DIFFERENCE METHOD; HEAT TREATMENT; NUMERICAL METHODS; PROBLEM SOLVING;

CFL CONDITION; DIFFERENCE METHODS; FRACTIONAL DERIVATIVES; FRACTIONAL HEAT EQUATIONS;

WAVE EQUATIONS;

EID: 44149091687     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2007.06.003     Document Type: Article

References (26)

1. Antimirov M.Ya., Kolyshkin A.A., and Vaillancourt R. Applied Integral Transforms (1993), American Mathematical Society, Providence, RI
2. Ben Nakhi Y., and Kalla S.L. Some boundary value problems of temperature fields in oil strata. Appl. Math. Comput. 146 (2003) 105-119
3. Boyadjiev L., Kamenov O., and Kalla S.L. On the Lauwerier formulation of the temperature field problem in oil strata. Int. J. Math. Math. Sci. 2005 (2005) 1577-1588
4. Boyadjiev L., and Scherer R. Fractional extensions of the temperature field problem in oil strata. Kuwait J. Sci. Eng. 31 2 (2004) 15-32
5. M. Ciesielski, J. Leszczynski, Numerical solutions of anomalous diffusion, in: Proc. CMM-2003, Computer Methods in Mechanics, Gliwice, Poland, 2003
6. Diethelm K., and Ford N.J. Numerical solution of the Bagley-Torvik equation. BIT 42 (2002) 490-507
7. Diethelm K., Ford N.J., and Freed A.D. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynam. 29 (2002) 3-22
8. Diethelm K., Ford N.J., and Freed A.D. Detailed error analysis for a fractional Adams method. Numer. Algorithms 36 (2004) 31-52
9. Diethelm K., Ford M.J., Ford N.J., and Weilbeer M. Pitfalls in fast numerical solvers for fractional differential equations. J. Comput. Appl. Math. 186 (2006) 482-503
10. Erdélyi A. Higher Transcendental Functions, vols. I-III (1955), McGraw-Hill, New York, NY
11. Fichtenholz G.M. Differential- und Integralrechnung II (1964), VEB Deutscher Verlag der Wissenschaften, Berlin
12. In: Hilfer R. (Ed). Applications of Fractional Calculus in Physics (2000), World Scientific, Singapore
13. Gradshteyn I.S., and Ryzhik I.M. Tables of Integrals, Series, and Products. 4th ed. (1980), Academic Press, New York
14. Kilbas A.A., Srivastava H.M., and Trujillo J.J. Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies vol. 204 (2006), Elsevier, Amsterdam
15. Kumar P., and Agrawal O.P. Numerical scheme for the solution of fractional differential equations of order greater than one. J. Comput. Nonlinear Dynam. 1 (2006) 178-185
16. Kumar P., and Agrawal O.P. An approximate method for numerical solution of fractional differential equations. Signal Processing 86 (2006) 2602-2610
17. Langlands T.A.M., and Henry B.I. The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205 (2005) 719-736
18. Mainardi F. The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9 (1996) 23-28
19. Meerschaert M.M., and Tadjeran C. A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220 (2007) 813-823
20. Miller K.S., and Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley, New York
21. Podlubny I. Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications (1999), Academic Press, San Diego, CA
22. Samko S.G., Kilbas A.A., and Marichev O.I. The Fractional Integrals and Derivatives. Theory and Applications (1993), Gordon and Breach, Amsterdam
23. Schneider W.R., and Wyss W. Fractional diffusion and wave equations. J. Math. Phys. 30 (1989) 134-144
24. Tadjeran C., and Meerschaert M.M. A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220 (2007) 813-823
25. Tadjeran C., Meerschaert M.M., and Scheffler H.-P. A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213 (2006) 205-213
26. Thomas J.W. Numerical Partial Differential Equations: Finite Difference Methods (1995), Springer-Verlag, New York