Rational solutions for the time-fractional diffusion equation

SIAM Journal on Applied Mathematics

Volumn 71, Issue 1, 2011, Pages 92-106

  Atkinson, Colin ^a   Osseiran, Adel ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Fourier Laplace transforms; Global rational approximation; Inverse Mittag Leffler function; Mellin transform of time fractional boundary value problem; Mittag Leffler function; Spectral decomposition of time fractional equation

Indexed keywords

FOURIER-LAPLACE TRANSFORM; GLOBAL RATIONAL APPROXIMATION; MELLIN TRANSFORM; MITTAG-LEFFLER FUNCTIONS; SPECTRAL DECOMPOSITION;

BOUNDARY VALUE PROBLEMS; FOURIER TRANSFORMS; LAPLACE EQUATION; LAPLACE TRANSFORMS;

RATIONAL FUNCTIONS;

EID: 79952331155     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/100799307     Document Type: Article

References (35)

1. C. Atkinson (1977), On stress singularities and interfaces in linear elastic fracture mechanics, Internat. J. Fracture, 13, pp. 807-820. (Pubitemid 8578362)
2. M. N. Berberan-Santos (2005), Properties of the Mittag-Leffler relaxation function, J. Math. Chem., 38, pp. 629-635.
3. M. Caputo (1967), Linear models of dissipation whose Q is almost frequency independent. II, Geophys. J. R. Astr. Soc., 13, pp. 529-539.
4. M. Caputo and F. Mainardi (1971), Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (Ser. II), 1, pp. 161-198.
5. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (1955), Higher Transcendental Functions, Vol. III, McGraw-Hill, New York.
6. W. Feller (1949), Fluctuation theory of recurrent events, Trans. Amer. Math. Soc., 67, pp. 98-119.
7. A. Freed, K. Diethelm, and Yu. Luchko (2002), Fractional-Order Viscoelasticity (FOV): Constitutive Development Using the Fractional Calculus: First Annual Report, NASA/TM-2002- 211914, Glenn Research Center, Cleveland, OH.
8. R. Gorenflo, J. Loutchko, and Y. Luchko (2002), Computation of the Mittag-Leffler function Eα,β(z) and its derivatives, Fract. Calc. Appl. Anal., 5, pp. 491-518.
9. R. Gorenflo and F. Mainardi (2009), Some recent advances in theory and simulation of fractional diffusion processes, J. Comput. Appl. Math., 229, pp. 400-415.
10. R. Hilfer and H. J. Seybold (2006), Computation of the generalized Mittag-Leffler function and its inverse in the complex plane, Integral Transforms Spec. Funct., 17, pp. 637-652. (Pubitemid 44180356)
11. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo (2006), Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
12. V. Kiryakova (1994), Generalized Fractional Calculus and Applications, Pitman Res. Notes Math. Ser. 301, Longman Scientific & Technical, Harlow, UK, John Wiley & Sons, New York.
13. F. Mainardi (1996a), Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals, 7, pp. 1461-1477.
14. F. Mainardi (1996b), The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9, pp. 23-28. (Pubitemid 126365654)
15. F. Mainardi (2004), Applications of integral transforms in fractional diffusion processes, Integral Transforms Spec. Funct., 15, pp. 477-484. (Pubitemid 39378880)
16. F. Mainardi (2010), Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London.
17. F. Mainardi and R. Gorenflo (2007), Time-fractional derivatives in relaxation processes: A tutorial survey, Fract. Calc. Appl. Anal., 10, pp. 269-308.
18. F. Mainardi, Yu. Luchko, and G. Pagnini (2001), The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4, pp. 153-192.
19. F. Mainardi, A. Mura, and G. Pagnini (2010), The M-Wright function in time-fractional diffusion processes: A tutorial survey, Int. J. Differ. Equ., Article ID 104505.
20. A. C. McBride (1979), Fractional Calculus and Integral Transforms of Generalized Functions, Res. Notes in Math. 31, Pitman, Boston, London.
21. A. C. McBride and G. Roach (1985), Fractional Calculus, Res. Notes in Math. 138, Pitman, Boston, London, Melbourne.
22. K. S. Miller and B. Ross (1993), An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York.
23. G. Mittag-Leffler (1903), Sur la nouvelle fonction Eα(x), C. R. Acad. Sci. Paris (Ser. II), 137, pp. 554-558.
24. K. B. Oldham and J. Spanier (1974), The Fractional Calculus, Academic Press, New York.
25. I. Podlubny (1998), Fractional Differential Equations, Academic Press, San Diego.
26. H. Pollard (1948), The completely monotonic character of the Mittag-Leffler function Eα(-x), Bull. Amer. Math. Soc., 54, pp. 1115-1116.
27. S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993), Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, Switzerland.
28. E. Scalas, R. Gorenflo, F. Mainardi, and M. Raberto (2003), Revisiting the derivation of the fractional diffusion equation. Scaling and disordered systems, Fractals, 11 (supp.), pp. 281-289. (Pubitemid 36446629)
29. W. R. Schneider and W. Wyss (1989), Fractional diffusion and wave equations, J. Math. Phys., 30, pp. 134-144.
30. S. T. Spencer, J. W. Hanneken, T. R. Ensley, and B. N. Narahari Achar (2007), Properties of the Mittag-Leffler function, American Physical Society March Meeting, Denver, CO, abstract R1.00249.
31. H. M. Srivastava and S. Owa, eds. (1989), Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood, Chichester, UK, Halsted Press, New York.
32. A. P. Starovoitov and N. A. Starovoitova (2007), Padé approximants of the Mittag-Leffler functions, Sb. Math., 198, pp. 1011-1023.
33. A. Wiman (1905), Über den Fundamentalsatz in der Teorie der Funktionen Eα(x), Acta Math., 29, pp. 191-201.
34. S. Winitzki (2003), Uniform approximations for transcendental functions, in Computational Science and Its Applications-ICCSA 2003. Part I, Lecture Notes in Comput. Sci. 2667, Springer-Verlag, Berlin, pp. 780-789.
35. W. Wyss (1986), The fractional diffusion equation, J. Math. Phys., 27, pp. 2782-2785.