Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation

Computers and Mathematics with Applications

Volumn 59, Issue 5, 2010, Pages 1766-1772

  Luchko, Yury ^a  

^a TECHNICAL UNIVERSITY OF APPLIED SCIENCES WILDAU   (Germany)

Author keywords

Caputo Dzherbashyan fractional derivative; Fourier method; Generalized solution; Generalized time fractional diffusion equation; Initial boundary value problems; Maximum principle

Indexed keywords

FOURIER METHODS; FRACTIONAL DERIVATIVES; GENERALIZED SOLUTION; INITIAL-BOUNDARY-VALUE PROBLEMS; TIME-FRACTIONAL DIFFUSION;

FOURIER TRANSFORMS; MAXIMUM PRINCIPLE; PARTIAL DIFFERENTIAL EQUATIONS;

DIFFUSION;

EID: 76449091249     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2009.08.015     Document Type: Article

References (24)

1. Chechkin A.V., Gorenflo R., and Sokolov I.M. Fractional diffusion in inhomogeneous media. J. Phys. A: Math. Gen. 38 (2005) 679-684
2. A. Freed, K. Diethelm, Yu. Luchko, Fractional-order viscoelasticity (FOV): Constitutive development using the fractional calculus, NASA's Glenn Research Center, Ohio, 2002
3. Gorenflo R., and Mainardi F. Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal. 1 (1998) 167-191
4. In: Hilfer R. (Ed). Applications of Fractional Calculus in Physics (2000), World Scientific, Singapore
5. Kilbas A.A., Srivastava H.M., and Trujillo J.J. Theory and Applications of Fractional Differential Equations (2006), Elsevier, Amsterdam
6. Mainardi F. Fractional diffusive waves in viscoelastic solids. In: Wagner J.L., and Norwood F.R. (Eds). IUTAM Symposium-Nonlinear Waves in Solids (1995), ASME/AMR, Fairfield, NJ 93-97
7. Mainardi F., and Tomirotti M. Seismic pulse propagation with constant Q and stable probability distributions. Ann. Geofis. 40 (1997) 1311-1328
8. Metzler R., and Klafter J. The random walk's guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 (2000) 1-77
9. Podlubny I. Fractional Differential Equations (1999), Academic Press, San Diego
10. Eidelman S.D., and Kochubei A.N. Cauchy problem for fractional diffusion equations. J. Differential Equations 199 (2004) 211-255
11. Gorenflo R., Luchko Yu., and Umarov S. On the Cauchy and multi-point problems for partial pseudo-differential equations of fractional order. Fract. Calc. Appl. Anal. 3 (2000) 249-277
12. Kochubei A.N. A Cauchy problem for evolution equations of fractional order. Differ. Equ. 25 (1989) 967-974
13. Kochubei A.N. Diffusion of fractional order. Differ. Equ. 26 (1990) 485-492
14. Mainardi F., Luchko Yu., and Pagnini G. The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4 (2001) 153-192
15. Voroshilov A.A., and Kilbas A.A. The Chauchy problem for the diffusion-wave equation with the Caputo partial derivative. Differ. Equ. 42 (2006) 638-649
16. Bazhlekova E.G. Duhamel-type representation of the solutions of nonlocal boundary value problems for the fractional diffusion-wave equation. In: Rusev P., Dimovski I., and Kiryakova V. (Eds). Proc. 2nd Int. Workshop "TMSF, Varna'96" (1998), IMI-Bulg. Academy of Sciences, Sofia 32-40
17. Chen J., Liu F., and Anh V. Analytical solution for the time-fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl. 338 (2008) 1364-1377
18. Metzler R., and Klafter J. Boundary value problems for fractional diffusion equations. Physica A 278 (2000) 107-124
19. Pskhu A.V. Partial Differential Equations of Fractional Order (2005), Nauka, Moscow (in Russian)
20. Zhang S. Existence of solution for a boundary value problem of fractional order. Acta Math. Sci. Ser. B 26 (2006) 220-228
21. Luchko Yu. Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351 (2009) 218-223
22. Vladimirov V.S. Equations of Mathematical Physics (1971), Nauka, Moscow
23. Luchko Yu. Operational method in fractional calculus. Fract. Calc. Appl. Anal. 2 (1999) 463-489
24. Luchko Yu., and Gorenflo R. An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnam. 24 (1999) 207-233