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Existence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition. (English) Zbl 1218.35245

Summary: Existence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition on a bounded domain with Lyapunov boundary is proved in the space of continuous functions up to boundary. Since a Green matrix of the problem is known, we may seek the solution as the linear combination of the single-layer potential, the volume potential, and the Poisson integral. Then the original problem may be reduced to a Volterra integral equation of the second kind associated with a compact operator. Classical analysis may be employed to show that the corresponding integral equation has a unique solution if the boundary data are continuous, the initial data are continuously differentiable, and the source term is Hölder continuous in the spatial variable. This in turn proves that the original problem has a unique solution.

MSC:

35R11 Fractional partial differential equations
45D05 Volterra integral equations
35C15 Integral representations of solutions to PDEs
35D30 Weak solutions to PDEs
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References:

[1] R. Metzler and J. Klafter, “The random walk’s guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1-77, 2000. · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[2] E. Bazhlekova, Fractional evolution equations in Banach spaces, dissertation, Technische Universiteit Eindhoven, 2001.
[3] S. D. Eidelman and A. N. Kochubei, “Cauchy problem for fractional diffusion equations,” Journal of Differential Equations, vol. 199, no. 2, pp. 211-255, 2004. · Zbl 1068.35037 · doi:10.1016/j.jde.2003.12.002
[4] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[5] Y. Luchko, “Maximum principle for the generalized time-fractional diffusion equation,” Journal of Mathematical Analysis and Applications, vol. 351, no. 1, pp. 218-223, 2009. · Zbl 1172.35341 · doi:10.1016/j.jmaa.2008.10.018
[6] Y. Luchko, “Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1766-1772, 2010. · Zbl 1189.35360 · doi:10.1016/j.camwa.2009.08.015
[7] J. Kemppainen and K. Ruotsalainen, “Boundary integral solution of the time-fractional diffusion equation,” Integral Equations and Operator Theory, vol. 64, no. 2, pp. 239-249, 2009. · Zbl 1185.31001 · doi:10.1007/s00020-009-1687-9
[8] A. N. Kochubeĭ, “Fractional Order Diffusion,” Journal of Difference Equations, vol. 26, no. 4, pp. 485-492, 1990. · Zbl 0729.35064
[9] W. R. Schneider and W. Wyss, “Fractional diffusion and wave equations,” Journal of Mathematical Physics, vol. 30, no. 1, pp. 134-144, 1989. · Zbl 0692.45004 · doi:10.1063/1.528578
[10] A. A. Kilbas and M. Saigo, H-Transforms: Theory and Applications, vol. 9 of Analytical Methods and Special Functions, CRC Press LLC, 2004, Theory and Application. · Zbl 1056.44001 · doi:10.1201/9780203487372
[11] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 3. More Special Functions, Overseas Publishers Association, Amsterdam, The Netherlands, 1990. · Zbl 0967.00503
[12] J. Kemppainen, “Properties of the single layer potential for the time fractional diffusion equation,” to appear in Journal of Integral Equations and Applications. · Zbl 1244.35160
[13] A. Friedman, Partial Differential Equations of Parabolic Type, Robert E. Krieger Publishing Company, Inc., Malabar, Conn, USA, 1983.
[14] R. Kress, Linear Integral Equations, vol. 82 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1999. · Zbl 0920.45001
[15] J. Kemppainen, “Existence and uniqueness of the solution for a time-fractional diffusion equation,” to appear in Fractional Calculus and Applied Analisys. · Zbl 1218.35245 · doi:10.1155/2011/321903
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