Weak solutions and convergence of the Galerkin method for the fractional diffusion equation
DOI:
https://doi.org/10.15407/dopovidi2015.03.032Keywords:
diffusion equation, Galerkin method, weak convergenceAbstract
We construct a semidiscrete Galerkin method for the time-fractional diffusion equation. We prove the weak convergence of the method in the case of the right-hand side from a negative space with respect to the space variable. The continuity of the solution with values in a space of square-integrable functions is proven.
Downloads
References
Metzler R., Klafter J. J. Phys. A: Math. Gen., 2004, 37: R161–R208. https://doi.org/10.1088/0305-4470/37/31/R01
Uchaikin V. V., The method of fractional derivatives, Ulianovsk: Artishok, 2008 (in Russian).
Sibatov R. T., Uchaikin V. V. Usp. fiz. nauk, 2009, 179, No 10: 1079–1104 (in Russian).
Sokolov I. M. Soft Matter, 2012, 42, No 8: 9043–9052. https://doi.org/10.1039/c2sm25701g
Bazhlekova E. Fractional evolution equations in Banach spaces, PhD Thesis., Eindhoven Univ. of Technology, 2001.
Ford N., Xiao J., Yan Y. Fract. Calc. Appl. An., 2011, 14, No 3: 454–474. https://doi.org/10.2478/s13540-011-0028-2
Jin B., Lazarov R., Pasciak J., Zhou Z. Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion, IMA J. Numer Anal., to appear., doi:10.1093/imanum/dru018. https://doi.org/10.1093/imanum/dru018
Chikrii A. A., Matichin I. I. Probl. upravleniia i informatiki, 2008, No 3: 133–142.
Alikhanov A. A. Dif. Equations, 2010, 46, No 5: 660–666. https://doi.org/10.1134/S0012266110050058
Evans L. C. Partial differential equations, Providence: Amer. Math. Soc., 1998.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Reports of the National Academy of Sciences of Ukraine
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
