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Fully discrete Galerkin scheme for a semilinear subdiffusion equation with nonsmooth data and time-dependent coefficient (2310.04246v1)
Published 6 Oct 2023 in math.NA and cs.NA
Abstract: We couple the L1 discretization of the Caputo fractional derivative in time with the Galerkin scheme to devise a linear numerical method for the semilinear subdiffusion equation. Two important points that we make are: nonsmooth initial data and time-dependent diffusion coefficient. We prove the stability and convergence of the method under weak assumptions concerning regularity of the diffusivity. We find optimal pointwise in space and global in time errors, which are verified with several numerical experiments.
- “The random walk’s guide to anomalous diffusion: a fractional dynamics approach” In Physics reports 339.1 Elsevier, 2000, pp. 1–77
- Łukasz Płociniczak “Analytical studies of a time-fractional porous medium equation. Derivation, approximation and applications” In Communications in Nonlinear Science and Numerical Simulation 24.1-3 Elsevier, 2015, pp. 169–183
- “Transient superdiffusion and long-range correlations in the motility patterns of trypanosomatid flagellate protozoa” In PLoS One 11.3 Public Library of Science San Francisco, CA USA, 2016, pp. e0152092
- “Single-molecule imaging reveals receptor–G protein interactions at cell surface hot spots” In Nature 550.7677 Nature Publishing Group, 2017, pp. 543
- “Anomalous diffusion probes microstructure dynamics of entangled F-actin networks” In Physical review letters 92.17 APS, 2004, pp. 178101
- Diego del-Castillo-Negrete, BA Carreras and VE Lynch “Nondiffusive transport in plasma turbulence: a fractional diffusion approach” In Physical Review Letters 94.6 APS, 2005, pp. 065003
- “Anomalous diffusion of magnetic elements across the solar surface” In Astrophysical Journal, Part 1 (ISSN 0004-637X), vol. 411, no. 1, p. 402-405. 411, 1993, pp. 402–405
- Ronald L Bagley and PJ Torvik “A theoretical basis for the application of fractional calculus to viscoelasticity” In Journal of Rheology 27.3 The Society of Rheology, 1983, pp. 201–210
- Łukasz Płociniczak “Error of the Galerkin scheme for a semilinear subdiffusion equation with time-dependent coefficients and nonsmooth data” In Computers & Mathematics with Applications 127, 2022, pp. 181–191 DOI: https://doi.org/10.1016/j.camwa.2022.09.028
- Keith B Oldham and Jerome Spanier “The Fractional Calculus, vol. 111 of Mathematics in science and engineering” Academic Press, New York, London, 1974
- Łukasz Płociniczak “A linear Galerkin numerical method for a quasilinear subdiffusion equation” In Applied Numerical Mathematics 185 Elsevier BV, 2023, pp. 203–220 DOI: 10.1016/j.apnum.2022.11.020
- Martin Stynes, Eugene O’Riordan and Jose Luis Gracia “Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation” In SIAM Journal on Numerical Analysis 55, 2016 DOI: 10.1137/16M1082329
- Natalia Kopteva “Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions” In Mathematics of Computation 88.319, 2019, pp. 2135–2155
- Bangti Jin, Buyang Li and Zhi Zhou “Subdiffusion with a time-dependent coefficient: analysis and numerical solution” In Mathematics of Computation 88.319, 2019, pp. 2157–2186
- Bangti Jin, Buyang Li and Zhi Zhou “Numerical analysis of nonlinear subdiffusion equations” In SIAM Journal on Numerical Analysis 56.1 SIAM, 2018, pp. 1–23
- Martin Stynes “A Survey of the L1 Scheme in the Discretisation of Time-Fractional Problems.” In Numerical Mathematics: Theory, Methods & Applications 15.4, 2022
- Bangti Jin, Buyang Li and Zhi Zhou “Subdiffusion with time-dependent coefficients: improved regularity and second-order time stepping” In Numerische Mathematik 145.4 Springer, 2020, pp. 883–913
- Kassem Mustapha “FEM for time-fractional diffusion equations, novel optimal error analyses” In Mathematics of Computation 87.313, 2018, pp. 2259–2272
- Vidar Thomée “Galerkin finite element methods for parabolic problems” Springer Science & Business Media, 2007
- Hong-lin Liao, Dongfang Li and Jiwei Zhang “Sharp Error Estimate of the Nonuniform L1 Formula for Linear Reaction-Subdiffusion Equations” In SIAM Journal on Numerical Analysis 56.2, 2018, pp. 1112–1133 DOI: 10.1137/17M1131829
- “Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems” In Journal of Mathematical Analysis and Applications 382.1 Elsevier, 2011, pp. 426–447
- “Numerical approximation of semilinear subdiffusion equations with nonsmooth initial data” In SIAM Journal on Numerical Analysis 57.3 SIAM, 2019, pp. 1524–1544
- “Existence and uniqueness for parabolic problems with Caputo time derivative” In Journal of Differential Equations 262.12 Elsevier, 2017, pp. 6018–6046