Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

By Yuri Luchko. 2011

194 citations recorded by the source corpus.

AbstractIn this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in the sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the onedimensional time-fractional diffusion equation $$(D_t^\alpha u)(t) = \frac{\partial } {{\partial x}}\left( {p(x)\frac{{\partial u}} {{\partial x}}} \right) - q(x)u + F(x,t), x \in (0,l), t \in (0,T)$$ the generalized solution to the initial-boundary-value problem with Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of this solution are investigated including its smoothness and asymptotics for some special cases of the source function.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/6dee48c1dda29712027df5ac057a7c0d5a16b185

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Citations in the corpus

Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications (2014, background): “The proof of the fact that the series (37) is indeed a solution of the initialboundary-value problem (22)-(23) for the one-dimensional time-fractional diffusion equation (21) closely follows the lines of the proof presented in [13] for the case of the one-dimensional fractional diffusion equation wi”
Maximum Principles for Multi-Term Space-Time Variable-Order Fractional Diffusion Equations and their Applications (2016, methodology): “By using of the variables separation and the Fourier method, in [12] Luchko obtained a formal solution in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem.”
Time-fractional diffusion equation in the fractional Sobolev spaces (2015, background): “We mention here also the papers [3], [14], [16]-[19], [25], where some recent developments regarding the partial fractional differential equations are presented.”
A variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures (2012, methodology): “In particular, the time-fractional derivative has been considered a powerful tool to model sub-diffusion processes [41, 27, 15].”
Numerical solution of time fractional diffusion systems (2017, background): “Some results of existence and uniqueness of solution can be found in [27, 28], where the analytical solution is also expressed in form of Fourier series.”