A modified quasi-boundary value method for the backward time-fractional diffusion problem

AbstractWe consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.

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Finite Difference Method on Non-Uniform Meshes for Time Fractional Diffusion Problem

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2020-0077 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Haili Qiao ◽

Aijie Cheng

Keyword(s):

Theoretical Analysis ◽

Fractional Derivative ◽

Time Discretization ◽

Summation Formula ◽

Fractional Diffusion ◽

Diffusion Problem ◽

Initial Moment ◽

Central Difference ◽

Ideal Convergence ◽

Derivative Term

AbstractIn this paper, we consider the time fractional diffusion equation with Caputo fractional derivative. Due to the singularity of the solution at the initial moment, it is difficult to achieve an ideal convergence rate when the time discretization is performed on uniform meshes. Therefore, in order to improve the convergence order, the Caputo time fractional derivative term is discretized by the {L2-1_{\sigma}} format on non-uniform meshes, with {\sigma=1-\frac{\alpha}{2}}, while the spatial derivative term is approximated by the classical central difference scheme on uniform meshes. According to the summation formula of positive integer k power, and considering {k=3,4,5}, we propose three non-uniform meshes for time discretization. Through theoretical analysis, different time convergence orders {O(N^{-\min\{k\alpha,2\}})} can be obtained, where N denotes the number of time splits. Finally, the theoretical analysis is verified by several numerical examples.

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A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems

Boundary Value Problems ◽

10.1155/bvp/2006/37524 ◽

2006 ◽

Vol 2006 ◽

pp. 1-8 ◽

Cited By ~ 4

Author(s):

M. Denche ◽

S. Djezzar

Keyword(s):

Boundary Value ◽

Boundary Value Method ◽

Ill Posed

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

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-012-0010-7 ◽

2012 ◽

Vol 15 (1) ◽

Cited By ~ 62

Author(s):

Yuri Luchko

Keyword(s):

Diffusion Equation ◽

Boundary Value Problems ◽

Boundary Value ◽

Initial Boundary ◽

Fractional Diffusion ◽

Fractional Diffusion Equation ◽

Initial Boundary Value Problems ◽

The Maximum Principle ◽

Time Fractional Diffusion Equation ◽

Initial Boundary Value

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.

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