Laplace Transform Method for Parabolic Problems with Time-Dependent Coefficients

Lee, Hyoseop; Lee, Jun Young; Sheen, Dongwoo

doi:10.1137/110824000

Cited by 15 publications

(6 citation statements)

References 23 publications

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“…There are a few relevant works on standard parabolic problems with a time-dependent coefficient [24,30,31,19]. For example, Luskin and Rannacher [24] proved optimal order error estimates for both spatially semidiscrete and fully discrete problems (by BE method) using a novel energy argument, and Sammon [30] analyzed fully discrete schemes with linear multistep methods.…”

Section: Introductionmentioning

confidence: 99%

Subdiffusion with a time-dependent coefficient: Analysis and numerical solution

Jin

Zhou

2019

Math. Comp.

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In this work, a complete error analysis is presented for fully discrete solutions of the subdiffusion equation with a time-dependent diffusion coefficient, obtained by the Galerkin finite element method with conforming piecewise linear finite elements in space and backward Euler convolution quadrature in time. The regularity of the solutions of the subdiffusion model is proved for both nonsmooth initial data and incompatible source term. Optimal-order convergence of the numerical solutions is established using the proven solution regularity and a novel perturbation argument via freezing the diffusion coefficient at a fixed time. The analysis is supported by numerical experiments.2010 Mathematics Subject Classification. Primary: 65M30, 65M15, 65M12.

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Section: Introductionmentioning

confidence: 99%

Subdiffusion with a time-dependent coefficient: Analysis and numerical solution

Jin

Zhou

2019

Math. Comp.

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“…Finally, time-dependent coefficients pose a challenging problem since taking the Laplace transform leads to a convolution. There are works that deal with this case [118], and it would be interesting to combine them with the techniques of this paper.…”

Section: Discussionmentioning

confidence: 99%

A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations

Colbrook,

Ayton

2021

Preprint

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We develop a rapid and accurate contour method for the solution of time-fractional PDEs. The method inverts the Laplace transform via an optimised stable quadrature rule, suitable for infinitedimensional operators, whose error decreases like exp(−cN/ log(N )) for N quadrature points. The method is parallisable, avoids having to resolve singularities of the solution as t ↓ 0, and avoids the large memory consumption that can be a challenge for time-stepping methods applied to time-fractional PDEs. The ODEs resulting from quadrature are solved using adaptive sparse spectral methods that converge exponentially with optimal linear complexity. These solutions of ODEs are reused for different times. We provide a complete analysis of our approach for fractional beam equations used to model small-amplitude vibration of viscoelastic materials with a fractional Kelvin-Voigt stress-strain relationship. We calculate the system's energy evolution over time and the surface deformation in cases of both constant and non-constant viscoelastic parameters. An infinite-dimensional "solve-then-discretise" approach considerably simplifies the analysis, which studies the generalisation of the numerical range of a quasi-linearisation of a suitable operator pencil. This allows us to build an efficient algorithm with explicit error control. The approach can be readily adapted to other time-fractional PDEs and is not constrained to fractional parameters in the range 0 < ν < 1.

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“…There have been many works on the theoretical and numerical study of classical parabolic and hyperbolic equations with general time-space dependent elliptic operator, e.g. [2,11,18]. To the best of our knowledge, taking the weak initial singularity into account, there is no study on the efficient numerical methods for time fractional evolution equations (sub-diffusion and diffusion-wave) where the elliptic operators include general time-space dependent coefficients, i.e., the elliptic operators take the form (1.4).…”

Section: Introductionmentioning

confidence: 99%

Second-order and nonuniform time-stepping schemes for time fractional evolution equations with time-space dependent coefficients

Lyu¹,

Vong²

2021

Preprint

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The numerical analysis of time fractional evolution equations with the second-order elliptic operator including general time-space dependent variable coefficients is challenging, especially when the classical weak initial singularities are taken into account. In this paper, we introduce a concise technique to construct efficient time-stepping schemes with variable time step sizes for two-dimensional time fractional sub-diffusion and diffusion-wave equations with general time-space dependent variable coefficients. By means of the novel technique, the nonuniform Alikhanov type schemes are constructed and analyzed for the sub-diffusion and diffusion-wave problems. For the diffusion-wave problem, our scheme is constructed by employing the recently established symmetric fractional-order reduction (SFOR) method. The unconditional stability of proposed schemes is rigorously discussed under mild assumptions on variable coefficients and, based on reasonable regularity assumptions and weak time mesh restrictions, the second-order convergence is obtained with respect to discrete H 1 -norm. Numerical experiments are given to demonstrate the theoretical statements.

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Laplace Transform Method for Parabolic Problems with Time-Dependent Coefficients

Cited by 15 publications

References 23 publications

Subdiffusion with a time-dependent coefficient: Analysis and numerical solution

Subdiffusion with a time-dependent coefficient: Analysis and numerical solution

A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations

Second-order and nonuniform time-stepping schemes for time fractional evolution equations with time-space dependent coefficients

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