Asymptotics for large time of solutions to nonlinear system associated with the penetration of a magnetic field into a substance

Jangveladze, Temur; Kiguradze, Zurab

doi:10.1007/s10492-010-0019-3

Cited by 8 publications

(6 citation statements)

References 19 publications

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“…In all these papers, it has been shown that the solution decays exponentially at a rate O e −t/2 . For theoretical results like existence, uniqueness and asymptotic behavior of a related model (1.3) with H = (0, 0, u) , we refer to [2,[8][9][10][11][12][13][14][15][16][17].…”

Section: A the Mathematical Modelmentioning

confidence: 99%

“…Earlier, the study of the has been initiated in , where the author has discussed the existence of a unique weak solution along with its asymptotic behavior for one dimensional version of the problem with

a (s) = {(1 + s)}^{p}, 0 < p \leq 1

and

f = 0.

When

a (s) = {(1 + s)}^{p},

for

- 1 / 2 < p < 0

and

0 < p \leq 1

and

Ω = (0, 1),

the results on stabilization have been derived in . In all these papers, it has been shown that the solution decays exponentially at a rate

O (e^{- t / 2}) .

For theoretical results like existence, uniqueness and asymptotic behavior of a related model with

H = (0, 0, u),

we refer to .…”

Section: Introductionmentioning

confidence: 99%

“…For numerical approximations of the problem and some related problems in one space dimension with

a (s) = (1 + s),

finite difference schemes have been investigated in . In all these articles, the authors have considered first order difference scheme in time, but second order central difference scheme in space and derived order of convergence

O (h + k),

where

h

and

k,

respectively are spatial and temporal discretizing parameters.…”

Section: Introductionmentioning

confidence: 99%

“…In all these articles, the authors have considered first order difference scheme in time, but second order central difference scheme in space and derived order of convergence

O (h + k),

where

h

and

k,

respectively are spatial and temporal discretizing parameters. The numerical approximations based on finite difference schemes for a related problem in one space dimension have been studied in . Moreover, for application of numerical schemes for a class of nonlinear integro‐differential equations, we refer to .…”

Section: Introductionmentioning

confidence: 99%

“…For the study of exponential decay properties of the exact as well as discrete solutions of the parabolic integro‐differential equations, in particular for Oldroyd model and Kelvin–Voigt model, we refer to . When

f = 0,

the results provide improved exponential decay compared with the decay property shown in for the problem in one space variable with a special case

a (s) = {(1 + s)}^{p}, 0 < p \leq 1.

Applying Galerkin method in spatial direction while keeping time variable as continuous, a semidiscrete method is analyzed. Compared with semidiscrete scheme applied to a 1D problem in , we not only discuss a general

a (s)

, but also derive optimal convergence rate with respect to

L^{2} (H^{1})

and

L^{\infty} (H^{1})

‐norms.…”

Section: Introductionmentioning

confidence: 99%

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Finite element Galerkin approximations to a class of nonlinear and nonlocal parabolic problems

Sharma

Khebchareon

Sharma

et al. 2016

Numerical Methods Partial

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In this article, a finite element Galerkin method is applied to a general class of nonlinear and nonlocal parabolic problems. Based on an exponential weight function, new a priori bounds which are valid for uniform in time are derived. As a result, existence of an attractor is proved for the problem with nonhomogeneous right hand side which is independent of time. In particular, when the forcing function is zero or decays exponentially, it is shown that solution has exponential decay property which improves even earlier results in one dimensional problems. For the semidiscrete method, global existence of a unique discrete solution is derived and it is shown that the discrete problem has an attractor. Moreover, optimal error estimates are derived in both L 2 ( H 0 1 ( Ω ) ) and L ∞ ( H 0 1 ( Ω ) ) ‐norms with later estimate is a new result in this context. For completely discrete scheme, backward Euler method with its linearized version is discussed and existence of a unique discrete solution is established. Further, optimal estimates in ℓ 2 ( H 0 1 ( Ω ) ) ‐norm are proved for fully discrete schemes. Finally, several numerical experiments are conducted to confirm our theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1232–1264, 2016

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Section: A the Mathematical Modelmentioning

confidence: 99%

a (s) = {(1 + s)}^{p}, 0 < p \leq 1

and

f = 0.

When

a (s) = {(1 + s)}^{p},

for

- 1 / 2 < p < 0

and

0 < p \leq 1

and

Ω = (0, 1),

the results on stabilization have been derived in . In all these papers, it has been shown that the solution decays exponentially at a rate

O (e^{- t / 2}) .

For theoretical results like existence, uniqueness and asymptotic behavior of a related model with

H = (0, 0, u),

we refer to .…”

Section: Introductionmentioning

confidence: 99%

“…For numerical approximations of the problem and some related problems in one space dimension with

a (s) = (1 + s),

O (h + k),

where

h

and

k,

respectively are spatial and temporal discretizing parameters.…”

Section: Introductionmentioning

confidence: 99%

“…In all these articles, the authors have considered first order difference scheme in time, but second order central difference scheme in space and derived order of convergence