On a class of nonlinear nonclassical parabolic equations

Cannon, John R.; Salman, Mohamed

doi:10.1080/00036810500277876

Cited by 12 publications

(9 citation statements)

References 11 publications

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“…Problems of this type (17) were previously mentioned in [23] and considered in [24], but afterwards they have been somewhat overlooked in the literature.…”

Section: Discussion On the Uniqueness Of Solutionmentioning

confidence: 99%

“…The differentiability in the φ and f components follows immediately from applying Theorem 1 and the estimate (6) to the initial boundary value problems (24) and (25), which imply that they have the unique solutions ∆u φ ∈ H 1,0 (Q) and ∆u f ∈ H 1,0 (Q) and that the mappings L 2 (Ω) ∋ ∆φ → ∆u φ ∈ H 1,0 (Q) and L 2 (Ω) ∋ ∆f → ∆u f ∈ H 1,0 (Q) define the bounded linear operators U φ and U f , which, by definition, they satisfy U φ ∆φ = ∆u φ = u(q, φ + ∆φ, f ) − u(q, φ, f ) and…”

Section: Variational Formulationmentioning

confidence: 96%

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Simultaneous reconstruction of the spatially-distributed reaction coefficient, initial temperature and heat source from temperature measurements at different times

Cao

Lesnic

2019

Computers & Mathematics with Applications

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“…Problems of this type (17) were previously mentioned in [23] and considered in [24], but afterwards they have been somewhat overlooked in the literature.…”

Section: Discussion On the Uniqueness Of Solutionmentioning

confidence: 99%

Section: Variational Formulationmentioning

confidence: 96%

Simultaneous reconstruction of the spatially-distributed reaction coefficient, initial temperature and heat source from temperature measurements at different times

Cao

Lesnic

2019

Computers & Mathematics with Applications

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“…Completion of the proof of Theorem 1.2 in this case: Our argument for long time existence in this case is simplified by axial symmetry, which reduces our problem to the setting of a scalar parabolic PDE with one spatial direction. While there are various results particular to such parabolic PDEs (see, for example, [14,24] and the references therein), here we use an argument more closely related to that in the previous section; when we fix time the resulting evolution equation is an ODE. Specifically, let us parametrise the evolving hypersurface as a radial graph by X :…”

Section: Proof Of Propositionmentioning

confidence: 99%

More Mixed Volume Preserving Curvature Flows

McCoy

2017

J Geom Anal

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We extend the results of McCoy (Calc Var Partial Differ Equ 24:131-154, 2005) to include several new cases where convex surfaces evolve to spheres under mixed volume preserving curvature flows, using recent results for unconstrained curvature flows and new regularity arguments in the constrained flow setting. We include results for speeds that are degree 1 homogeneous in the principal curvatures and indicate how, with sufficient curvature pinching conditions on the initial hypersurfaces, some results may be extended to speed homogeneous of degree α>1. In particular, these extensions require lower speed bounds that are obtained here without using estimates for equations of porous medium type, in contrast to most previous work.ABSTRACT. We extend the results of [28] to include several new cases where convex surfaces evolve to spheres under mixed volume preserving curvature flows, using recent results for unconstrained curvature flows and new regularity arguments in the constrained flow setting. We include results for speeds that are degree 1 homogeneous in the principal curvatures and indicate how, with sufficient curvature pinching conditions on the initial hypersurfaces, some results may be extended to speeds homogeneous of degree α > 1. In particular, these extensions require lower speed bounds that are obtained here without using estimates for equations of porous medium type, in contrast to most previous work.

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“…[1,6,7,16,17,28]. We name (1) by nonclassical parabolic equation after [16,17,26,28,30] since the term u xxt is involved.…”

Section: Introductionmentioning

confidence: 99%

Error Analysis of Chebyshev-Legendre Pseudo-spectral Method for a Class of Nonclassical Parabolic Equation

Zhao

2011

J Sci Comput

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Many physical phenomena are modeled by nonclassical parabolic initial boundary value problems which involve a nonclassical term u xxt in the governed equation. Combining with the Crank-Nicolson/leapfrog scheme in time discretization, ChebyshevLegendre pseudo-spectral method is applied to space discretization for numerically solving the nonclassical parabolic equation. The proposed approach is based on Legendre Galerkin formulation while the Chebyshev-Gauss-Lobatto (CGL) nodes are used in the computation. By using the proposed method, the computational complexity is reduced and both accuracy and efficiency are achieved. The stability and convergence are rigorously set up. The convergence rate shows 'spectral accuracy'. Numerical experiments are presented to demonstrate the effectiveness of the method and to confirm the theoretical results.

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On a class of nonlinear nonclassical parabolic equations

Cited by 12 publications

References 11 publications

Simultaneous reconstruction of the spatially-distributed reaction coefficient, initial temperature and heat source from temperature measurements at different times

Simultaneous reconstruction of the spatially-distributed reaction coefficient, initial temperature and heat source from temperature measurements at different times

More Mixed Volume Preserving Curvature Flows

Error Analysis of Chebyshev-Legendre Pseudo-spectral Method for a Class of Nonclassical Parabolic Equation

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