Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form

Broadbridge, Philip

doi:10.3390/e10030365

Cited by 18 publications

(27 citation statements)

References 27 publications

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“…In some modelling applications, such as when u is a probability density or a temperature, increasing entropy is a constraint of the validity of a physical solution. This is another feature that we have come to expect from generic behaviour of second-order diffusion equations but which is not always valid for fourth-order equations [9,18]. There are indeed many cases of dissipative fourth order equations that satisfy neither increasing entropy nor maintenance of positivity [12].…”

Section: Introductionmentioning

confidence: 96%

Fourth Order Diffusion Equations with Increasing Entropy

Tehseen

Broadbridge

2012

Entropy

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The general quasi-linear autonomous fourth order diffusion equation ut = −[G(u)uxxx + h(u, ux, uxx)]x with positive variable diffusivity G(u) and lower-order flux component h is considered on the real line. A direct algorithm produces a general class of equations for which the Shannon entropy density obeys a reaction-diffusion equation with a positive irreducible source term. Such equations may have any positive twice-differentiable diffusivity function G(u). The forms of such equations are the indicators of more general conservation equations whose entropy equation may be expressed in an alternative reaction-diffusion form whose source term, although reducible, is positive

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Section: Introductionmentioning

confidence: 96%

Fourth Order Diffusion Equations with Increasing Entropy

Tehseen

Broadbridge

2012

Entropy

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“…Unlike standard second-order diffusion problems, fourth-order diffusion problems do not in general obey a maximum principle and they do not have increasing Shannon information [20]. Solutions θ(x, t) to standard fourth-order diffusion problems are not one-to-one functions of x (e.g.…”

Section: Resultsmentioning

confidence: 99%

Temperature-dependent surface diffusion near a grain boundary

Broadbridge

Goard

2009

J Eng Math

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Metal surface evolution is described by a nonlinear fourth-order partial differential equation for curvature-driven flow. The standard boundary conditions for grain-boundary grooving, at a grain-grain-fluid triple intersection, involve a prescribed slope at the groove axis. The well-known similarity reduction is no longer valid when the dihedral angle and surface diffusivity depend on time due to variation of the surface temperature. We adapt a nonlinear fourth-order model that can be discerned from symmetry analysis to be integrable, equivalent to the fourth-order linear diffusion equation. The connection between classical symmetries and separation of variables allows us to develop the correction to the self-similar approximation as a power series in a time-like variable.

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“…Examples are molecular beam epitaxy (Barabási and Stanley 1995) and surface coating with viscous fluids (Schwartz and Roy 2004, Myers 1998, Mullins 1957. Fourth order term also appears in the partial differential equation for the determination of the entropy evolution connected to the Shannon information density (Broadbridge 2008).…”

Section: Discussionmentioning

confidence: 99%

A new analytical formulation of retention effects on particle diffusion processes

Bevilácqua

Galeão

Costa

2011

An. Acad. Bras. Ciênc.

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The ultimate purpose of this paper is to present a new analytical formulation to simulate diffusion with retention in a reactive medium under stable thermodynamic conditions. The analysis of diffusion with retention in a continuum medium is developed after the solution of an equivalent problem using a discrete approach. The new law may be interpreted as the reduction of all diffusion processes with retention to a unifying phenomenon that can adequately simulate the retention effect namely a circulatory motion. It is remarkable that the governing equation requires a fourth order differential term as suggested by the discrete approach. The relative fraction of diffusion particles β is introduced as a control parameter in the diffusion-retention law as suggested by the discrete approach. This control parameter is essential to avoid retention isolated from the diffusion process. Two matrices referring to material properties are introduced and related to the real phenomenon through the circulation hypothesis. The governing equation may be highly non-linear even if the material properties are constant, but the retention effect is a function of the concentration level, that is, β is a function of the concentration.

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Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form

Cited by 18 publications

References 27 publications

Fourth Order Diffusion Equations with Increasing Entropy

Fourth Order Diffusion Equations with Increasing Entropy

Temperature-dependent surface diffusion near a grain boundary

A new analytical formulation of retention effects on particle diffusion processes

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