The persistence of logconcavity for positive solutions of the one dimensional heat equation

AbstractConsider positive solutions of the one dimensional heat equation. The space variable x lies in (–a, a): the time variable t in (0,∞). When the solution u satisfies (i) u (±a, t) = 0, and (ii) u(·, 0) is logconcave, we give a new proof based on the Maximum Principle, that, for any fixed t > 0, u(·, t) remains logconcave. The same proof techniques are used to establish several new results related to this, including results concerning joint concavity in (x, t) similar to those considered in Kennington [15].

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Discrete Maximum Principle in One-Dimensional Heat Equation

Journal of Advanced College of Engineering and Management ◽

10.3126/jacem.v2i0.16093 ◽

2016 ◽

Vol 2 ◽

pp. 5

Author(s):

Durga Jang K.C. ◽

Ganesh Bahadur Basnet

Keyword(s):

Maximum Principle ◽

Heat Equation ◽

Discrete Maximum Principle ◽

Finite Difference Schemes ◽

Discrete Version ◽

One Dimensional ◽

The Maximum Principle ◽

Linear Partial Differential Equations ◽

The One ◽

College Of Engineering

The maximum principle plays key role in the theory and application of a wide class of real linear partial differential equations. In this paper, we introduce ‘Maximum principle and its discrete version’ for the study of second-order parabolic equations, especially for the one-dimensional heat equation. We also give a short introduction of formation of grid as well as finite difference schemes and a short prove of the ‘Discrete Maximum principle’ by using different schemes of heat equation.Journal of Advanced College of Engineering and Management, Vol. 2, 2016, Page: 5-10

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Stress-driven diffusion in a drying liquid paint layer

European Journal of Applied Mathematics ◽

10.1017/s0956792598003532 ◽

1998 ◽

Vol 9 (5) ◽

pp. 447-461 ◽

Cited By ~ 4

Author(s):

B. W. VAN DE FLIERT ◽

R. VAN DER HOUT

Keyword(s):

Maximum Principle ◽

Diffusion Process ◽

Viscoelastic Material ◽

Polymer Network ◽

Paint Layer ◽

Dimensional Model ◽

One Dimensional ◽

The Maximum Principle ◽

The One ◽

Liquid Paint

A model is presented for the diffusion-driven drying of a polymeric solution such as liquid paint. Included is a stress build-up and relaxation in the polymer network of the viscoelastic material, which influences the diffusion process. The behaviour of the (one-dimensional) model is analysed by means of the maximum principle and illustrated with numerical calculations.

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The Maximum Principle Approach to the Optimum One-Dimensional Journal Bearing

Journal of Lubrication Technology ◽

10.1115/1.3451447 ◽

1970 ◽

Vol 92 (3) ◽

pp. 482-487 ◽

Cited By ~ 20

Author(s):

C. J. Maday

Keyword(s):

Maximum Principle ◽

Film Thickness ◽

Journal Bearing ◽

Maximum Load ◽

Load Direction ◽

One Dimensional ◽

The Maximum Principle ◽

Minimum Film Thickness ◽

Constant Viscosity ◽

The One

Pontryagin’s Maximum Principle is used to determine the journal bearing which supports the maximum load for a given minimum film thickness and a specified load direction. The one-dimensional configuration which uses a constant-viscosity, incompressible lubricant is considered. Comparison shows that the optimum bearing carries a load about 13.5 percent greater than the maximum carried by the usual full-Sommerfeld bearing and about 121 percent greater than that carried by the half-Sommerfeld unit. The problem is formulated subject to the constraints of a fixed load direction and a specified minimum film thickness while the only boundary condition imposed is that the pressure must vanish at the inlet and at the outlet. The actual extent of the bearing is determined in the optimization process and it is shown that this extent is 360 deg. Further, the bearing is stepped with only two regions of different but constant film thickness.

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