249 research outputs found
Value Functions for Bolza Problems with Discontinuous Lagrangians and Hamilton-Jacobi Inequalities
We investigate the value function of the Bolza problem of the Calculus of
Variations with a lower semicontinuous Lagrangian and a
final cost , and show that it is locally Lipschitz for whenever
is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a
generalized sense.
When the Lagrangian is continuous, then the value function is the unique
lower semicontinuous solution to the corresponding Hamilton-Jacobi equation,
while for discontinuous Lagrangian we characterize the value function by using
the so called contingent inequalities.Comment: 33 pages. Control, Optimization and Calculus of Variations, to appea
A moving mesh method with variable relaxation time
We propose a moving mesh adaptive approach for solving time-dependent partial
differential equations. The motion of spatial grid points is governed by a
moving mesh PDE (MMPDE) in which a mesh relaxation time \tau is employed as a
regularization parameter. Previously reported results on MMPDEs have invariably
employed a constant value of the parameter \tau. We extend this standard
approach by incorporating a variable relaxation time that is calculated
adaptively alongside the solution in order to regularize the mesh appropriately
throughout a computation. We focus on singular problems involving self-similar
blow-up to demonstrate the advantages of using a variable relaxation ime over a
fixed one in terms of accuracy, stability and efficiency.Comment: 21 page
Numerical analysis of a nonlocal parabolic problem resulting from thermistor problem
We analyze the spatially semidiscrete piecewise linear finite element method
for a nonlocal parabolic equation resulting from thermistor problem. Our
approach is based on the properties of the elliptic projection defined by the
bilinear form associated with the variational formulation of the finite element
method. We assume minimal regularity of the exact solution that yields optimal
order error estimate. The full discrete backward Euler method and the
Crank-Nicolson-Galerkin scheme are also considered. Finally, a simple algorithm
for solving the fully discrete problem is proposed
Some results on blow up for semilinear parabolic problems
The authors describe the asymptotic behavior of blow-up for the semilinear heat equation ut=uxx+f(u) in R×(0,T), with initial data u0(x)>0 in R, where f(u)=up, p>1, or f(u)=eu. A complete description of the types of blow-up patterns and of the corresponding blow-up final-time profiles is given. In the rescaled variables, both are governed by the structure of the Hermite polynomials H2m(y). The H2-behavior is shown to be stable and generic. The existence of H4-behavior is proved. A nontrivial blow-up pattern with a blow-up set of nonzero measure is constructed. Similar results for the absorption equation ut=uxx−up, 0<p<1, are discussed
Explosive solutions of a stochastic non-local reaction–diffusion equation arising in shear band formation
This is the peer reviewed version of the following article: Kavallaris, N. I. (2015). Explosive solutions of a stochastic non-local reaction–diffusion equation arising in shear band formation. Mathematical Methods in the Applied Sciences 38(16): 3564-3574, which has been published in final form at DOI: 10.1002/mma.3514. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.In this paper, we consider a non-local stochastic parabolic equation which actually serves as a mathematical model describing the adiabatic shear-banding formation phenomena in strained metals. We first present the derivation of the mathematical model. Then we investigate under which circumstances a finite-time explosion for this non-local SPDE, corresponding to shear-banding formation, occurs. For that purpose some results related to the maximum principle for this non-local SPDE are derived and afterwards the Kaplan's eigenfunction method is employed
A time discretization scheme for a nonlocal degenerate problem modelling resistance spot welding
This is the author's PDF version of an article published in Mathematical Modelling of Natural Phenomena© 2015. The definitive version is available at http://www.mmnp-journal.org/articles/mmnp/abs/2015/06/mmnp2015106p90/mmnp2015106p90.htmlIn the current work we construct a nonlocal mathematical model describing the phase transition occurs during the resistance spot welding process in the industry of metallurgy. We then consider a time discretization scheme for solving the resulting nonlocal moving boundary problem. The scheme consists of solving at each time step a linear elliptic partial differential equation and then making a correction to account for the nonlinearity. The stability and error estimates of the developed scheme are investigated. Finally some numerical results are presented confirming the efficiency of the developed numerical algorithm
Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach
Five types of blow-up patterns that can occur for the 4th-order semilinear
parabolic equation of reaction-diffusion type
u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1,
\quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For
the semilinear heat equation , various blow-up patterns
were under scrutiny since 1980s, while the case of higher-order diffusion was
studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure
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