3,212 research outputs found
Diffusion and wave behaviour in linear Voigt model
A boundary value problem related to a third- order parabolic equation with a
small parameter is analized. This equation models the one-dimensional evolution
of many dissipative media as viscoelastic fluids or solids, viscous gases,
superconducting materials, incompressible and electrically conducting fluids.
Moreover, the third-order parabolic operator regularizes various non linear
second order wave equations. In this paper, the hyperbolic and parabolic
behaviour of the solution is estimated by means of slow time and fast time. As
consequence, a rigorous asymptotic approximation for the solution is
established
Mathematical Contributions to the Dynamics of the Josephson Junctions: State of the Art and Open Problems
Mathematical models related to some Josephson junctions are pointed out and
attention is drawn to the solutions of certain initial boundary problems and to
some of their estimates. In addition, results of rigorous analysis of the
behaviour of these solutions when the time tends to infinity and when the small
parameter tends to zero are cited. These analyses lead us to mention some of
the open problems.Comment: 11 page
The effect on Fisher-KPP propagation in a cylinder with fast diffusion on the boundary
In this paper we consider a reaction-diffusion equation of Fisher-KPP type
inside an infinite cylindrical domain in , coupled with a
reaction-diffusion equation on the boundary of the domain, where potentially
fast diffusion is allowed. We will study the existence of an asymptotic speed
of propagation for solutions of the Cauchy problem associated with such system,
as well as the dependence of this speed on the diffusivity at the boundary and
the amplitude of the cylinder.
When the domain reduces to a strip between two straight lines. This
models the effect of two roads with fast diffusion on a strip-shaped field
bounded by them.Comment: 31 pages, 3 figure
On asymptotic effects of boundary perturbations in exponentially shaped Josephson junctions
A parabolic integro differential operator L, suitable to describe many
phenomena in various physical fields, is considered. By means of equivalence
between L and the third order equation describing the evolution inside an
exponentially shaped Josephson junction (ESJJ), an asymptotic analysis for
(ESJJ) is achieved, explicitly evaluating, boundary contributions related to
the Dirichlet problem
Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries
In this work we study the behavior of a family of solutions of a semilinear
elliptic equation, with homogeneous Neumann boundary condition, posed in a
two-dimensional oscillating thin region with reaction terms concentrated in a
neighborhood of the oscillatory boundary. Our main result is concerned with the
upper and lower semicontinuity of the set of solutions. We show that the
solutions of our perturbed equation can be approximated with ones of a
one-dimensional equation, which also captures the effects of all relevant
physical processes that take place in the original problem
Error Estimates for Approximations of Distributed Order Time Fractional Diffusion with Nonsmooth Data
In this work, we consider the numerical solution of an initial boundary value
problem for the distributed order time fractional diffusion equation. The model
arises in the mathematical modeling of ultra-slow diffusion processes observed
in some physical problems, whose solution decays only logarithmically as the
time tends to infinity. We develop a space semidiscrete scheme based on the
standard Galerkin finite element method, and establish error estimates optimal
with respect to data regularity in and norms for both smooth
and nonsmooth initial data. Further, we propose two fully discrete schemes,
based on the Laplace transform and convolution quadrature generated by the
backward Euler method, respectively, and provide optimal convergence rates in
the norm, which exhibits exponential convergence and first-order
convergence in time, respectively. Extensive numerical experiments are provided
to verify the error estimates for both smooth and nonsmooth initial data, and
to examine the asymptotic behavior of the solution.Comment: 25 pages, 2 figure
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
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