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In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter
dependence to problems involving (a) nonaffine dependence on the
parameter, and (b) nonlinear dependence on the field variable.
The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational
decomposition. We first review the coefficient function...
Für die Lösungen seminlinearer parabolischer Differentialgleichungen werden Einschliessungsaussagen hergeleitet. Hierbei werden Aussagen zur Stabilität von Lösungen ermittelt. Die Resultate werden am Beispiel der Fitzhugh-Nagumo Gleichungen diskutiert.
The phase relaxation model is a diffuse interface model with
small parameter ε which
consists of a parabolic PDE for temperature
θ and an ODE with double obstacles
for phase variable χ.
To decouple the system a semi-explicit Euler method with variable
step-size τ is used for time discretization, which requires
the stability constraint τ ≤ ε. Conforming piecewise
linear finite elements over highly graded simplicial meshes
with parameter h are further employed for space discretization.
A posteriori...
We investigate a system describing electrically charged particles in the whole space ℝ². Our main goal is to describe large time behavior of solutions which start their evolution from initial data of small size. This is achieved using radially symmetric self-similar solutions.
In this work we study the problem in , in , on , in ,
is a bounded regular domain such that , , , , and are positive functions such...
The Cauchy problem for an infinite system of parabolic type equations is studied. General operators of parabolic type of second order with variable coefficients are considered and the system is weakly coupled. We prove the existence and uniqueness of a bounded solution under Carathéodory type conditions and its differentiability, as well as the existence and uniqueness in the class of functions satisfying a natural growth condition. Both results are obtained by the fixed point method.
We consider the Cauchy problem for nonlinear parabolic equations with functional dependence. We prove Schauder-type existence results for unbounded solutions. We also prove existence of maximal solutions for a wide class of differential functional equations.
We study the initial-value problem for parabolic equations with time dependent coefficients and with nonlinear and nonlocal right-hand sides. Nonlocal terms appear in the unknown function and its gradient. We analyze convergence of explicit finite difference schemes by means of discrete fundamental solutions.
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