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Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations

Martin A. Grepl, Yvon Maday, Ngoc C. Nguyen, Anthony T. Patera (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

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...

Eine parabolische Gleichung mit vergänglichen Lösungen.

J.H. Sampson (1982)

Manuscripta mathematica

Einschliessungsaussagen bei Systemen semilinearer parabolischer Differentialgleichungen

Wilhelm Heinrichs (1991)

Applications of Mathematics

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.

Ergodic properties of multidimensional Brownian motion with rebirth.

Grigorescu, Ilie, Kang, Min (2007)

Electronic Journal of Probability [electronic only]

Error control and adaptivity for a phase relaxation model

Zhiming Chen, Ricardo H. Nochetto, Alfred Schmidt (2000)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Error Control and Andaptivity for a Phase Relaxation Model

Zhiming Chen, Ricardo H. Nochetto, Alfred Schmidt (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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...

Estimates for the maximum modulus of solutions to the Burgers system of equations.

Nalimov, V.I. (2004)

Sibirskij Matematicheskij Zhurnal

Estimation of a temporally and spatially varying diffusion coefficinet in a parabolic system by an augmented Lagrangian technique.

K. Kunisch, G. Peichl (1991)

Numerische Mathematik

Estimations uniformes à l’explosion pour les équations de la chaleur non linéaires et applications

Frank Merle, Hatem Zaag (1996/1997)

Séminaire Équations aux dérivées partielles

Euclid΄s algorithm and accelerated iterative methods

Elias A. Lipitakis (1990)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

Existence and asymptotics of solutions of the Debye-Nernst-Planck system in ℝ²

Agnieszka Herczak, Michał Olech (2009)

Banach Center Publications

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.

Existence and nonexistence of global solutions of the quasilinear parabolic equations with inhomogeneous terms.

Kobayashi, Yasumaro (2010)

Advances in Difference Equations [electronic only]

Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities

Boumediene Abdellaoui, Eduardo Colorado, Ireneo Peral (2004)

Journal of the European Mathematical Society

In this work we study the problem $u_{t} {- div (| x |}^{- 2 γ} \nabla u) = λ \frac{u^{α}}{{| x |}^{2 (γ + 1)}} + f$ in $Ω \times (0, T)$ , $u \geq 0$ in $Ω \times (0, T)$ , $u = 0$ on $\partial Ω \times (0, T)$ , $u (x, 0) = u_{0} (x)$ in $Ω$ , $Ω \subset ℝ^{N}$ $(N \geq 2)$ is a bounded regular domain such that $0 \in Ω$ , $λ > 0$ , $α > 0$ , $- \infty < γ < (N - 2) / 2$ , $f$ and $u_{0}$ are positive functions such...

Existence of global attractors in $L^{p}$ for $m$ -Laplacian parabolic equation in $ℝ^{N}$ .

Chen, Caisheng, Shi, Lanfang, Wang, Hui (2009)

Boundary Value Problems [electronic only]

Existence of multiple solutions for a nonlinearly perturbed elliptic parabolic system in $ℝ^{2}$ .

Ishiwata, Michinori, Ogawa, Takayoshi, Takahashi, Futoshi (2009)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence of solutions for infinite systems of parabolic equations with functional dependence

Anna Pudełko (2005)

Annales Polonici Mathematici

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.

Existence of solutions with exponential growth for nonlinear differential-functional parabolic equations

Agnieszka Bartłomiejczyk, Henryk Leszczyński (2014)

Annales Polonici Mathematici

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.

Expansion of embedded curves with tuning angle greater than - ... .

B. Chow, Lii-Perng Liou, Dong-Ho Tsai (1996)

Inventiones mathematicae

Explicit difference schemes for nonlinear differential functional parabolic equations with time dependent coefficients-convergence analysis

A. Poliński (2006)

Annales Polonici Mathematici

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.

Exponential convergence for a periodically driven semilinear heat equation

Nikos Zygouras (2009)

Annales de l'I.H.P. Analyse non linéaire

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