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Borel summable solutions of the Burgers equation

Grzegorz Łysik (2009)

Annales Polonici Mathematici

We give necessary and sufficient conditions for the formal power series solutions to the initial value problem for the Burgers equation t u - x ² u = x ( u ² ) to be convergent or Borel summable.

Cauchy problem for semilinear parabolic equations with initial data in Hps(Rn) spaces.

Francis Ribaud (1998)

Revista Matemática Iberoamericana

We study local and global Cauchy problems for the Semilinear Parabolic Equations ∂tU - ΔU = P(D) F(U) with initial data in fractional Sobolev spaces Hps(Rn). In most of the studies on this subject, the initial data U0(x) belongs to Lebesgue spaces Lp(Rn) or to supercritical fractional Sobolev spaces Hps(Rn) (s > n/p). Our purpose is to study the intermediate cases (namely for 0 < s < n/p). We give some mapping properties for functions with polynomial growth on subcritical Hps(Rn)...

Cauchy problems in weighted Lebesgue spaces

Jan W. Cholewa, Tomasz Dłotko (2004)

Czechoslovak Mathematical Journal

Global solvability and asymptotics of semilinear parabolic Cauchy problems in n are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over n , n . In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.

Computational technique for treating the nonlinear Black-Scholes equation with the effect of transaction costs

Hitoshi Imai, Naoyuki Ishimura, Hideo Sakaguchi (2007)

Kybernetika

We deal with numerical computation of the nonlinear partial differential equations (PDEs) of Black–Scholes type which incorporate the effect of transaction costs. Our proposed technique surmounts the difficulty of infinite domains and unbounded values of the solutions. Numerical implementation shows the validity of our scheme.

Continuous dependence estimates for the ergodic problem of Bellman-Isaacs operators via the parabolic Cauchy problem

Claudio Marchi (2012)

ESAIM: Control, Optimisation and Calculus of Variations

This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacs operators. We establish such an estimate for the parabolic Cauchy problem in the whole space  [0, +∞) × ℝn and, under some periodicity and either ellipticity or controllability assumptions, we deduce a similar estimate for the ergodic constant associated to the operator. An interesting byproduct of the latter result will be the local uniform convergence for some classes of singular perturbation problems.

Continuous dependence of the entropy solution of general parabolic equation

Mohamed Maliki (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

We consider the general parabolic equation : u t - Δ b ( u ) + d i v F ( u ) = f in Q = ] 0 , T [ × N , T > 0 with u 0 L ( N ) , for a ....

Control of Traveling Solutions in a Loop-Reactor

Y. Smagina, M. Sheintuch (2010)

Mathematical Modelling of Natural Phenomena

We consider the stabilization of a rotating temperature pulse traveling in a continuous asymptotic model of many connected chemical reactors organized in a loop with continuously switching the feed point synchronously with the motion of the pulse solution. We use the switch velocity as control parameter and design it to follow the pulse: the switch velocity is updated at every step on-line using the discrepancy between the temperature at the front...

Coupling of transport and diffusion models in linear transport theory

Guillaume Bal, Yvon Maday (2002)

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

This paper is concerned with the coupling of two models for the propagation of particles in scattering media. The first model is a linear transport equation of Boltzmann type posed in the phase space (position and velocity). It accurately describes the physics but is very expensive to solve. The second model is a diffusion equation posed in the physical space. It is only valid in areas of high scattering, weak absorption, and smooth physical coefficients, but its numerical solution is much cheaper...

Coupling of transport and diffusion models in linear transport theory

Guillaume Bal, Yvon Maday (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is concerned with the coupling of two models for the propagation of particles in scattering media. The first model is a linear transport equation of Boltzmann type posed in the phase space (position and velocity). It accurately describes the physics but is very expensive to solve. The second model is a diffusion equation posed in the physical space. It is only valid in areas of high scattering, weak absorption, and smooth physical coefficients, but its numerical solution is...

Deep learning for gradient flows using the Brezis–Ekeland principle

Laura Carini, Max Jensen, Robert Nürnberg (2023)

Archivum Mathematicum

We propose a deep learning method for the numerical solution of partial differential equations that arise as gradient flows. The method relies on the Brezis–Ekeland principle, which naturally defines an objective function to be minimized, and so is ideally suited for a machine learning approach using deep neural networks. We describe our approach in a general framework and illustrate the method with the help of an example implementation for the heat equation in space dimensions two to seven.

Currently displaying 41 – 60 of 285