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Applications of approximate gradient schemes for nonlinear parabolic equations

Robert Eymard, Angela Handlovičová, Raphaèle Herbin, Karol Mikula, Olga Stašová (2015)

Applications of Mathematics

We develop gradient schemes for the approximation of the Perona-Malik equations and nonlinear tensor-diffusion equations. We prove the convergence of these methods to the weak solutions of the corresponding nonlinear PDEs. A particular gradient scheme on rectangular meshes is then studied numerically with respect to experimental order of convergence which shows its second order accuracy. We present also numerical experiments related to image filtering by time-delayed Perona-Malik and tensor diffusion...

Applications of the Carathéodory theorem to PDEs

Konstanty Holly, Joanna Orewczyk (2000)

Annales Polonici Mathematici

We discuss and exploit the Carathéodory theorem on existence and uniqueness of an absolutely continuous solution x: ℐ (⊂ ℝ) → X of a general ODE ẋ ( * ) = ℱ ( t , x ) for the right-hand side ℱ : dom ℱ ( ⊂ ℝ × X) → X taking values in an arbitrary Banach space X, and a related result concerning an extension of x. We propose a definition of solvability of (*) admitting all connected ℐ and unifying the cases “dom ℱ is open” and “dom ℱ = ℐ × Ω for some Ω ⊂ X”. We show how to use the theorems mentioned above to get approximate...

Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls

Alexander Khapalov (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the one dimensional semilinear reaction-diffusion equation, governed in Ω = (0,1) by controls, supported on any subinterval of (0, 1), which are the functions of time only. Using an asymptotic approach that we have previously introduced in [9], we show that such a system is approximately controllable at any time in both L2(0,1)( and C0[0,1], provided the nonlinear term f = f(x,t, u) grows at infinity no faster than certain power of log |u|. The latter depends on the regularity...

Approximate controllability by birth control for a nonlinear population dynamics model

Otared Kavian, Oumar Traoré (2011)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we analyse an approximate controllability result for a nonlinear population dynamics model. In this model the birth term is nonlocal and describes the recruitment process in newborn individuals population, and the control acts on a small open set of the domain and corresponds to an elimination or a supply of newborn individuals. In our proof we use a unique continuation property for the solution of the heat equation and the Kakutani-Fan-Glicksberg fixed point theorem.

Approximate controllability by birth control for a nonlinear population dynamics model

Otared Kavian, Oumar Traoré (2011)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we analyse an approximate controllability result for a nonlinear population dynamics model. In this model the birth term is nonlocal and describes the recruitment process in newborn individuals population, and the control acts on a small open set of the domain and corresponds to an elimination or a supply of newborn individuals. In our proof we use a unique continuation property for the solution of the heat equation and the Kakutani-Fan-Glicksberg fixed point theorem.

Approximate controllability of linear parabolic equations in perforated domains

Patrizia Donato, Aïssam Nabil (2001)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we consider an approximate controllability problem for linear parabolic equations with rapidly oscillating coefficients in a periodically perforated domain. The holes are ε -periodic and of size ε . We show that, as ε → 0 , the approximate control and the corresponding solution converge respectively to the approximate control and to the solution of the homogenized problem. In the limit problem, the approximation of the final state is alterated by a constant which depends on the proportion...

Approximate Controllability of linear parabolic equations in perforated domains

Patrizia Donato, Aïssam Nabil (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we consider an approximate controllability problem for linear parabolic equations with rapidly oscillating coefficients in a periodically perforated domain. The holes are ε-periodic and of size ε. We show that, as ε → 0, the approximate control and the corresponding solution converge respectively to the approximate control and to the solution of the homogenized problem. In the limit problem, the approximation of the final state is alterated by a constant which depends on the proportion...

Approximate solution of an inhomogeneous abstract differential equation

Emil Vitásek (2012)

Applications of Mathematics

Recently, we have developed the necessary and sufficient conditions under which a rational function F ( h A ) approximates the semigroup of operators exp ( t A ) generated by an infinitesimal operator A . The present paper extends these results to an inhomogeneous equation u ' ( t ) = A u ( t ) + f ( t ) .

Approximate solutions of abstract differential equations

Emil Vitásek (2007)

Applications of Mathematics

The methods of arbitrarily high orders of accuracy for the solution of an abstract ordinary differential equation are studied. The right-hand side of the differential equation under investigation contains an unbounded operator which is an infinitesimal generator of a strongly continuous semigroup of operators. Necessary and sufficient conditions are found for a rational function to approximate the given semigroup with high accuracy.

Currently displaying 381 – 400 of 491