Browsing by Author "Perjan, Andrei"

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CONVERGENCE ESTIMATES FOR ABSTRACT SECOND ORDER DIFFERENTIAL EQUATIONS WITH TWO SMALL PARAMETERS AND LIPSCHITZIAN NONLINEARITIES
(Centrul Universitar Nord din Baia Mare, 2022) Perjan, Andrei; Rusu, Galina
In a real Hilbert space H we consider the following singularly perturbed Cauchy problem { εu′′ εδ (t) + δ u′ εδ (t) + Auεδ (t) + B(uεδ (t)) = f (t), t ∈ (0, T ), uεδ (0) = u0, u′ εδ (0) = u1, where u0, u1 ∈ H, f : [0, T ] 7 → H, ε, δ are two small parameters, A is a linear self-adjoint operator and B is a nonlinear A1/2 Lipschitzian operator. We study the behavior of solutions uεδ in two different cases: ε → 0 and δ ≥ δ0 > 0; ε → 0 and δ → 0, relative to solution to the corresponding unperturbed problem. We obtain some a priori estimates of solutions to the perturbed problem, which are uniform with respect to parameters, and a relationship between solutions to both problems. We establish that the solution to the unperturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of t = 0
CONVERGENCE ESTIMATES FOR SOME ABSTRACT LINEAR SECOND ORDER DIFFERENTIAL EQUATIONS WITH TWO SMALL PARAMETERS
(IOS Press, 2016) Perjan, Andrei; Rusu, Galina
In a real Hilbert space H we consider the following singularly perturbed Cauchy problem (Equation presented) where u0, u1 ∈ H, f : [0, T ] → H and ϵ, δ are two small parameters. We study the behavior of the solutions uϵδ to the problem (Pϵδ) in two different cases: (i) when ϵ → 0 and δ ≥ δ0 > 0; (ii) when ϵ → 0 and δ → 0. We obtain a priori estimates of the solutions to the perturbed problem, which are uniform with respect to the parameters, and a relationship between the solutions to both problems. We establish that the solution to the unperturbed problem has a singular behavior with respect to the parameters in the neighborhood of t = 0. We describe the boundary layer and the boundary layer function in both cases.
CONVERGENCE ESTIMATES FOR SOME ABSTRACT SECOND ORDER DIFFERENTIAL EQUATIONS IN HILBERT SPACES
("VALINEX", 2019-09-28) Perjan, Andrei; Rusu, Galina
n a real Hilbert space H we consider the following perturbed Cauchy problem ( " u′′ " (t) + u′ " (t) + Au " (t) + B(u " (t)) = f(t), t ∈ (0, T ), u " (0) = u0, u′ " (0) = u1, (P " ) where u0, u1 ∈ H, f : [0, T ] 7→ H and ", are two small parameters, A is a linear self-adjoint operator, B is a locally Lipschitz and monotone operator. We study the behavior of solutions u " to the problem (P " ) in two different cases: (i) when " → 0 and ≥ 0 > 0; (ii) when " → 0 and → 0. We establish that the solution to the unperturbed problem has a singular behavior, relative to the parameters, in the neighborhood of t = 0. We show the boundary layer and boundary layer function in both cases.
LARGE-TIME BEHAVIOR OF THE DIFFERENCE OF SOLUTIONS OF TWO EVOLUTION EQUATION
(CEP USM, 2017) Perjan, Andrei; Rusu, Galina
In a real Hilbert space H we consider a linear self-adjoint positive definite operator A : V = D ( A ) ⊂ H → H and investigate the behavior of the difference u − v of solutions to the problems u ′′ ( t ) + u ′ ( t ) + Au ( t ) = f ( t ) , t > 0 , u (0) = u 0 , u ′ (0) = u 1 , v ′ ( t ) + Av ( t ) = f ( t ) , t > 0 , v (0) = u 0 , where u 0 , u 1 ∈ H, f : [0 , + ∞ ) → H.
LIMITS OF SOLUTIONS TO THE SEMILINEAR PLATE EQUATION WITH SMALL PARAMETER
(Academia de Ştiinţe a Moldovei, 2022) Perjan, Andrei; Rusu, Galina
We study the existence of the limits of solutions to the semilinear plate equation with boundary Dirichlet condition with a small parameter coefficient of the second order derivative in time. We establish the convergence of solutions to the perturbed problem and their derivatives in spacial variables to the corresponding solutions to the unperturbed problem as the small parameter tends to zero.
LIMITS OF SOLUTIONS TO THE SEMILINEAR PLATE EQUATION WITH SMALL PARAMETER
(2022) Perjan, Andrei; Rusu, Galina
We study the existence of the limits of solutions to the semilinear plate equation with boundary Dirichlet condition with a small parameter coefficient of the second order derivative in time. We establish the convergence of solutions to the perturbed problem and their derivatives in spacial variables to the corresponding solutions to the unperturbed problem as the small parameter tends to zero.
LIMITS OF SOLUTIONS TO THE SEMILINEAR WAVE EQUATION WITH SMALL PARAMETER
(Academia de Ştiinţe a Moldovei, 2006) Perjan, Andrei
We study the existence of the limits of solution to singularly perturbed initial boundary value problem of hyperbolic - parabolic type with boundary Dirichlet condition for the semilinear wave equation. We prove the convergence of solutions and also the convergence of gradients of solutions to perturbed problem to the corresponding solutions to the unperturbed problem as the small parameter tends to zero. We show that the derivatives of solution relative to time-variable possess the boundary layer function of the exponential type in the neighborhood of t = 0
LIMITS OF SOLUTIONS TO THE SINGULARLY PERTURBED ABSTRACT HYPERBOLIC-PARABOLIC SYSTEM
(2014) Perjan, Andrei; Rusu, Galina
We study the behavior of solutions to the problem εu′′ε(t) +u′ε(t) +A(t)uε(t) =fε(t), t∈(0, T), uε(0) =u0ε, u′ε(0) =u1ε,in the Hilbert space H asε→0, whereA(t), t∈(0,∞),is a family of linear self-adjoint
LINEAR SINGULAR PERTURBATIONS OF HYPERBOLIC-PARABOLIC TYPE
(Academia de Ştiinţe a Moldovei, 2003) Perjan, Andrei
We study the behavior of solutions of the problem εu′′(t)+u′(t)+Au(t) =f (t), u(0) = u0, u′(0) = u1 in the Hilbert space H as ε → 0, where A is a linear,symmetric, strong positive operator.
The Regular Perturbations of the Nonviscous Semilinear 1D Canh-Hilliard Equation [Articol]
(2024) Perjan, Andrei
We study the behavior of solutions to the relaxation non-viscous semilinear one space dimension Canh-Hilliard equation when the relaxations tend to zero.
SINGULAR PERTURBATIONS OFHYPERBOLIC-PARABOLIC TYPE
(Institutul de Matematică şi Informatică al AŞM, 2003) Perjan, Andrei
We study the behavior of solutions of the problemεu′′(t)+u′(t)+Au(t) =f(t), u(0) =u0, u′(0) =u1in the Hilbert spaceHasε→0, whereAis a linear,symmetric, strong positive operator.
SINGULARLY PERTURBED CAUCHY PROBLEM FOR ABSTRACTLINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDERIN HILBERT SPACES
(Institutul de Matematică şi Informatică al AŞM, 2008) Perjan, Andrei; Rusu, Galina
We study the behavior of solutions to the problem(ε“u′′ε(t) +A1uε(t)”+u′ε(t) +A0uε(t) =f(t), t >0,uε(0) =u0, u′ε(0) =u1,in the Hilbert space H asε7→0, whereA1andA0are two linear selfadjoint operators.
TWO PARAMETER SINGULAR PERTURBATION PROBLEMS FOR SINE-GORDON TYPE EQUATIONS
(2022) Perjan, Andrei; Rusu, Galina
In the real Sobolev space H1 0 (Ω) we consider the Cauchy-Dirichlet problem for sine-Gordon type equation with strongly elliptic operators and two small parameters. Using some a priori estimates of solutions to the perturbed problem and a relationship between solutions in the linear case, we establish convergence estimates for the difference of solutions to the perturbed and corresponding unperturbed problems. We obtain that the solution to the perturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of t = 0