Browsing by Author "Rusu, Galina"
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Item 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, GalinaIn 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 = 0Item CONVERGENCE ESTIMATES FOR SOME ABSTRACT LINEAR SECOND ORDER DIFFERENTIAL EQUATIONS WITH TWO SMALL PARAMETERS(IOS Press, 2016) Perjan, Andrei; Rusu, GalinaIn 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.Item CONVERGENCE ESTIMATES FOR SOME ABSTRACT SECOND ORDER DIFFERENTIAL EQUATIONS IN HILBERT SPACES("VALINEX", 2019-09-28) Perjan, Andrei; Rusu, Galinan 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.Item LARGE-TIME BEHAVIOR OF THE DIFFERENCE OF SOLUTIONS OF TWO EVOLUTION EQUATION(CEP USM, 2017) Perjan, Andrei; Rusu, GalinaIn 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.Item LIMITS OF SOLUTIONS TO THE SEMILINEAR PLATE EQUATION WITH SMALL PARAMETER(Academia de Ştiinţe a Moldovei, 2022) Perjan, Andrei; Rusu, GalinaWe 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.Item LIMITS OF SOLUTIONS TO THE SEMILINEAR PLATE EQUATION WITH SMALL PARAMETER(2022) Perjan, Andrei; Rusu, GalinaWe 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.Item LIMITS OF SOLUTIONS TO THE SINGULARLY PERTURBED ABSTRACT HYPERBOLIC-PARABOLIC SYSTEM(2014) Perjan, Andrei; Rusu, GalinaWe 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-adjointItem SINGULAR LIMITS OF SOLUTIONS TO THE CAUCHY PROBLEM FOR SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS IN HILBERT SPACES(2009) Rusu, GalinaWe 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 asε→0, whereA1andA0are two linear selfadjoint operators.Item SINGULAR LIMITS OF SOLUTIONS TO THE CAUCHY PROBLEMFOR SECOND ORDER LINEAR DIFFERENTIAL EQUATIONSIN HILBERT SPACES(Institutul de Matematică şi Informatică al AŞM, 2009) Rusu, GalinaWe 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 spaceHasε→0, whereA1andA0are two linear selfadjoint operators.Item SINGULARLY PERTURBED CAUCHY PROBLEM FOR ABSTRACTLINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDERIN HILBERT SPACES(Institutul de Matematică şi Informatică al AŞM, 2008) Perjan, Andrei; Rusu, GalinaWe 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.Item STRATEGII DE DIMINUARE A STRESULUI PROFESIONAL LA CADRELE DIDACTICE(CEP USM, 2020) Rusu, GalinaLe stress reste toujours un probleme actuel, bien qu’il soit etudie depuis 100 ans. La recherche dans ce domaine montre que le stress au travail est un phenomene qui touche presque toutes les professions. Les organisations les plus touchees par le stress sont les prestataires de services, dont l’education. De plus en plus les recherchs sont consacrees a ce phenomene sous plusieurs angles psychologique, pedagogique, sociaux. Le stress a été etudie par plusieurs chercheurs, dont Hans Selye.Item TWO PARAMETER SINGULAR PERTURBATION PROBLEMS FOR SINE-GORDON TYPE EQUATIONS(2022) Perjan, Andrei; Rusu, GalinaIn 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