2. Articole

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Subject: search.filters.subject.singular perturbation

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    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.
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    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.
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    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.
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    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.
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    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
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    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