- Journal Article
Author: Le Thi Phuong Ngoc (2011) - In this paper, we consider a nonlinear wave equation associated with the Dirichlet boundary condition. First, the existence and uniqueness of a weak solution are proved by using the Faedo-Galerkin method. Next, we present an asymptotic expansion of high order in many small parameters of a weak solution. This extends recent corresponding results where an asymptotic expansion of a weak solution in two or three small parameters is established.
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- Journal Article
Author: Le Thi Phuong Ngoc (2016) - In this paper, a nonlinear Love equation with a mixed homogeneous condition is studied. The uniqueness and existence of a weak solution is proved with the help of an a priori estimate and the Galerkin method. Furthermore, a new result related exponential decay of a weak solution is also established.
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- Journal Article
Author: Pham Hong Danh (2015) - In this paper, we consider a nonlinear functional integral equation with variable delays. Using tools of functional analysis and Banach's fixed point theorem in a Fréchet space, the existence of a unique solution for the above equation is proved. Nontrivial examples are also given to illustrate our result.
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- Journal Article
Author: Le XuanTruong (2011) - This paper is devoted to studying a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under the suitable conditions, we prove that any weak solutions with negative initial energy will blow up in finite time. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.
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- Journal Article
Author: Tran Minh Thuyet (2013) - The paper is devoted to the study of a Volterra integral equation of the second kind related to the macroeconomic models. Using contraction mapping principle and some techniques of nonlinear analysis, the uniqueness existence, approximation and asymptotic expansions of solutions with espect to small parameters appeared in the model are established.
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- Journal Article
Author: Le Thi Phuong Ngoc; Huynh Thi Hoang Dung; Pham Hong Danh; Nguyen Thanh Long (2014) - This paper is devoted to the study of the following perturbed system of nonlinear functional equations f i (x)=∑ k=1 m ∑ j=1 n ϵa ijk Ψx,f j (R ijk (x)),∫ 0 X ijk (x) f j (t)dt+b ijk f j (S ijk (x))+g i (x),(E) x∈Ω=[-b,b], i=1,⋯,n, where ϵ is a small parameter, a ijk , b ijk are the given real constants, R ijk , S ijk , X ijk :Ω→Ω, g i :Ω→ℝ, Ψ:Ω×ℝ 2 →ℝ are the given continuous functions and f i :Ω→ℝ are unknown functions. First, by using the Banach fixed point theorem, we find sufficient conditions for the unique existence and stability of a solution of (E). Next, in the case of Ψ∈C 2 (Ω×ℝ 2 ;ℝ), we investigate the quadratic convergence of (E). Finally, in the case of Ψ∈C N (Ω×ℝ 2 ;ℝ)...
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- Journal Article
Author: Le Thi Phuong Ngoc (2016) - This paper is devoted to the study of a nonlinear Carrier wave equation in an annular membrane associated with Robin-Dirichlet conditions. Existence and uniqueness of a weak solution are proved by using the linearization method for nonlinear terms combined with the Faedo-Galerkin method and the weak compact method. Furthermore, an asymptotic expansion of a weak solution of high order in a small parameter is established.
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- Journal Article
Author: Le Thi Phuong Ngoc (2017) - This paper is devoted to the study of a nonlinear Carrier wave equation in the annular associated with Robin-Dirichlet conditions. Using a high order iterative scheme, the existence of a local unique weak solution is proved. Moreover, the sequence established here converges to a unique weak solution at a rate of order N with N ≥ 2.
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- Journal Article
Author: Le Thi Phuong Ngoc (2014) - Motivated by recent known results about the solvability of nonlinear functionalintegral equations in one, two or n variables, this paper establishes the existence of asymp-totically stable solutions for a Volterra-Hammerstein integral equation in three variables.The proofs are completed via a fixed point theorem of Krasnosel'skii type, a condition forthe relative compactness of a subset in certain space and integral inequalities with explicit estimates.
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- Journal Article
Author: Le Thi Phuong Ngoc (2013) - Motivated by the well-posedness results in [Nonlinear Anal. Ser. B: RWA. 4(3) (2003), 483, 501; Nonlinear Anal. Ser. B: RWA. 11(5) (2010), 3453-3462] for the models describing the propagation of high frequency electromagnetic waves in nonlinear dielectric media, because of their mathematical context, we study a similar model and prove results about existence, uniqueness, the asymptotic behavior and an asymptotic expansion of the solution up to order N in a small parameter λ with error λ^{N+½}
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- Journal Article
Author: Nguyen Huu Nhan (2016) - In this paper, we consider the initial and boundary value problem for a nonlinear wave equation, with the source term containing a nonlinear integral, associated with homo- geneous Dirichlet boundary conditions. We establish here a high order iterative scheme in order to get a convergent sequence at a rate of order N to a local unique weak solution of the above problem. This scheme shows that the convergence can be obtained with a high rate if the nonlinear term in the original equation is smooth enough.
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- Journal Article
Author: Le Thi Phuong Ngoc (2017) - This paper is devoted to study of a nonlinear heat equation with a viscoelastic term associated with Robin conditions. At first, by the Faedo–Galerkin and the compactness method, we prove existence, uniqueness, and regularity of a weak solution. Next, we prove that any weak solution with negative initial energy will blow up in finite time. Finally, by the construction of a suitable Lyapunov functional, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions.
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- Journal Article
Author: Le Thi Phuong Ngoc (2011) - Consider the initial-boundary value problem for the nonlinear wave equation utt − µ(t)uxx + K|u| p−2u + λ|ut| q−2ut = F(x, t), 0 < x < 1, 0 < t < T, µ(t)ux(0, t) = K0u(0, t) + Rt 0 k (t − s) u (0, s) ds + g(t), −µ(t)ux(1, t) = K1u(1, t) + λ1|ut(1, t)| α−2ut(1, t), u(x, 0) = ue0(x), ut(x, 0) = ue1(x), where p, q, α ≥ 2; K0, K1, K ≥ 0; λ, λ1 > 0 are given constants and µ, F, g, k, ue0, ue1, are given functions. First, the existence and uniqueness of a weak solution are proved by using the Galerkin method. Next, with α = 2, we obtain an asymptotic expansion of the solution up to order N in two small parameters λ, λ1 with error p λ2 + λ 2 1 N+ 1 2 .
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- Journal Article
Author: Le Xuan Truong (2011) - The paper is devoted to the study a system of nonlinear wave equations associated with the mixed nonhomogeneous conditions. Existence of a weak solution is proved by using the Faedo–Galerkin method. Uniqueness, regularity and decay properties of solutions are also discussed.
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- Journal Article
Author: Pham Hong Danh (2017) - This paper is devoted to the study of existence results and some properties of solutions of m-order nonlinear integrodifferential equations in two variables. The main tools are the Banach fixed point theorem or Schauder fixed point theorem coupled with the definitions of suitable Banach spaces and adding appropriate conditions which are useful to yield relatively compact subsets in these space. To our knowledge, these techniques have not been used before. In order to illustrate the results obtained here, two examples are given.
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- Journal Article
Author: Le Thi Phuong Ngoc (2016) - We study the following initial-boundary value problem: ⎧ ⎨ ⎩ ut – (μ + α ∂ ∂t )( ∂2u ∂x2 + 1 x ∂u ∂x ) + f(u) = f1(x,t), 1 < x < R,t > 0, ux(1,t) = h1u(1,t) + g1(t), u(R,t) = gR(t), u(x, 0) = u˜ 0(x), () where μ > 0, α > 0, h1 ≥ 0, R > 1 are given constants and f, f1, g1, gR, u˜ 0 are given functions. First, we use the Galerkin and compactness method to prove the existence of a unique weak solution u(t) of Problem (1) on (0, T), for every T > 0. Next, we study the asymptotic behavior of the solution u(t) as t → +∞. Finally, we prove the existence and uniqueness of a weak solution of Problem (1)1,2 associated with a ‘(N + 1)-points condition in time’ case, u(x, 0) = N i=1 ηiu(x, Ti),...
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- Journal Article
Author: Nguyen Huu Nhan (2017) - In this paper, we consider the Robin–Dirichlet problem for a nonlinear wave equation with the source term containing a nonlinear integral. Using the Faedo–Galerkin method and the linearization method for nonlinear terms, we prove the existence and uniqueness of a weak solution. We also discuss an asymptotic expansion of high order in a small parameter of a weak solution.
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- Journal Article
Author: Pham Hong Danh (2014) - Using tools of functional analysis and a fixed point theorem of Krasnosel'skiitype, this paper proves solvability and asymptotically stable of a mixed functional integral equation in N variables. Furthermore, the set of solutions is compact. In order to illustratethe results obtained here, an example is given.
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