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Alain Pham Ngoc Dinh

Publications and source records attributed to Alain Pham Ngoc Dinh.

14 recordsLinked to original sources
arXiv2011

Ice formation in the Arctic during summer: false-bottoms

The only source of ice formation in the Arctic during summer is a layer of ice called false-bottoms between an under-ice melt pond and the underlying ocean. Of interest is to give a mathematical model in order to determine the simultaneous growth and ablation of false-bottoms, which is governed by both of heat fluxes and salt fluxes. In one dimension, this problem may be considered mathematically as a two-phase Stefan problem with two free boundaries. Our main result is to prove the existence and uniqueness of the solution from the initial condition.

math.AP↗
arXiv2011

Large time behavior of differential equations with drifted periodic coefficients modeling Carbon storage in soil

This paper is concerned with the linear ODE in the form $y'(t)=λρ(t)y(t)+b(t)$, $λ<0$ which represents a simplified storage model of the carbon in the soil. In the first part, we show that, for a periodic function $ρ(t)$, a linear drift in the coefficient $b(t)$ involves a linear drift for the solution of this ODE. In the second part, we extend the previous results to a classical heat non-homogeneous equation. The connection with an analytic semi-group associated to the ODE equation is considered in the third part. Numerical examples are given.

math.AP↗
arXiv2011

Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type

This paper is devoted to study a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under 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

math.AP↗
arXiv2010

On a nonlinear heat equation associated with Dirichlet -- Robin conditions

This paper is devoted to the study of a nonlinear heat equation associated with Dirichlet-Robin conditions. At first, we use the Faedo -- Galerkin and the compactness method to prove existence and uniqueness results. Next, we consider the properties of solutions. We obtain that if the initial condition is bounded then so is the solution and we also get asymptotic behavior of solutions as. Finally, we give numerical results

math.AP↗
arXiv2009

A new stability results for the backward heat equation

In this paper, we regularize the nonlinear inverse time heat problem in the unbounded region by Fourier method. Some new convergence rates are obtained. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. Especially, the optimal convergence of the approximate solution at t = 0 is also proved. This work extends to many earlier results in (f2,f3, hao1,Quan,tau1, tau2, Trong3,x1).

math.AP↗
arXiv2008

Determination of the body force of a two-dimensional isotropic elastic body

Let $Ω$ represent a two$-$dimensional isotropic elastic body. We consider the problem of determining the body force $F$ whose form $ϕ(t)(f_1(x),f_2(x))$ with $ϕ$ be given inexactly. The problem is nonlinear and ill-posed. Using the Fourier transform, the methods of Tikhonov's regularization and truncated integration, we construct a regularized solution from the data given inexactly and derive the explicitly error estimate. Numerical part is given

math.AP↗
arXiv2008

Determine the spacial term of a two-dimensional heat source

We consider the problem of determining a pair of functions $(u,f)$ satisfying the heat equation $u_t -Δu =φ(t)f (x,y)$, where $(x,y)\in Ω=(0,1)\times (0,1)$ and the function $φ$ is given. The problem is ill-posed. Under a slight condition on $φ$, we show that the solution is determined uniquely from some boundary data and the initial temperature. Using the interpolation method and the truncated Fourier series, we construct a regularized solution of the source term $f$ from non-smooth data. The error estimate and numerical experiments are given.

math.AP↗
arXiv2008

Determine the source term of a two-dimensional heat equation

Let $Ω$ be a two-dimensional heat conduction body. We consider the problem of determining the heat source $F(x,t)=φ(t)f(x,y)$ with $φ$ be given inexactly and $f$ be unknown. The problem is nonlinear and ill-posed. By a specific form of Fourier transforms, we shall show that the heat source is determined uniquely by the minimum boundary condition and the temperature distribution in $Ω$ at the initial time $t=0$ and at the final time $t=1$. Using the methods of Tikhonov's regularization and truncated integration, we construct the regularized solutions. Numerical part is given.

math.AP↗
arXiv2007

Mathematical Model for the Evaporation of a Liquid Fuel Droplet, Subject to Nonlinear Constraints

We study the mathematical evolution of a liquid fuel droplet inside a vessel. In particular, we analyze the evolution of the droplet radius on a finite time interval. The model problem involves an hyperbolic system coupled with the pressure and velocity of the surrounding gas. Existence of bounded solutions for the mass fraction of the liquid, submitted to nonlinear constraints, is shown. Numerical simulations are given, in agreement with known physical experiments.

math.AP↗
arXiv2007

Existence and Decay of Solutions of a Nonlinear Viscoelastic Problem with a Mixed Nonhomogeneous Condition

We study the initial-boundary value problem for a nonlinear wave equation given by u_{tt}-u_{xx}+\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u) , 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+ηu(1,t)=g(t), u(x,0)=û_{0}(x), u_{t}(x,0)={û}_{1}(x), where η\geq 0, q\geq 2 are given constants {û}_{0}, {û}_{1}, g, k, f are given functions. In part I under a certain local Lipschitzian condition on f, a global existence and uniqueness theorem is proved. The proof is based on the paper [10] associated to a contraction mapping theorem and standard arguments of density. In Part} 2, under more restrictive conditions it is proved that the solution u(t) and its derivative u_{x}(t) decay exponentially to 0 as t tends to infinity.

math.AP↗
arXiv2007

Laguerre polynomials and the inverse Laplace transform using discrete data

We consider the problem of finding a function defined on $(0,\infty)$ from a countable set of values of its Laplace transform. The problem is severely ill-posed. We shall use the expansion of the function in a series of Laguerre polynomials to convert the problem in an analytic interpolation problem. Then, using the coefficients of Lagrange polynomials we shall construct a stable approximation solution.

math.AP↗