Dang Duc Trong

Nguyen Huy Tuan, Dang Duc Trong, Le Duc Thang et al.

In this paper, we investigate the Cauchy problem for both linear and semi-linear elliptic equations. In general, the equations have the form \[ \frac{\partial^{2}}{\partial t^{2}}u\left(t\right)=\mathcal{A}u\left(t\right)+f\left(t,u\left(t\right)\right),\quad t\in\left[0,T\right], \] where $\mathcal{A}$ is a positive-definite, self-adjoint operator with compact inverse. As we know, these problems are well-known to be ill-posed. On account of the orthonormal eigenbasis and the corresponding eigenvalues related to the operator, the method of separation of variables is used to show the solution in series representation. Thereby, we propose a modified method and show error estimations in many accepted cases. For illustration, two numerical examples, a modified Helmholtz equation and an elliptic sine-Gordon equation, are constructed to demonstrate the feasibility and efficiency of the proposed method.

Dang Duc Trong, Phan Thanh Nam, Phung Trong Thuc

Let a three-dimensional isotropic elastic body be described by the Lamé system with the body force of the form $F(x,t)=ϕ(t)f(x)$, where $ϕ$ is known. We consider the problem of determining the unknown spatial term $f(x)$ of the body force where the surface stress history is given as the overdetermination. This inverse problem is ill-posed. Using the interpolation method and truncated Fourier series, we construct a regularized solution from approximate data and provide explicit error estimates. AMS 2010 Subject Classification: 35L20, 35R30. Keywords: Body force, elastic, ill$-$posed problem, interpolation, Fourier series.

Dang Duc Trong, Truong Trung Tuyen

Let $ϕ$ be a nontrivial function of $L^1(\RR)$. For each $s\geq 0$ we put \begin{eqnarray*} p(s)=-\log \int_{|t|\geq s}|ϕ(t)|dt. \end{eqnarray*} If $ϕ$ satisfies \begin{equation} \lim_{s\to \infty}\frac{p(s)}{s}=\infty ,\label{170506.1} \end{equation} we obtain asymptotic estimates of the size of small-valued sets $B_ε=\{x\in\RR : |\hatϕ(x)|\leq ε, |x|\leq R_ε\}$ of Fourier transform \begin{eqnarray*} \hatϕ(x)=\int_{-\infty}^{\infty}e^{-ixt}ϕ(t)dt, x\in \RR, \end{eqnarray*} in terms of $p(s)$ or in terms of its Young dual function \begin{eqnarray*} p^{*}(t)=\sup_{s\geq 0}[st-p(s)], t\geq 0. \end{eqnarray*} Applying these results, we give an explicit estimate for the error of Tikhonov's regularization for the solution $f$ of the integral convolution equation \begin{eqnarray*} \int_{-\infty}^{\infty}f(t-s)ϕ(s)ds =g(t), \end{eqnarray*} where $f,g \in L^2(\RR)$ and $ϕ$ is a nontrivial function of $L^1(\RR)$ satisfying condition (\ref{170506.1}), and $g,ϕ$ are known non-exactly. Also, our results extend some results of \cite{tld} and \cite{tqd}.