Let $Omega$ be a two-dimensional heat conduction body. We consider the problem of determining the heat source $F(x,t)=varphi(t)f(x,y)$ with $varphi$ 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 $Omega$ at the initial time $t=0$ and at the final time $t=1$. Using the methods of Tikhonovs regularization and truncated integration, we construct the regularized solutions. Numerical part is given.
In this paper we shall consider the growth at infinity of a sequence $(P_n)$ of entire functions of bounded orders. Our results extend the results in cite{trong-tuyen2} for the growth of entire functions of genus zero. Given a sequence of entire func
tions of bounded orders $P_n(z)$, we found a nearly optimal condition, given in terms of zeros of $P_n$, for which $(k_n)$ that we have begin{eqnarray*} limsup_{ntoinfty}|P_n(z)|^{1/k_n}leq 1 end{eqnarray*} for all $zin mathbb C$ (see Theorem ref{theo5}). Exploring the growth of a sequence of entire functions of bounded orders lead naturally to an extremal function which is similar to the Siciaks extremal function (See Section 6).
We compute the degree complexity of a family of birational mappings of the plane with high order singularities.
Our main result in this paper is the following: Given $H^m, H^n$ hyperbolic spaces of dimensional $m$ and $n$ corresponding, and given a Holder function $f=(s^1,...,f^{n-1}):partial H^mto partial H^n$ between geometric boundaries of $H^m$ and $H^n$.
Then for each $epsilon >0$ there exists a harmonic map $u:H^mto H^n$ which is continuous up to the boundary (in the sense of Euclidean) and $u|_{partial H^m}=(f^1,...,f^{n-1},epsilon)$.
