Recent Developments in Inverse Problems - Abstract

Rahimov, Anar

Numerical solution to inverse problems for hyperbolic equation

We consider the following inverse problem with respect to a hyperbolic type equation: beginequation labeleq1 beginarrayl fracpartial ^2v(x,t)partial t^2 = a_0(x,t)fracpartial ^2v(x,t)partial x^2 + a_1(x,t)fracpartial v(x,t)partial x + a_2 (x,t)fracpartial v(x,t)partial t +
+ a_3(x,t)v(x,t) + f(x,t) + F(x,t),,,,,F(x,t) = sumlimits_s = 1^L B_s(x,t)C_s(x),
left( x,t right) in Omega = left left( x,t right):0 < x < l,,,0 < t \leqslant T} \right\}, \\ \end{array} \end{equation} \noindent under initial, boundary, and additional conditions: \begin{equation} \label{eq3} v(x,0) = \varphi_0(x),\,\,\,\frac{\partial v(x,0)}{\partial t} = \varphi_1 (x),\,\,\,\,x \in \left[ {0,l} \right], \end{equation} \begin{equation} \label{eq4} v(0,t) = \psi_0(t),\,\,\,\,\,v(l,t) = \psi_1(t),\,\,\,\,t \in \left[ {0,T} \right], \end{equation} \begin{equation} \label{eq5} v(x,\bar {t}_s) = \varphi_{2s}(x),\,\,\,\,x \in \left[ {0,l} \right],\,\,\,\bar {t}_s \in \left( {0,T} \right],\,\,\,s = 1,...,L. \end{equation} \noindent Here $L > 0$ is given integer number, $bar t_s in left( 0,T right],,,,s = 1,...,L$ are given time instants; the functions $a_0(x,t) > 0,,a_1(x,t),,a_2(x,t),,a_3 (x,t)$, $f(x,t),,B_s(x,t)$, $varphi_0(x),,varphi _1(x)$, $psi_0(t),,psi_1(t)$, $varphi _2s(x),,,s = 1,...L$ are given, continuous with respect to $x$ and $t$; $B_s(x,t)$ are linear independent functions; the functions $varphi _0(x),,varphi_2s (x),,psi_0(t),,psi_1(t)$ satisfy the consistency conditions. The problem (1)-(4) consists in determining the unknown continuous $L$-dimension vector-function $C(x) =(C_1(x),...,C_L(x))$ and the corresponding solution to the boundary value problem $v(x,t)$, which is twice continuously differentiable with respect to $x$ and $t$. We propose an approach to numerical solution to the problem (1)-(4), which is based on the use of the method of lines. The initial problem is reduced to a system of ordinary differential equations with unknown parameters. To solve this system we propose an approach based on the sweep method [1]. We also consider the case when $F( x,t) = sumlimits_s = 1^L B_s (t)C_s (x,t) $, and instead of conditions in (4), there are the conditions [ v( bar x_s ,t) = psi _2s (t),,,,,bar x_s in left( 0,l right),,,,t in left[ 0,T right],,,,s = 1,...,L. ] All the necessary computational schemes, formulae, and results of the carried out numerical experiments will be given in the report. bigskip References [1] Aida-zade K.R., Rahimov A.B. An approach to numerical solution of some inverse problems for parabolic equations, Inverse Problems in Science and Engineering, 2014, Vol. 22, No 1, pp. 96-111.