Session S08 - Inverse Problems and Applications

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Solution of an integro-differential equation with conditions of Dirichlet using techniques of the inverse moments problem.

María Beatriz Pintarelli

Universidad Nacional de La Plata, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We want to find $w(x,t)$ such that \[w_{t}= \int_{0}^{t}k(t-s)w_{xx}(x,s)ds+f(x,t)\] about a domain $E= \left\lbrace (x,t),\quad 00 \right\rbrace $, with conditions \[w(x,0)=h_{1}(x); \qquad w(0,t)=k_{1}(t); \qquad w(L,t)=k_{2}(t)\] where $k(x)$ has continuous derivative on $x=0$, the value of $k(0)$ is known, $f(x,t)$ known and derivative with respect to $ t $ continuous, using the problem generalized moments techniques.

We will see that an approximate solution of the equation integro-differential can be found using the techniques of generalized inverse moments problem and bounds for the error of the estimated solution.

First the problem is reduced to solving a hyperbolic or parabolic partial derivative equation considering the unknown source. The method consists of two steps. In each one an integral equation is solved numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.

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