We consider linear and non-linear ill posed inverse problems for parabolic
equation. The linear case consists in reconstruction of the temperature
field from a given Cauchy data on the part of boundary solution
domain. Numerical
solution for this problem is based on Landweber method in every iteration of
which two direct well posed mixed initial boundary value problem are solved.
The non-linear problem is concerned with reconstructing a part of boundary
from the given Cauchy data on the known boundary. The Newton method is used
for corresponding non-linear operator equation and as result the parabolic
initial boundary value problems need to be solved on every iteration step.
The numerical solution of direct well posed non-stationary problems is
realized by boundary integral equation method.
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