Abstract:
We consider two inverse problems for determining the coefficients for a
one-dimensional nonlinear diffusion-reaction equation of the
Fisher–Kolmogorov–Petrovsky–Piskunov type. The first problem consists in
determining the kinetic coefficient for a nonlinear lower term, depending
only on the time variable, according to a given integral condition. And the
second problem consists in determining the time-dependent diffusion
coefficient, again according to a given integral condition.
To solve both problems, the time derivative of the derivative is first
sampled. In the first problem, the diffusion term is approximated in time
according to the implicit scheme, and the nonlinear minor term in the
semi-explicit scheme. And in the second problem, the diffusion term is
approximated in time in an explicitly implicit scheme, and the nonlinear
minor term is again in a semi-explicit scheme. As a result, both problems
reduce to differential-difference problems with respect to functions
depending on the spatial variable. For numerical solution of the problems
obtained, a non-iterative computational algorithm is proposed, based on
reduction of the differential-difference problem to two direct
boundary-value problems and a linear equation with respect to the unknown
coefficient. On the basis of the proposed numerical method, numerical
experiments were performed for model problems.