An inverse problem for the Euler-Bernoulli equation and a new scheme for solving a hierarchical size-structured model with nonlinear growth, mortality, and reproduction rates

A special technique has been developed for identification of the solution and unknown coefficient in the Euler-Bernoulli equation. The original problem of unknown coefficient identification from over-posed data is transformed into a higher-order well-possed problem following the idea of the Method of Variational Imbedding.;Two new schemes based on characteristic lines for solving a hierarchically structured population model with nonlinear growth, mortality and reproduction rate are proposed. The schemes are stable and have second order of approximation in x and t. The idea of the developed method is not to follow the characteristics from the initial condition, but for each time-step to recover through the characteristics the position at the previous time level of the functional values that arrive at a grid point on the net tile level. Numerical results confirm second order of convergence of the new schemes. The schemes are validated for three exact solution: a two continuous and one discontinuous. In addition, we compare the results of the new schemes with two known numerical numerical schemes for the model under consideration.