In this paper, we investigate the Cauchy problem for the modified Helmholtz equation. We consider the data completion problem in a bounded cylindrical domain on which the Neumann and the Dirichlet conditions are given in a part of the boundary. Since this problem is ill-posed, we reformulate it as an optimal control problem with an appropriate cost function. The method of factorization of boundary value problems is used to immediately obtain an approximation of the missing boundary data. In order to regularize this problem, we firstly scrutinize two classical regularizations for the cost function. Then we propose a new numerical regularization named “adaptive Runge–Kutta regularization”, which does not require any penalization term. Finally, we compare them numerically.

The resolution of a system for modeling the competition between opponents in a chemostat when one of these can produce a toxin has been studied. We propose a novel method to overcome the analytical difficulties of standard mathematical methods. The method is based on the variational iteration method and combined with the Gauss-Seidel technique for increasing the convergence rate. Numerical examples are considered to demonstrate the practicality and improve the convergence of the proposed method.