| Popis: |
Motivated by a recent report by R. E. Mickens, we design an efficient, non-standard, two-step, nonlinear, explicit, exact finite-difference method to approximate solutions of a population equation with squareroot reaction law. Mickens’ report establishes the existence of nonnegative, traveling-wave solutions of that model which are bounded from above by 1, and which are spatially and temporally monotone. As its analytic counterpart, the computational technique proposed in the present manuscript is capable of preserving the non-negativity and the boundedness of initial profiles under suitable and flexible conditions on the computational parameters. We provide theoretical results on the existence and uniqueness of non-negative and bounded solutions of the method, and we establish that our technique conditionally preserves the spatial and temporal monotonicity of the approximations. The numerical simulations obtained through a computer implementation of our finite-difference scheme support the fact that the method preserves all of the mathematical characteristics of approximations mentioned above. |
