Piotr Zwierkowski CONVERGENCE OF A FINITE DIFFERENCE SCHEME FOR VON FOERSTER EQUATION WITH FUNCTIONAL DEPENDENCE

We analyse a finite difference scheme for von Foerster–McKendrick type equations with functional dependence forward in time and backward with respect to one dimensional spatial variable. Some properties of solutions of a scheme are given. Convergence of a finite difference scheme is proved. The presented theory is illustrated by a numerical example. Introduction Von Foerster–McKendrick type models are well known models of mathematical biology, describing a population with a structure of its members, given for example by their age [3], size [1] or level of maturation of individuals [7]. Existence, uniqueness and other properties of solutions for above mentioned models are studied in the literature. We are interested in some class of initial problems, originating in [7], which presents erytroid production model, based on a continuous maturation-proliferation mechanism. Far-reaching generalization of this problem is presented in [9]. In this paper we deal with the problem considered in [9] with one dimensional spatial variable. Let T > 0, τ0, τ1 ∈ R+, where R+ = [0,+∞). Let us introduce I0 = [−τ0, 0], I = [0, T ], B = [−τ0, 0]× [−τ1, τ1], E0 = [−τ0, 0]× R+, E = [0, T ]× R+. For a given function q : I0 ∪ I → R, t ∈ I define the Hale operator qt : I0 → R by qt(s) = q(t+ s), s ∈ I0, 2000 Mathematics Subject Classification: 65M06, 65M12, 35R10.