Jun 2, 1999 - Louisiana, Lafayette, Louisiana. Abstract. In this paper we develop a monotone approximation method, based on an upper and lower solutions ...
QUARTERLY OF APPLIED MATHEMATICS VOLUME LVII, NUMBER 2 JUNE 1999, PAGES 261-267
A MONOTONE APPROXIMATION FOR THE NONAUTONOMOUS SIZE-STRUCTURED POPULATION MODEL By AZMY S. ACKLEH and KENG DENG Department
of Mathematics,
University
of Southwestern
Louisiana,
Lafayette,
Louisiana
Abstract. In this paper we develop a monotone approximation method, based on an upper and lower solutions technique, for solving the nonautonomous size-structured model. Such a technique results in the existence and uniqueness of solutions for this equation. Furthermore, we establish a first-order convergence of the method and present a numerical example.
1. Introduction. bolic initial-boundary population:
(1.1)
In this paper we consider the following first-order nonlocal hypervalue problem that describes the dynamics of a size-structured
v,t + (g{t, x)u)x = —m(t, x) u,
0 < t < T,
g(t,a)u(t,a) = C(t) + u(0,x) = uo(x),
0 X) TO(T'X) W(T>X) dx dT' Letting
x) — e_7t£(£, x), where 7 (> 0) is chosen so that 7 —m > 0 on Dt, then we
have that
e7' f^ w(t, x)((t, x)dx
< fl e7TC(r,a) f* q{r, x)w(t, x) dx dr + Jo /a W(T' x)e7T[Cr + 9{r, x)(x] dx dr
+ Jo fa eJT((T>x) (7 - "i(r, a1))W(T> dx drWe now set up the following backward
problem:
Cr + sCx =0, £(r, b) —0, C(M) = X(z),
(2.4)
0 < t < i, 0< r
0 < s < t,
Note that the initial and boundary
Substituting
a < x < b,
a < x < b.
values for £ imply that 0 < C < 1 on Dt-
such a ( in (2.3) yields nb
(2.6)
ft
/ w(t,x)x(x)dx Ja
pb
< L / / w(t,x)+ dxdr, Jo J a
where L = max [q(t, x) + 7 — m(t, a;)]. Dt
Since this inequality verging to
holds for every x, we can choose a sequence
1 if w(t, x) > 0,
X={ Consequently,
{xn} °n (a, b) con-
0
otherwise.
we find that fb
nt
/ w(t,x)+dx
< L /
Ja
nb
/ w(T,x)+dxdr,
J0 Ja
which by Gronwall's inequality leads ds to
f rb
w(t, x)+dx = 0
J a
Thus, the conclusion follows.
Remark
Dt-
2.2. If we assume that u,v £ L°°(Dt),
then we obtain that u > v a.e. on
264
AZMY S. ACKLEH and
KENG DENG
3. Monotone approximation scheme. We start this section by constructing monotone sequences of upper and lower solutions. To this end, let a°(t,x) and /3°(t,x) be a lower solution and an upper solution of (1.1), respectively. We now define our two sequences {a^}^ and as follows:
at + (^(^i X)ak)x = —m(t,x)
(3.1)
ak,
g(t,a)ak(t,a) = Rk~l(t), ak(0,x) = uq(x),
where Rk~1(t) = C(t) +
0
