Fractional reaction-diffusion equation for species ... - ResearchGate

Fractional reaction-diffusion equation 13
Therefore, by (36) and Proposition 3,
for all x and
0 ≤ [T ( t n )S N ( t n )]n u 0 (x) = [T ( t n )S( t n )]n u 0 (x) ≤ NK (37)
0 ≤ [S N ( t n )T ( t n )]n u 0 (x) = [S( t n )T ( t n )]n u 0 (x) ≤ NK (38)
for all x. This also shows that 0 ≤ [u N (t)](x) ≤ NK for all x in view of (35).
Therefore u N (t) is a mild solution of (27), too, since f N (u(x)) = f(u(x)) for
0 ≤ u(x) ≤ NK. Since f is locally Lipschitz, a well known result [44, Chapter
6, Theorem 1.4] implies that (27) has a unique local mild solution and since
u N (t) is defined for all t > 0 it follows that u N (t) is the unique global mild
solution of (27) and is given by the Trotter product formula (32) in view of
(35), (37) and (38). Finally, an easy computation shows that the function
y ↦→ [ ˜S(t)](y) in (14) is concave down on y > 0 for any t > 0. Therefore, by
Jensen’s inequality [21, pp. 153–154],
[∫ ∞
]
[S(t)T (t)u 0 ](x) = ˜S(t) k t (y)u 0 (x − y) dy
−∞
Thus,
≥
=
∫ ∞
−∞
∫ ∞
−∞
k t (y) ˜S(t) [u 0 (x − y)] dy
k t (y) [S(t)u 0 ] (x − y) dy = [T (t)S(t)u 0 ](x).
S N (t)T (t)u 0 = S(t)T (t)u 0 ≥ T (t)S(t)u 0 = T (t)S N (t)u 0
which finishes the proof by Proposition 1, (37) and (38).
Corollary 1 Under the assumptions of Theorem 1, if u 0 and its first two
derivatives in x exist and and belong to C 0 (R), then (27,28) has a unique
strong solution u(t) = W (t)u 0 for all u 0 ≥ 0 in X given by the Trotter
Product Formula (32).
Proof In this case Proposition 2 shows that u 0 is in the domain of the operator
A, and since f : X → X is continuously differentiable, u is also a strong
solution by [44, Chapter 6, Theorem 1.5].
⊓⊔
Theorem 1 shows that the fractional reaction-diffusion equation (11) in
continuous time can be solved numerically by computing solutions to one of
its discrete time counterparts (15) or (16) with τ = t/n. The approximate
solutions u n (x, t) converge to the unique solution u(x, t) of the fractional
reaction-diffusion equation at any time t > 0 for any smooth initial population
density u 0 (x) as n → ∞. This result links the continuous time partial
differential equation model with the corresponding discrete time integrodifference
equation model. Furthermore, the approximation (15) gives a lower
bound to the exact solution while the approximation (16) gives an upper
bound. Hence these two approximation can be compared to yield exact error
bounds. See Section 4 for an illustration.
⊓⊔