Mathematics

Homework

Consider the reaction-diffusion system of the form
@u1
@t = Du1 @@x 2u21 + @@y 2u21 + f(u1; u2) in ;
@u
2
@t = Du2 @@x 2u22 + @@y 2u22 + s(u1; u2) in ;
@v
1
@t = h1(u1; u2; v1; v2) on @;
@v
2
@t = h2(u1; u2; v1; v2) on @;
D
u1
@u1
@x = h1(u1; u2; v1; v2) on@;
D
u2
@u2
@y = h2(u1; u2; v1; v2) on@;
(1)
where
h1(u1; u2; v1; v2) = c1u1 d1v1;
h
2(u1; u2; v1; v2) = c2u2 d2v2; (2)
where
f(u1; u2) = a1u1 b1u1u2;
s
(u1; u2) = a2u2 + b2u1u2: (3)
u1(x; y; t), u2(x; y; t), v1(x; y; t) and v2(x; u; t) at position (x; y) 2 Ω and time t 2 [0; 1),
and
Du , Dv are diffusion coefficient which are assumed constant. Parameters c1, c2 , d1
and d2 are positive parameters. The functions f and s are usual polynomials or rational
functions.
0.1 Question : Find the Steady state solution for (1)
First, the dependent variables u1 and v1
Integrating (1) for u1 and v1 with respect to x and y over Ω we find
ZZ

@u1
@t dx dy+ZZ
@
@v1
@t dx dy = ZZ

D
u1 @@x 2u21 + @@y 2u21 dx dy+ZZ

f(u1; u2) dx dy
you have to using the Divergence theorem.
Similarly
1

second, the dependent variables u2 and v2
Integrating (1) for u2 and v2 with respect to x and y over Ω we find
ZZ

@u2
@t dx dy+ZZ
@
@v2
@t dx dy = ZZ

D
u2 @@x 2u22 + @@y 2u22 dx dy+ZZ

s(u1; u2) dx dy
you have to using the Divergence theorem.
reference :
A Primer on PDEs Models, Methods, Simulations Authors: Salsa, S., Vegni, F., Zaretti,
A., Zunino, P.
(unfortunately I do not have pdf for this book).
I want it done by 24 hour.
2

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