Homework

Consider the reaction-diffusion system of the form

@u1

@t = Du1 @@x 2u21 + @@y 2u21 + f(u1; u2) in Ω;

@u2

@t = Du2 @@x 2u22 + @@y 2u22 + s(u1; u2) in Ω;

@v1

@t = h1(u1; u2; v1; v2) on @Ω;

@v2

@t = h2(u1; u2; v1; v2) on @Ω;

–D

u1

@u1

@x = h1(u1; u2; v1; v2) on@Ω;

–D

u2

@u2

@y = h2(u1; u2; v1; v2) on@Ω;

(1)

where

h1(u1; u2; v1; v2) = c1u1 – d1v1;

h2(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

Ω

@u1

@t dx dy+ZZ

@Ω

@v1

@t dx dy = ZZ

Ω

D

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

Ω

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

Ω

@u2

@t dx dy+ZZ

@Ω

@v2

@t dx dy = ZZ

Ω

D

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

Ω

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

# Mathematics

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