The equations of a reaction diffusion system are expressed in the form partial d

Question

The equations of a reaction diffusion system are expressed in the form partial derivative u/partial derivative t = au (1 - u^2) - bv + D_1 nabla^2 u partial derivative u/partial derivative t = cu - dv + D_2 nabla^2 v where D_1 and D_2 are diffusion coefficients. Consider a 1-D spatial domain so that dimensionless state variables u(x, t) and v(x, t) depend on position (x, m) and time (t, sec). What are the units of the parameters a, b, c, d, D_1 and D_2 ? In the absence of diffusion, what uniform equilibrium solutions (u(x, t) = u_e, v(x, t) = v_e) exist ? (under what conditions) Suppose the following parameter values are given: a = 0.5, b = 1, c = 1, d = 1, D_1 = 0.001, D_2 = 0.1 Is the uniform equilibrium condition (u_e = 0, v_e = 0) stable? I.e., will small perturbations, consisting of spatial waves of wave number k, grow or decay? For which values of k? You may solve part c) numerically - it is not necessary to have a closed form solution.

Explanation / Answer

a) Unit is 1/S

b) A t = f(A, B) + DA 2A x2 , (6.2a) B t = g(A, B) + DB 2B x2 ,

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