代写Mathematical Biology Homework Assignment 5 2024–25代写Java程序
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Homework Assignment 5
2024–25
Please submit solutions to the following two questions as
Homework Assignment 5
by 16:00pm on Monday, November 25, 2024.
I. Consider the reaction-diffusion equation
under the additional conditions that u(0, t) = 0 = ∂x/∂u (π, t), where a is some positive constant.
(a) Assuming that Equation (1) admits separable solutions of the form u(x, t) = X(x)T(t), show that X(x) and T(t) have to satisfy the differential equations
X'' = λX and T˙ = (λ - a)T, (2)
where λ is a real constant.
(b) Solve Equation (2) for the functions X(x) and T(t) under the given conditions.
(c) Deduce that any function of the form.
with n = 0, 1, 2, . . . and Cn constant, is a solution for (1).
II. The Burgers-Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) advection-reaction- diffusion equation can be written in rescaled form. as
ut + kuux = uxx + u(1 - u), (4)
where k > 0 is a real constant.
(a) Determine the homogeneous – i.e. time- and space-independent – rest states of Equation (4)
(b) Let z = x - ct, with c positive, and derive the travelling wave equation corre- sponding to (4) that is satisfied by U(z).
(c) Rewrite that equation as the first-order system
U, = V, (5a)
V, = -cV + kUV - U(1 - U); (5b)
then, determine the equilibria thereof, and decide their stability.
(d) Given your findings in item (c), deduce that monotonic front solutions to (4) only exist for c > 2.
(e) Verify that, for c = 2/k + k/2 with k > 2, a heteroclinic connection between the equilibria of (4) is given explicitly by
V(U) = -2/KU(1 - U) (6)