Math 4600, Homework 10
1. One of the well-known phenomenological (capturing the phenomena, but
not necessarily the mechanisms) models of cancer is represnted by Gompertz
equation
dN
= −bN ln(N/K).
dt
a) Solve this equation with N (0) = N0 using the substitution u = ln(N/K), to
obtain the solution N (t) = K exp(−Ae−bt ), where A = − ln(N0 /K)
b) (computing) Plot the solution as a function of time for K = 1, N0 = 0.1,
b = 1. Illustrate with a plot and describe in words what happens to the tumor
dynamics if the growth rate b is varied. Compare with the logistic dynamics
Ṅ = −bN (N − K)
2. Do the following exercises for these functions:
a) f = y 2 sin x,
b) f = exp(−0.1(x2 + y 2 ))
∂2 f
∂2 f
- find ∂f
∂x , ∂x∂y , ∂y∂x , gradient of f
- (computing) Visualize the surface and the level curves given by the equation
z = f (x, y)
in Matlab. Explore different functions for plotting: mesh, meshz, meshc, surface
and see what they do. Type
help mesh
or similar to see the available options. Here is a sample code to help you:
[X,Y] = meshgrid(-5:.2:5, -6:.2:6);
z=sin(X).*cos(Y);
meshc(X,Y,z)
xlabel(’x’)
ylabel(’y’)
3. Concentration of nutients in a Petri dish (shallow pool of nutrients) can
be described by c(x, y) = C0 exp(−α(x2 + y 2 )). A bacteria is sitting at the
point (1, −1) and moves in the direction of increasing nutrients concentration.
In which direction will it move? Draw a picture. Where will it end up?
The following two problems refer to the one-dimensional diffusion equation
ut = Duxx
.
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4. Find the steady state solution of the one-dimensional diffusion equation
with boundary conditions ux (0, t) = c1 , u(L, t) = c2 , x ∈ [0, L]. Plot it.
5. In an experiment a substance of concentration A is released into a narrow
tube at x = 0, t = 0. It difuses along the tube with diffusion constant D = 1.
Detectors are setup along the tube at all locations x > 0. They can detect the
substance if the concentration is above 5% of A.
a) What is the furthest location X(t) where the detector will be responding at
time t? (Hint: You will need to solve the diffusion equation before you find
X(t). Use fundamental solution approach.)
b) Sketch the time evolution of X(t)
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