Page 460 - Applied Numerical Methods Using MATLAB

PROBLEMS 449
y 4
(e) ∂u(x, y)/∂x =− cos y, ∂u(x, y)/∂x =−e sin 4 − e cos y
x=0 x=4
(P9.3.10)
4 x
∂u(x, y)/∂y = cos x, u(x, 4) = e cos x − e cos 4 (P9.3.11)
y=0

y 4
(f) ∂u(x, y)/∂x =− cos y, u(4,y) = e cos 4 − e cos y(P9.3.12)
x=0
4 x
∂u(x, y)/∂y = cos x, ∂u(x, y)/∂y = e cos x + e sin 4
y=0 y=4
(P9.3.13)
4
y
y
(g) u(0,y) = e − cos y, ∂u(x, y)/∂x =−e sin 4 − e cos y(P9.3.14)
x=4
4 x
∂u(x, y)/∂y = cos x, ∂u(x, y)/∂y = e cos x + e sin 4
y=0 y=4
(P9.3.15)
y 4
(h) ∂u(x, y)/∂x =− cos y, ∂u(x, y)/∂x =−e sin 4 − e cos y
x=0 x=4
(P9.3.16)
4 x
∂u(x, y)/∂y = cos x, ∂u(x, y)/∂y = e cos x + e sin 4
y=0 y=4
(P9.3.17)
9.4 Parabolic PDEs: Heat Equations
Modify the program “solve_heat.m” (in Section 9.2.3) so that it can solve
the following PDEs by using the explicit forward Euler method, the implicit
backward Euler method, and the Crank–Nicholson method.

2
∂ u(x, t) ∂u(x, t)
(a) = for 0 ≤ x ≤ 1, 0 ≤ t ≤ 0.1 (P9.4.1)
∂x 2 ∂t
with the initial/boundary conditions

4
u(x, 0) = x , u(0,t) = 0, u(1,t) = 1 (P9.4.2)
(i) With the solution region divided into M × N = 10 × 20 sections,
does the explicit forward Euler method converge? What is the value
2
of r = A t/( x) ?
(ii) If you increase M and N to make M × N = 20 × 40 for better
accuracy, does the explicit forward Euler method still converge?
2
What is the value of r = A t/( x) ?
(iii) What is the number N of subintervals along the t axis that we should
choose in order to keep the same value of r for M = 20? With that
value of r, does the explicit forward Euler method converge?