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4.6 Exercises
149
Exercise 4.5 Consider the following initial-boundary value problem
t > 0,
x ∈ (0, 1),
for
u t = u xx
u(1,t)=u r (t),
u(0,t)= u (t),
u(x, 0) = f(x).
Here u (t),u r (t), and f(x) are bounded functions satisfying u (0) = f(0)
and u r (0) = f(1).
(a) Derive an explicit scheme for this problem.
(b) Derive an implicit scheme for this problem and show that the linear
system that arises can be solved by Gaussian elimination.
Exercise 4.6 Derive an explicit scheme for the following Neumann prob-
lem:
u t = u xx for x ∈ (0, 1), t > 0,
u x (0,t)= u x (1,t)=0,
u(x, 0) = f(x).
Use the analytical solution given in Example 3.5 on page 101 to check the
quality of your approximations.
Exercise 4.7 Repeat Exercise 4.6 by deriving an implicit approximation
of the problem. Compare the numerical solutions provided by the explicit
and the implicit schemes.
Exercise 4.8 Consider the problem

u t = αu xx for x ∈ (− , ), t > 0,
u(− , t)= a, u( , t)=b,
u(x, 0) = f(x).

where a, b and , α > 0 are given constants.
(a) Derive an explicit scheme.

(b) Derive an implicit scheme.
(c) Find the exact solution when α =2, = π, a = −π, b = π, and
f(x)= x + sin (3x).
(d) Implement the schemes derived in (a) and (b) and compare the results
with the analytical solution derived in (c).