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4. Finite Difference Schemes For The Heat Equation
150
Exercise 4.9 Consider the problem
u t =4u xx − 10u + q(x, t)
u( 2 ,t)=b(t),
u( 1 ,t)= a(t),
u(x, 0) = f(x),
where 2 > 1 are given constants, and a(t),b(t), and q(x, t) are given
functions.
(a) Derive an explicit scheme.
(b) Derive an implicit scheme. for x ∈ ( 1 , 2 ), t > 0,
(c) Suppose
t
t
1 = −2, 2 =3, a(t)= e − 2, b(t)= e +3, f(x)=1 + x,
and
t
q(x, t)=11e +10x.
Show that
t
u(x, t)= e + x
is an exact solution of the problem.

(d) Implement the schemes derived in (a) and (b) and compare the results
with the analytical solution derived in (c).

Exercise 4.10 Consider the problem

u t =(α(x, t)u x ) x + c(x, t)u x + q(x, t) for x ∈ ( 1 , 2 ), t > 0,
u( 1 ,t)= a(t), u( 2 ,t)=b(t),
u(x, 0) = f(x),

where 2 > 1 are given constants, and a(t),b(t),α(x, t),c(x, t), and q(x, t)
are given smooth functions.

(a) Derive an explicit scheme.
(b) Derive an implicit scheme.