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book Mobk070 March 22, 2007 11:7
STURM–LIOUVILLE TRANSFORMS 151
Problems
Use an appropriate Sturm–Liouville transform to solve each of the following problems:
1. Chapter 3, Problem 1.
2. Chapter 2, Problem 2.
3. Chapter 3, Problem 3.
∂u 1 ∂ ∂u
= r + G(constant t)
∂t r ∂r ∂r
4. u(r, 0) = 0
u(1, t) = 0
u bounded
5. Solve the following using an appropriate Sturm–Liouville transform:
2
∂ u ∂u
=
∂x 2 ∂t
u(t, 0) = 0
u(t, 1) = 0
u(0, x) = sin(πx)
6. Find the solution for general ρ(t):
2
∂u ∂ u
=
∂t ∂x 2
u(t, 0) = 0
u(t, 1) = ρ(t)
u(0.x) = 0
FURTHER READING
V. S. Arpaci, Conduction Heat Transfer, Reading, MA: Addison-Wesley, 1966.
R. V. Churchill, Operational Mathematics, 3rd ed. New York: McGraw-Hill, 1972.
I. H. Sneddon, The Use of Integral Transforms, New York: McGraw-Hill, 1972.
