Suppose that u(x, t) satisfies the heat equation Ut = α-Uxx, a²uxx, 0 0, with a² being the thermal diffusivity. The boundary conditions u(0, t) = 0, ux(1, t) +2u(1, t) = 0, t> 0, are imposed. (a) Using the separation of variables u(x, t) = X(x)T(t), derive the ordinary differential equations X"(x) + AX(x) = 0, İ(t) + Aa²T (t) = 0, where A is the separation constant; and express the boundary conditions in terms of X(x). (b) Show that there are no non-trivial solutions for A ≤ 0. (A graphical explanation would suffice for A< 0). (c) Show that non-trivial solutions exist for λ = µ²> 0 provided that tan μ == 2 By considering the graphs of tan µ and -µ/2, explain why there are an infinite num- ber of positive eigenvalues A = An, (n = 1,2,3,...).
Suppose that u(x, t) satisfies the heat equation Ut = α-Uxx, a²uxx, 0 0, with a² being the thermal diffusivity. The boundary conditions u(0, t) = 0, ux(1, t) +2u(1, t) = 0, t> 0, are imposed. (a) Using the separation of variables u(x, t) = X(x)T(t), derive the ordinary differential equations X"(x) + AX(x) = 0, İ(t) + Aa²T (t) = 0, where A is the separation constant; and express the boundary conditions in terms of X(x). (b) Show that there are no non-trivial solutions for A ≤ 0. (A graphical explanation would suffice for A< 0). (c) Show that non-trivial solutions exist for λ = µ²> 0 provided that tan μ == 2 By considering the graphs of tan µ and -µ/2, explain why there are an infinite num- ber of positive eigenvalues A = An, (n = 1,2,3,...).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:2. Suppose that u(x, t) satisfies the heat equation
Ut
= α²
a²uxx, 0<x<1, t>0,
with a² being the thermal diffusivity. The boundary conditions
u(0, t) = 0, ux(1, t) +2u(1, t) = 0, t> 0,
are imposed.
(a) Using the separation of variables u(x, t) = X(x)T(t), derive the ordinary differential
equations
X"(x) + AX(x) = 0,
İ(t) + Aa²T (t) = 0,
where A is the separation constant; and express the boundary conditions in terms of
X(x).
(b) Show that there are no non-trivial solutions for A ≤ 0. (A graphical explanation would
suffice for A < 0).
(c) Show that non-trivial solutions exist for A
=
tan u
=
µ² > 0 provided that
μ
2
By considering the graphs of tan µ and —µ/2, explain why there are an infinite num-
ber of positive eigenvalues λ = An, (n = 1,2,3,...).
(d) Determine the general solution u(x, t) in terms of the (unknown) eigenvalues An.
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