Partial differential equations
The temperature distribution Θ(x, t) along an insulated metal rod of length L is described by the differential equation
∂2Θ/∂x2 =1/D*∂Θ/∂t for (0<x<L, t>0)
where D ( is not equal to zero) is a constant. The rod is held at a fixed temperature of 0◦C at one end and is insulated at the other end, which gives rise to the boundary conditions Θ(0, t) = 0 and Θx(L, t) = 0, for t > 0.
The initial temperature distribution in the rod is given by
Θ(x, 0) = 0.4 sin (3πx/2L) (0 ≤ x ≤ L).
(a) Use the method of separation of variables, with Θ(x,t) = X(x)T(t),
to show that the function X(x) satisfies the differential equation X′′ − μX = 0
for some constant μ. Write down the corresponding differential equation that T (t) must satisfy.
(b) Find the boundary conditions that X(x) must satisfy.
(c) Show that if μ = 0, then the only solution of equation (1) that
satisfies the boundary conditions is the trivial solution X(x) = 0.
(d) Show that if μ = c2 with c > 0, then the only solution of equation (1)
that satisfies the boundary conditions is the trivial solution X(x) = 0.
(e) Suppose that μ < 0, so μ = −k2 for some k > 0. Find the non-trivial
solutions of equation (1) that satisfy the boundary conditions, stating
clearly what values k is allowed to take.
(f) Solve the differential equation found in part (a) that the function T (t)
(g) Use your answers to write down a family of product solutions
Θn(x, t) = Xn(x) Tn(t) that satisfy the first two boundary conditions. Hence show the general solution of the partial differential equation
(h) Find the particular solution that satisfies the given initial temperature distribution.
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