Question

The heat equation is: ∂u/∂t = α∇^2u

where α is the thermal diffusivity of a substance, and u(x, y, z, t) describes the temperature of a substance as a function of space and time. Show that the separation of variables procedure can be successfully applied to this partial differential equation to separate the space and time variables. You should obtain, as your final answer, a series of ordinary differential equations.

Answer #1

let u=f(x)*g(y)*h(z)*T(t)

then LHS of the equation=f*g*h*T'

RHS of the equation becomes:

hence the resultant equation becomes:

dividing both sides by f*g*h*T,

we get

now each term is independently function of their own variable only.

hence assigning

we get 4 ODEs and one constraint i.e. k1=k2+k3+k4

(a) Separate the following partial differential equation into
two ordinary differential equations: Utt + 4Utx − 2U = 0. (b) Given
the boundary values U(0,t) = 0 and Ux (L,t) = 0, L > 0, for all
t, write an eigenvalue problem in terms of X(x) that the equation
in (a) must satisfy.

1. Solve fully the heat equation problem: ut = 5uxx u(0, t) =
u(1, t) = 0 u(x, 0) = x − x ^3 (Provide all the details of
separation of variables as well as the needed Fourier
expansions.)

(a) Separate the following partial differential equation into
two ordinary differential equations: e 5t t 6 Uxx + 7t 2 Uxt − 6t 2
Ut = 0. (b) Given the boundary values Ux (0,t) = 0 and U(2π,t) = 0,
for all t, write an eigenvalue problem in terms of X(x) that the
equation in (a) must satisfy. That is, state (ONLY) the resulting
eigenvalue problem that you would need to solve next. You do not
need to actually solve...

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