Question

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.)

Answer #1

Solve the heat equation and find the steady state solution :
uxx=ut 0<x<1, t>0,
u(0,t)=T1, u(1,t)=T2, where T1 and T2 are
distinct constants, and u(x,0)=0

Using separation of variables to solve the heat equation, ut =
kuxx on the interval 0 < x < 1 with boundary conditions ux
(0, t ) = 0 and ux (1, t ) = 0, yields the general solution,
∞
u(x,t) = A0 + ?Ane−kλnt cos?nπx? (with λn = n2π2)
n=1DeterminethecoefficientsAn(n=0,1,2,...)whenu(x,0)=f(x)= 0,
1/2≤x<1 .

uxx = ut - u (0<x<1, t>0),
boundary conditions: u(1,t)=cost, u(0,t)= 0
initial conditions: u(x,0)= x
i) solve this problem by using the method of separation of
variables. (Please, share the solution step by step)
ii) graphically present two terms(binomial) solutions for
u(x,1).

Solve the heat equation ut = k uxx, 0 < x < L, t >
0
u(0, t) = u(L, t) = 0, t > 0
u(x, 0) = f(x), 0 < x < L
a) f(x) = 6 sin 9πx L
b) f(x) = 1 if 0 < x ≤ L/2 2 if L/2 < x < L

Solve the heat equation and find the steady state solution:
uxx = ut, 0 < x < 1, t > 0,
u(0,t) = T1, u(1,t) = T2, where T1
and T2 are distinct
constants, and u(x,0) = 0

Solve the below boundary value equation
1. Ut=2uxx o<x<pi 0<t
2. u(0,t) = ux(pi,t) 0<t
3. u(x,0) = 1-2x 0<x<pi

Solve the wave equation:
utt = c2uxx, 0<x<pi, t>0
u(0,t)=0, u(pi,t)=0, t>0
u(x,0) = sinx, ut(x,0) = sin2x, 0<x<pi

Please solve the following:
ut=kuxx+sin3πx, 0<x<1, t>0
u(0,t)=u(1,t)=0, t>0
u(x,0)=sinπx, 0<x<1

Use the Fourier sine transform to derive the solution formula
for the heat equation ut = c2 uxx
on the half-infinite bar (0 ≤ x < ∞) with Dirichlet boundary
condition u(0, t) = a, for some constant a, and initial condition
u(x, 0) = f(x).

Solve ut=uxx, 0 < x < 3, given the
following initial and boundary conditions:
- u(0,t) = u(3,t) = 1
- u(x,0) = 0
Please write clearly and explain your reasoning.

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