Question

u_{xx} = u_{t} - 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).

Answer #1

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.

Determine the solution of the following initial boundary-value
problem
Uxx=4Utt 0<x<Pi t>0
U(x,0)=sinx 0<=x<Pi
Ut(x,0)=x 0<=x<Pi
U(0,t)=0 t>=0
U(pi,t)=0 t>=0

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 wave equation on the whole line (no boundary
conditions) with initial conditions: u(x,0) = 0, ut
(x,0)=xe^(-x^2)

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.)

(PDE)
WRITE down the solutions to the ff initial boundary problem for
wave equation in the form of Fourier series :
1. Utt = Uxx ; u( t,0) = u(t,phi) = 0 ; u(0,x)=1 , Ut( (0,x) =
0
2. Utt = 4Uxx ; u( t,0) = u(t,1) = 0 ; u(0,x)=x , Ut( (0,x) =
-x

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 wave equation Utt - C^2 Uxx = 0 with initial condtions
:
1) u(x,0) = log (1+x^2), Ut(x,0) = 4+x
2) U(x,0) = x^3 , Ut(x,0) =sinx
(PDE)

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 following initial/boundary value problem:
∂u(t,x)/∂t = ∂^2u(t,x)/∂x^2 for t>0, 0<x<π,
u(t,0)=u(t,π)=0 for t>0,
u(0,x)=sin^2x for 0≤x≤ π.
if you like, you can use/cite the solution of Fourier sine
series of sin^2(x) on [0,pi] = 1/4-(1/4)cos(2x)
please show all steps and work clearly so I can follow your
logic and learn to solve similar ones myself.

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