Question

Solve the heat equation and find the steady state solution :

u_{xx}=u_{t} 0<x<1, t>0,
u(0,t)=T_{1}, u(1,t)=T_{2}, where T1 and T2 are
distinct constants, and u(x,0)=0

Answer #1

Doubt in this then comment below...i will explain you.

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**firstly we make boundary conditions homoegeneous by
taking a function v= Ax+B ...**

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 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

Find 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) using the Fourier sine
transform.

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).

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.)

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.

In each of Problems 1 through 8, find the steady-state solution
of the heat conduction equation α2uxx = ut that satisfies the given
set of boundary conditions.
1. ux (0, t) = 0, u( L, t) = 0
2. u(0, t) = 0, ux ( L, t) = 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)

8. Find the solution of the following PDE:
utt − 9uxx = 0
u(0, t) = u(3π, t) = 0
u(x, 0) = sin(x/3)
ut (x, 0) = 4 sin(x/3) − 6 sin(x)
9. Find the solution of the following PDE:
utt − uxx = 0
u(0, t) = u(1, t) = 0
u(x, 0) = 0
ut(x, 0) = x(1 − x)
10. Find the solution of the following PDE:
(1/2t+1)ut − uxx = 0
u(0,t) = u(π,t) =...

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).

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