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Use superposition to solve:

with boundary conditions:

and initial condition

.

Solution:

Let's consider a more general problem of heat equation of form:
( 1)
with initial condition:
( 2)
The solution of this equation can be expressed as a superposition of 2 elemantary solutions:
( 3)
where
u1 = the general solution of homogenous associated equation of (1)
u2 = a particular solution of non-homogenous equation (1)
There are several methods to determine the solution of (1), classical and modern, using the theory of distributions.
I prefer to use the theory of distributions, so that I will try to sketch the steps to determine the complete solution.
We will use the following definitions and integral transforms:

1) The "fundamental solution" of (1), denoted by (E) is the solution of associated equation:
...

Solution Summary

A heat equation with a moving source is investigated using the Dirac Impulse Function. The solution is detailed and well presented.

You have a mass-spring system, a unit impulse is applied to this system (at equilibrium,at rest) and the response is recorded and determined to be
(10e^-0.1t)- (10e^-0.2t)
In general terms what does the form of the impulse response function tell you about the system?

(a) Write an expression for the electric charge density p (r) of a point charge q at r/. Make
sure that the volume integral of p equals q.
(b) What is the charge density of an electric dipole, consisting of a point charge -q at the
origin and a point charge +q at a?
(c) What is the charge density of a uniform, infini

The heat flow in one dimension is governed by the partial differential equation [see the attachment for equation]
where [attached] is the temperature in space as a function of time. Using the Fourier transform in x solve this equation given the condition [attached]; C is a constant.

The heat transfer in a semi-infinite rod can be described by the following PARTIAL differential equation:
∂u/∂t = (c^2)∂^2u/∂x^2
where t is the time, x distance from the beginning of the rod and c is the material constant. Function
u(t,x) represents the temperature at the given time t and p

In the dirac notation for a quantum state of a system, an eigenstate wave function u_n is replaced by the vector |n>, and a general state wave function (see attached)
a) Translate the following mathematical statements to the corresponding forms n wave mechanics: i) = (see attached), where A is an hermitian operator.

Find the solution u(x,t) of the heatequation:
ut = 1/2 uxx
(a) with initial data u(x,0) = x
(b) with initial data u(x,0) = x^2
(c) with initial data u(x,0) = sinx
(d) with initial data u(x,0) = 0 x < 0 and u(x,0) = 1 or x >/= 0
I know the solution of the heatequation with given initial data is unique. So if you happe