Partial Differential Equations - Heat Equation
Not what you're looking for?
The causal solution f(x t) of the one-dimensional diffusion equation with source term f
See attached
Purchase this Solution
Solution Summary
The inhomogeneous heat equation (partial differential equation) in 1D can be solved by means of Green function
Solution Preview
The causal solution of the one-dimensional diffusion equation with source term f
(1)
is given by
(2)
where is the causal Green function satisfying
(3)
a) Change variables in (3) to , . Letting with the Heaviside step function , show (3) is solved if satisfies
(4)
Solve by Fourier transforming with respect to the position variable. Hence obtain an expression for
b) The temperature of a long uniform insulating metal rod satisfies
A heating element is used to create an initial temperature distribution Writing obtain an equation for , and combining (1), (2) and the answer to a) identify the temperature variation at the center of the rod, .
Solution:
a) After changing the variables, equation (3) becomes:
(5)
and putting
(6)
one yields:
(7)
where the following property of Dirac delta distribution was used:
(7')
The other partial derivative will be:
...
Purchase this Solution
Free BrainMass Quizzes
Intro to the Physics Waves
Some short-answer questions involving the basic vocabulary of string, sound, and water waves.
Classical Mechanics
This quiz is designed to test and improve your knowledge on Classical Mechanics.
Variables in Science Experiments
How well do you understand variables? Test your knowledge of independent (manipulated), dependent (responding), and controlled variables with this 10 question quiz.
Basic Physics
This quiz will test your knowledge about basic Physics.
The Moon
Test your knowledge of moon phases and movement.