Tagged Questions
16
votes
1answer
276 views
Seeking Fourier series solution on Laplace equation…still looking, am I on track?
Okay, I've been working at this a couple of days now, I will try to give relevant details but will omit some intermediate steps.
The problem as given says:
Consider the BVP for $u=u(x,y)$:
...
1
vote
1answer
34 views
Steady Temperature Distribution Pipe
I was wondering if anyone can show me what approach to finding the steady state temperature distribution in this problem. The image is in the link below.
...
1
vote
1answer
86 views
Laplace heat equation
An infinite straight metal pipe has annular cross-section $a \leq r \leq b$. The temperature of the inner surface of the pipe is equal to $\cos(\theta)$, and the outer surface is thermally insulted.
...
1
vote
1answer
65 views
Alternative complete bases for Fourier Series.
Knowing that
$$\left\{ \sin\left(kx\right)\right\} _{k\in\mathbb{N}}$$
and
$$\left\{ \cos\left(kx\right)\right\} _{k\in\mathbb{N\cup}\left\{ 0\right\} }$$
are complete systems in $L^2(0,\pi)$. How ...
5
votes
3answers
209 views
A question related to Wave Equation
Let $L>0$. Suppose $f, g$ are $C^2$ functions on $\mathbb{R}$ such that
$$f(t)+f(-t)+\int_{-t}^t g(s)\,ds=0$$ and
$$f(L+t)+f(L-t)+\int_{L-t}^{L+t} g(s)\,ds=0$$
for all $t\in \mathbb{R}.$
Does it ...
2
votes
0answers
36 views
What are the connections between spectral expansion and differential operator?
For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion,
$$f(x)=\sum_n \hat{f}(n) e^{inx},$$
where the exponentials are eigenfunctions for differential operator ...
1
vote
1answer
100 views
Theoretical Understanding of Fourier Series
I understand mathematically how it is derived, and how it works (ie how it is applied and the significance of Fourier Series in solving PDEs). What I want to know is why it works, theoretically and ...
3
votes
3answers
178 views
Fourier Series: Integral of a Sum or Sum of Integrals?
While touching on Fourier series in a PDEs course, our professor basically waved her hands at the concept that
$$
...
1
vote
2answers
200 views
Wave equation with initial and boundary conditions - is this function right?
If $y(x,t)$ satisfies the 1-dimensional wave equation
$$\frac{\partial^2y}{\partial t^2}=c^2\frac{\partial^2y}{\partial x^2}\quad\text{for }0\leq x \leq l$$
with boundary conditions
...
1
vote
0answers
115 views
Expansion in cosine Fourier series
In solving the following problem with the method of separation of variables
$$
u_{tt}=u_{xx} \quad 0<x<\pi, t>0 \\
u(x,0)=g \quad 0 \leq x \leq \pi \\
u_x(t,0)=u_x(t,\pi)=0 \quad t>0
$$
...
1
vote
3answers
341 views
Solution of Laplace's equation in an annulus with constant Dirichlet conditions?
What's the solution to Laplace's equation $\nabla^2V=0 $ in the annulus with centre 0, inner radius 1, and outer radius 2, with boundary conditions $V=0$ on the inner boundary and $V=1$ on the outer ...
3
votes
1answer
247 views
heat equation solution
This is the question, I have solved it but I need someone to double check my solution. Question: Find the temperature $u(x, t)$ in a rod of length $L$ if the initial temperature is $f(x)$ throughout ...
1
vote
1answer
231 views
Differential equations, HEAT equation with insulated ends.
This is the question, I have solved it but I need someone to double check my solution.
Question: Find the temperature $u(x, t)$ in a rod of length $L$ if the initial temperature is $f (x)$ throughout ...
6
votes
1answer
196 views
An elegant non-technical account on the work of Joseph Fourier.
It would seem difficult for a naive person to understand the beauty of work done by Fourier. So as far as I know, one can use the Fourier transforms, analysis and series to apply them for heat ...
2
votes
0answers
617 views
Finding coefficients of a double Fourier series
This is the end of a PDE (heat equation in 2D) I am trying to solve with bounds from $0 < x < L$ and $0 < y < H$. It is a Newmann condition problem (i.e. all derivatives of $x$ and $y$ at ...
8
votes
3answers
1k views
The mathematics of music - why sine waves?
Of course, the Fourier transform is an extremely elegant mathematical method of overwhelming simplicity, and this straight away puts sine waves (or complex exponentials) on a high pedestal.
But what ...
