0
votes
1answer
20 views

Show that Fourier series arising in solution of differential eqn. converges uniformly

Let $f \in L_2(0,\pi)$ have the Fourier expansion $f(x) = \sum_{n=2}^{\infty} f_n\sin(nx)$. Compute (formally) the boundardy value problem $$ u''(x) + u(x) = f(x) \qquad \mbox{ for } 0 < x < ...
2
votes
1answer
28 views

$ U_{xx}+U_{yy}=0$ with rectangular boundary conditions

When solving $ U_{xx}+U_{yy}=0$ with $u(0,y)=u(a,y)=u(x,b)=0,u(x,0)=f(x)$. $0<=x<=a$ , $0<=y<=b$ by the method of separation of variables I have $-X''(x)-\lambda X(x)=0 $ ...
0
votes
1answer
22 views

Fourier series of coshx using fourier of $e^{x}$.

I have to find the Fourier series of $coshx$ on $(-l,l)$.What I did was I found the Fourier series of $e^{x}=\sum _{n=-\infty}^{\infty }{(-1)^n (\ell^2+in\pi)\over{l^2+n^2\pi^2}}\sinh(\ell)e^{{in\pi ...
2
votes
3answers
38 views

Finding fourier sine series using another cosine series

I have to find the sine series of $x^3$ using the the cosine series of $x^2/2$. $${x^2 \over 2}={l^2 \over 6}+{2 l^2 \over \pi^2}\left[\sum{(-1)^n \over n^2}\cos\left({n\pi x \over l}\right)\right]$$ ...
2
votes
1answer
41 views

Finding Fourier of $x^3$ by Fourier of $x^2$

I found the cosine series of $x^2/2$ to be (by first finding Fourier sine series for $x$ on $(0,l)$ and then integrating that term by term) $${x^2 \over 2}={l^2 \over 6}+{2 l^2 \over ...
0
votes
1answer
45 views

Separation of variable of PDE

For any $u_0,u_1\in L^2(0,\pi)$ and $f\in L^2((0,\pi)\times(0,+\infty))$ find using separation of variables and Fourier series a formal explicit expression of the solution of the problem ...
2
votes
1answer
36 views

Dirichlet vs Neumann Boundary Conditions of a PDE?

Say I have the PDE $$u_{tt}=4u_{xx}, u(x,0)=f(x),u_t(x,0)=g(x), 0<x<L$$ How does the solution change if I am given the boundary conditions $$u(0,t)=u(1,t)=0$$ versus $$u_x(0,t)=u_x(1,t)=0$$? ...
3
votes
1answer
97 views

Fourier series for $e^x$

I'm trying to teach myself partial differential equations from Strauss' book. I have run into a very bizarre problem - I cannot figure out what is the Fourier series of $e^x$! And not even Google has ...
1
vote
1answer
77 views

Use Fourier's method of separation of variables to solve the boundary value problem

Use Fourier's method of separation of variables to solve the boundary value problem comprising the following PDE and BC: PDE: $x \sin(y) u_x + \cos(y) u_y = -2 \sin(y) u $, $u = u(x,y)$ Boundary ...
7
votes
2answers
234 views

Compute $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$.

Compute the Fourier series for $x^3$ and use it to compute the value of $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$. I determined the coefficients of the Fourier series, which are $$a_0 = ...
0
votes
0answers
44 views

How to solve this differential equation of the second order

Do you know how to solve this equation? I'm a physicist student and I have initial equation, condition and answer. Unfortunately I need an explanation how this answer was got. I am mew to such ...
0
votes
1answer
70 views

Solving the heat equation using Fourier series; specific questions

Like this previous question, Solving the heat equation using Fourier series, I too am reading the same wikipedia article, ...
0
votes
0answers
17 views

2D Wave propagating in duct with height change

Suppose that we have a two - dimensional rigid wall duct cosisting of two semi - infinite regions $x<0,\ 0\leq y\leq a\ $ and $x>0,\ a<y\leq b$ (this means exactly that there is a height ...
0
votes
2answers
80 views

Using fourier analysis in order to solve differential equations.

http://www.enm.bris.ac.uk/admin/courses/EMa2/Lecture%20Notes%2009-10/LSPDE5.pdf The above PDF teaches us the separation of variables method. However, there are some things I dont understand, that I ...
0
votes
1answer
31 views

Boundary Value Problem (Fourier Series)

Consider the following boundary value problem: \begin{equation} \begin{cases} u_{tt}-u_{xx}=x, &0\leq x \leq \pi,t\geq0 \\ u(0,t)=u_{x}(\pi,t)=0 \\ u(x,0)=\sin(\frac{3}{2}x), & ...
2
votes
0answers
25 views

Convergence of the series $\sum_{\xi\in\mathbb Z^n} e^{2\pi ix\cdot \xi} a(x, \xi)\hat{f}(\xi)$?

I need some help with the following problem: let $a:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb C$ be a smooth function and suppose there are constantes $C_{\alpha, \beta}$ and $M(\alpha, \beta)$ ...
2
votes
1answer
133 views

How to properly prepare for a graduate level PDE course using the books by Evans and Strauss

For my undergrad background, I have Calculus 1-3, Linear Algebra, one semester of ODE, one semester of real analysis. Never had any PDE before. Thus I know this background is hardly enough to do well ...
2
votes
0answers
59 views

Need help on computing odd, even extensions of a function

OK I am going over d'Alembert solutions. And I came across the following example. $$ f(x) = \begin{cases} \frac{3}{10}x &0 \le x \le \frac{1}{3} \\ \frac{3(x-1)}{20} & \frac{1}{3} \le x \le ...
0
votes
1answer
56 views

Separation of Variables (Partial Differential Equation)

Does Separation of Variables work for the following PDE ? $$\nabla^2 W(x,y) \pm \alpha W(x,y) = \beta,$$ where $\alpha$ and $\beta$ are constants.
0
votes
0answers
36 views

Checking on a double-Fourier sum problem

The problem is as follows, with the following initial conditions: $f(x,y) = \sin \pi x \sin \pi y$ $g(x) = \sin \pi x$, and $a=b=1$ and $c=\frac{1}{\pi}$ The initial conditions are $u(x,y,0) = ...
3
votes
1answer
36 views

Compare mixed derivatives to laplacian

Suppose $u,f$ periodic and smooth in $Q=[0,1]^n$ such that $\Delta u=f$. Show that for each $i,j$, $$\int_Q \left| \frac{\partial^2 u}{\partial x_i \, \partial x_j} \right|^2 \leq C \int_Q |f|^2.$$ ...
2
votes
2answers
52 views

$C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in ...
0
votes
1answer
123 views

Fourier Sine Series extension

If $\phi(x)$ is any function on $(0, l)$, derive the expansion $\displaystyle\phi(x) = \sum c_n \sin\left(\left(n + \frac{1}{2}\right) \frac{\pi x}{l}\right)$ for $0 < x < l$ by the following ...
1
vote
1answer
230 views

steady state solution to differential equation - checking my work

EDIT: fixed a stray negative sign. The problem as given: $y'' + 2y' + 5y = 10\cos t$ We want to find the general solution and the steady-state solution. We're using $\mu y'' + c y' + k y = F(t)$ ...
3
votes
0answers
75 views

An estimate For the Laplacian semi-group

Let $S(t)$ be the semi-group generated by the Dirichlet Laplacian in $L^2(0,1)$, which is given, for $y\in L^2(0,1)$, by $$S(t)y=\displaystyle\sum_{n=1}^\infty e^{-n^2\pi^2 t} \langle y,\sin(n\pi x) ...
3
votes
1answer
89 views

A PDE problem about $(\partial_x^2 + \partial_t^2)u = 0$ using Fourier series.

I'm trying to solve the following initial value problem (from Folland, pg. 277, exercise $48$a) using Fourier series: Let $x \mapsto f(x)$ and $x \mapsto u(x, t)$ be periodic functions on ...
3
votes
2answers
128 views

Laplace equation with weird boundary condition

So, guys, here's my problem. I have this differential equation $$ U''_{xx}+U''_{yy}=0 $$ with these boundary conditions: $$ U'_{y}(x,0)=0 $$ $$ U'_{y}(x,\pi)=0 $$ $$ U(0,y)=0 $$ $$ ...
18
votes
1answer
521 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
97 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
389 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
86 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
241 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
55 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
129 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
359 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
247 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
149 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
690 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
339 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
396 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
230 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 ...
4
votes
0answers
1k 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 ...
10
votes
3answers
2k 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 ...