# Tagged Questions

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### 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 ...
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### 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 $$u_x(0,t)=u_x(1,t)=0$$? ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### Find the first four terms in the Fourier series for a solution of the wave equation

The question is to find the first four terms in the Fourier series for $u(x,t), t>0$. It is for a plucked string of length L, has zero initial displacement (I.e. $u(x,0)=0, 0<x<L$) and ...
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### 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 $\mu y'' + c y' + k y = F(t)$ ...
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### 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 $u=u(x,y)$: ...
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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. ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...