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1answer
22 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 < ...
1
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1answer
28 views

Relative error when computing derivatives via FFT

I want to compute a discrete derivative via the FFT. This amounts to multiplication by the wave number in Fourier space, as detailed in the stack exchange answer here. When I increase the ...
1
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1answer
42 views

Proof of Fourier series Theorem (k-continuous derivatives)

Here's the theorem: Theorem: If $f$ is periodic with Fourier coefficients $a_n,b_n$ and if the series $$\sum_{n=1}^\infty (|n^{k}a_n|+|n^{k}b_n|)$$ converges for some integer $k \geq 1$, then f ...
1
vote
1answer
46 views

Proving this Corollary regarding Fourier Series

Okay so here's the the problem: Let $k \in \mathbb{N}$. If $f$ is periodic, with Fourier coefficients $a_n,b_n$ and the series $\sum_{n=1}^\infty{(|a_n| + |b_n|)n^k}$ converges for some $k$, then ...
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2answers
58 views

In search of periodic solutions of a system of ODEs by means of Fourier series

Consider the following non-linear system of ODEs : \begin{cases} x' = y \\ y' = x^2-\lambda x. \end{cases} In search of a solution such that $y(0) = y(2 \pi) = 0$, I am being told to seek $x$ and $y$ ...
1
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1answer
38 views

Termwise Integration of Fourier Series

This is a question from Edwards and Penney 4th edition Differential Equations and Boundary value problems from section 9.3. Suppose that $f(t)$ is a piecewise continuous period $2L$ funtion. ...
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
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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 ...
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1answer
119 views

Fourier Series and Solving Differential Equations

I am getting stuck on how to use Fourier Series to solve ODE's. Take the problem where \begin{equation} E(t)=200t(\pi^2-t^2), \end{equation} for $t$ between $-\pi$ and $\pi$ (period of $2\pi$), ...
0
votes
2answers
319 views

Step function Fourier series

How to do a step function based Fourier series? What I am confused about is how to calculate the time period since the step function doesn't end? And I don't really know the period since the ...
1
vote
1answer
199 views

Can we express all doubly periodic functions as one of doubly periodic function?

Singly Periodic Functions $e^{x},\cos(x),\sin(x),\tan(x), .. etc.$ Euler's identity is $$e^{i\alpha}=\cos(\alpha)+i\sin(\alpha)$$ $$e^{-i\alpha}=\cos(\alpha)-i\sin(\alpha)$$ Thus, we can express ...
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1answer
145 views

Green' s function for harmonic oscillator

Does someone know how to get a solution of differential equation for Green's function $(-d^2/dt^2 + \omega^2) G(t, s) = \delta(t-s) $? There is a periodicity of G, actually $\Delta (t-s) = G(t,s)$ ...
3
votes
1answer
161 views

Using Fourier series techniques to solve $x'' + 3x = 7$ with $x'(0) = x'(5) = 0$

$$x'' + 3x= 7$$ Given conditions $x'(0)=x'(5)=0$. I checked the list and I went through three books. I am doing intro to differential equations. I just don't know how to get the extensions... I was ...
1
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1answer
98 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. ...
3
votes
0answers
350 views

Solve a differential equation using Fourier series

Assume I have a second order differential equation $\ddot{x} = F(x,\dot{x})$ (or an equivalent equation of first order) and that I know there is a periodic solution to it (for simplicity's sake, ...
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 ...
2
votes
2answers
108 views

The difference between m and n in calculating a Fourier series

I am studying for an exam in Differential Equations, and one of the topics I should know about is Fourier series. Now, I am using Boyce 9e, and in there I found the general equation for a Fourier ...
1
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0answers
90 views

Partial Differential Equation Eigenvalue of zero question

In the event that I'm solving a partial differential equation through separation of variables, if I end up with an eigenvalue of zero, what do I do with the corresponding eigenfunction? That is to ...
1
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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 ...
2
votes
0answers
130 views

Check my solution - Modelling of a spring with Differential Equation

I am doing some work with differential equations. I have solved the following problem but am uncertain if I'm doing it correctly. Could someone look over it for me and check if I'm doing something ...
0
votes
2answers
101 views

Fourier transform help

Find the Fourier tranform of $f(x)=x^2e^{-x^2}$ In a previous question when I found the Fourier transform of $f(x) =e^{-x^2}$, I used the formulas $F(f')=i\omega F$ and $F(xf)=iF'$. Will they be ...
1
vote
3answers
448 views

Solving $f''+f=\sin x$ with Fourier series

Does there exist a twice differentiable periodic function $f$ such that $f''(x) + f(x) =\sin(x)$ for all $x \in [-\pi, \pi]$? How to solve this differential equation using Fourier series? I ...