MATH 414 Lecture 6

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Fourier Series

Emerged from Fourier's theory of heat.

Suppose we have heat distribution over a bar of length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} such that $u(0,t)=0$ to $u(\pi ,t)=0$, where $u$ is a function of position $x$ and time $t$: $u(x,t)$.

Heat flow equation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}}

Our boundary conditions are that $u(0,t)=u(\pi ,t)=0$

Separation of Variables

Keep the homogeneous equations

Then look for separation solutions (normal modes, esp. in vibrations problems)

Assume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(x,t) = X(x) \, T(x)} (hence the "separation" in the name)

Get equations of the form: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X'' + \lambda \, X = 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X(0) = X(\pi) = 0} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T' = -\lambda \, T} .

1. the last is easy to solve: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(t) = A \, \mathrm{e}^{-\lambda \, t}}
2. The first two set up an eigenvalue problem. There are solutions for only certain values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 1, \ldots, n} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_n = n^2} (eigenvalues); Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_n(x) = \sin{\left( n \, x \right)}} (eigenfunctions)

Hence the separated solutions are of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^{-n^2 \, t} \, \sin{\left( n \, x \right)}}

We expect $u(x,t)=\sum _{n=1}^{\infty }b_{n}\,\mathrm {e} ^{-n^{2}\,t}\,\sin {\left(n\,x\right)}$

Initial conditions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(x,0) = f(x)} implies

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \sum_{n=1}^\infty b_n \, \sin{\left( n \, x \right)}}

This series shall henceforth be called the sine series

Problems

1. How do we get the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_n} 's?
2. Does the series represent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} in any sense?
3. Does the series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(x,t) = \sum_{n=1}^\infty b_n \, \mathrm{e}^{-n^2 \, t} \, \sin{\left( n \, x \right)}} solve the heat flow problem?

The Fourier Series

Let's look at heat flow in a ring of radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} . We get the same sort of equations.

Given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} , we need a series of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = a_0 + \sum_{n=1}^\infty a_n \, \cos{\left( n \, x \right)} + b_n \, \sin{\left( n \, x \right)}}

This is the Fourier Series

Fourier found that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0 = \frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x) \,\mathrm{d}x}

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \ge 1} ,

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, \cos{\left( n \, x \right)} \,\mathrm{d}x}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, \sin{\left( n \, x \right)} \,\mathrm{d}x}

Orthogonality Relations

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\pi}^{\pi} \cos{\left( n \, x \right)} \, \cos{\left( m \, x \right)} \,\mathrm{d}x = \begin{cases} 2 \pi & n = m = 0 \\ \pi & n = m \ge 1 \\ 0 & n \ne m \end{cases}}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\pi}^{\pi} \sin{\left( n \, x \right)} \, \sin{\left( m \, x \right)} \,\mathrm{d}x = \begin{cases} \pi & n = m \\ 0 & n \ne m \end{cases}}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\pi}^{\pi} \cos{\left( n \, x \right)} \, \sin{\left( m \, x \right)} \,\mathrm{d}x = 0} . This is because $\cos {x}$ is even, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin{x}} is odd, and the product of an even function and an odd function is odd. So by symmetry the integral is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} .

Using these relations, we can integrate the Fourier series definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} :

${\begin{aligned}f(x)&=a_{0}+\sum _{n=1}^{\infty }a_{n}\,\cos {\left(n\,x\right)}+b_{n}\,\sin {\left(n\,x\right)}\int _{-\pi }^{\pi }f(x)\,\mathrm {d} x&=a_{0}\,\int _{-\pi }^{\pi }\,\mathrm {d} x+\sum _{n=1}^{\infty }a_{n}\int _{-\pi }^{\pi }\cos {\left(n\,x\right)}\,\cos {\left(0\,x\right)}\,\mathrm {d} x+b_{n}\int _{-\pi }^{\pi }\sin {\left(n\,x\right)}\,\sin {\left(0\,x\right)}\,\mathrm {d} x\end{aligned}}$

We want to show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\pi}^{\pi} \sin{\left( m \, x \right)} \, \sin{\left( n \, x \right)} \,\mathrm{d}x = \pi \, \delta_{m,n}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int_{-\pi}^{\pi} \sin{\left( m \, x \right)} \, \sin{\left( n \, x \right)} \,\mathrm{d}x &= \frac{1}{2} \int_{-\pi}^{\pi} \left( \cos{\left( (m-n) \, x \right)} - \cancel{\cos{\left( (m+n) \, x \right)}} \right) \,\mathrm{d}x \end{align}}

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \ne n} , we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\pi}^{\pi} \cos{\left( (m-n) \,x \right)} \,\mathrm{d}x = \left. \frac{1}{m-n} \, \sin{\left( (m-n) \, x \right)} \right|_{-\pi}^{\pi} = 0}

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = n} , then

$\int _{-\pi }^{\pi }\sin ^{2}{\left(n\,x\right)}\,\mathrm {d} x={\frac {1}{2}}\,\int _{-\pi }^{\pi }\cos {\left(0\,x\right)}\,\mathrm {d} x={\frac {1}{2}}\,\int _{-\pi }^{\pi }\,\mathrm {d} x=\pi$