Finding Complete Orthogonal Systems for Boundary Value Problems

[evinda]

Hello! (Wave)

Suppose that we have a $C^{\infty}$ function $f: [0, \pi] \to \mathbb{R}$ for which it holds that $f(0)=f(\pi)=0$.

How can we find a complete orthogonal system of this space? (Thinking)

 

[I like Serena]

Hello! (Wave)

Suppose that we have a $C^{\infty}$ function $f: [0, \pi] \to \mathbb{R}$ for which it holds that $f(0)=f(\pi)=0$.

How can we find a complete orthogonal system of this space? (Thinking)

Hey! (Smile)

Have you considered a Fourier Series? (Wondering)

 

[evinda]

Hey! (Smile)

Have you considered a Fourier Series? (Wondering)

So do we find the coefficients of the Fourier series

$$f \sim \frac{a_0}{2}+ \sum_{k=1}^{\infty} ( a_k \cos(kx)+ b_k \sin(kx)) $$

and from this we find an orthogonal system? How can we do it? (Thinking)

 

[I like Serena]

So do we find the coefficients of the Fourier series

$$f \sim \frac{a_0}{2}+ \sum_{k=1}^{\infty} ( a_k \cos(kx)+ b_k \sin(kx)) $$

and from this we find an orthogonal system? How can we do it? (Thinking)

First off, we know that the Fourier series is an orthogonal system. (Nerd)

Since each function in the system must fit the boundary conditions, it follows that $a_k=0$.
So we're left with:
$$f(x) = \sum_{k=1}^{\infty} b_k \sin(kx)$$
(Thinking)

That is, the orthogonal system is:
$$\{ x \mapsto \sin(kx) \mid k \in \mathbb Z_{>0} \}$$
(Mmm)

 

[evinda]

First off, we know that the Fourier series is an orthogonal system. (Nerd)

Since each function in the system must fit the boundary conditions, it follows that $a_k=0$.
So we're left with:
$$f(x) = \sum_{k=1}^{\infty} b_k \sin(kx)$$
(Thinking)

That is, the orthogonal system is:
$$\{ x \mapsto \sin(kx) \mid k \in \mathbb Z_{>0} \}$$
(Mmm)

So the complete orthogonal system is the set of the eigenfunctions, right? (Thinking)

 

[I like Serena]

So the complete orthogonal system is the set of the eigenfunctions, right? (Thinking)

Which eigenfunctions?

 

[evinda]

Which eigenfunctions?

Oh, this holds only if we are given an initial value problem, right? (Thinking)

So if we are given a function and boundary conditions, do we find the Fourier series of the function and apply the boundary conditions and the orthogonal complete system will consist of the functions $1$ / $\sin kx$ / $\cos kx$ that will remain? (Thinking)

 

[I like Serena]

Oh, this holds only if we are given an initial value problem, right? (Thinking)

So if we are given a function and boundary conditions, do we find the Fourier series of the function and apply the boundary conditions and the orthogonal complete system will consist of the functions $1$ / $\sin kx$ / $\cos kx$ that will remain? (Thinking)

Basically, yes. (Smile)