Basis functions for a Galerkin procedure

Question

For a Galerkin procedure, I am trying to construct a set of linerly independent functions $\{\varphi_n\}_{n = 1}^N$ satisfying $$ \varphi_n(0) = 0, ~~ \varphi_n'(1) = 0, $$ for all $n \geq 1$.

A trivial set satisfying the above boundary considitions is $$ \varphi_n(t) = t^n (1 - t)^{n + 1}, $$ which are linearly independent: the Wronskian vanishes only at $t = 0, 0.5, 1$.

Are there other families of $\varphi_n$? Any advice or direction is appreciated.

That family is trivially l.i. because all of the polynomials have different degrees. — Gonzalo A. Benavides, Jun 28 '18 at 04:17
Definitely right! I realized it a bit later. Are there any other families? — Asatur Khurshudyan, Jun 28 '18 at 04:20

Gonzalo A. Benavides · Answer 1 · 2018-07-04T18:25:21.177

2

The family

$$ \psi_n(x)=\frac{2}{(2n+1)\pi}\sin\left(\frac{2n+1}{2}\pi x\right) $$ may work for you.

Edit

If $\{f_n\}$ is a set of differentiable linearly independent functions that $t \not \in \text{span}(\{\varphi_n\})$ and satisfies $f'_n(1)\neq 0$, you can define the linearly independent family $\{\varphi_n\}$ by $$ \varphi_n(t)=\frac{1}{f_n'(1)} (f_n(t)-f_n(0))-t, $$ which satisfies $\varphi_n(0)=\varphi_n'(1)=0$.

edited Jul 04 '18 at 18:25

answered Jun 28 '18 at 04:33

Gonzalo A. Benavides

993

Maybe you meant $\varphi_n(t) = \sin\left(\frac{2n + 1}{2}\pi t\right)$? – Asatur Khurshudyan Jun 28 '18 at 04:36
$\psi_n'(x)=\cos(2\pi n x)$, am I right? Then $\psi_n'(1)=\cos(2 \pi n)=1$. – Gonzalo A. Benavides Jun 28 '18 at 04:38
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I am so sorry to have a typo in the second boundary condition. It should be $\varphi_n'(1) = 0$. – Asatur Khurshudyan Jun 28 '18 at 04:38
It's ok, maybe now it would help :) – Gonzalo A. Benavides Jun 28 '18 at 04:40
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I have checked, the Wronskian is 0 only when $t = 0$ and $t = 1$. – Asatur Khurshudyan Jun 28 '18 at 04:44
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Good! I will think in another family. I'll let you know. – Gonzalo A. Benavides Jun 28 '18 at 04:47
I added a general answer. – Gonzalo A. Benavides Jun 30 '18 at 06:22
Thanks! Are $\varphi_n$ linearly independent for arbitrary $f_n$? – Asatur Khurshudyan Jul 04 '18 at 00:18

Basis functions for a Galerkin procedure

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