Constructing a function space to automatically satisfy BCs

[member 428835]

Suppose we have a piecewise function $$f(x) =
\begin{cases}
f_1(x) & \text{if } -1\leq x \leq \xi_1 \\
0 & \text{if } -\xi_1\leq x \leq \xi_2 \\
f_2(x) & \text{if } -\xi_2 \leq x \leq 1
\end{cases}$$

where $\xi_1,\xi_2$ are known constants and $f_1(x),f_2(x)$ are unknown functions. $f(x)$ is subject to the following boundary conditions$$\int_{-1}^{\xi_1}f_1(x)\,dx + \int_{\xi_2}^{1}f_2(x)\,dx=0\\
f_1(\xi_1)=0\\
f_2(\xi_2)=0.$$

We assume $f(x)$ takes the following form: $$f_1(x) = \sum_{k=0}^N b_k P_k(x)\\
f_2(x) = \sum_{k=0}^N c_k P_k(x)
$$
where $P_k(x)$ is the $k$th Legendre polynomial. In order to satisfy the above boundary conditions for any $\xi_1<\xi_2\in[-1,1]$ I necessarily solve for 3 constants, a combination of $b_k$ and $c_k$, right? My text reads "There are $2(N+1)−3 = 2N−1$ linearly independent coefficient vectors that solve [the boundary conditions]"; what does this mean? I thought it meant there are $2N-1$ undetermined coefficients.

See, ultimately I am trying to solve the system of $j$ algebraic equations $$-\lambda^2\sum_{i=1}^nM_{ij}a_i=\sum_{i=1}^nK_{ij}a_i$$ where $K_{ij},M_{ij}$ are differential operators, functions of only $P_k(x),P_i(x)$ and $\lambda^2$ is an eigenvalue to be determined. So am I understanding this correct, that we are using the system of equations to solve for the remaining $2N-1$ constants $b_k,c_k$?

Thanks so much for any help you have to offer!

 

[jvicens]

Hi there,

Yes, you are correct in your understanding that the boundary conditions and the given form of $f_1(x)$ and $f_2(x)$ allow us to solve for $2N-1$ undetermined coefficients. This is because we have 3 boundary conditions and 2 functions with $N+1$ coefficients each, resulting in a total of $2(N+1)-3=2N-1$ undetermined coefficients.

As for the text stating that there are $2N-1$ linearly independent coefficient vectors that solve the boundary conditions, this means that there are $2N-1$ combinations of $b_k$ and $c_k$ that satisfy the boundary conditions and are not linearly dependent on each other. This is important because it ensures that the solution we obtain is unique and not just one of many possible solutions.

In terms of solving the system of equations you mentioned, it seems like you are on the right track. The system of equations allows us to solve for the remaining undetermined coefficients and determine the eigenvalue $\lambda^2$.

I hope this helps clarify things for you. Let me know if you have any other questions. Good luck with your research!