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A Constructing a function space to automatically satisfy BCs

  1. Jul 14, 2017 #1

    joshmccraney

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    Gold Member

    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!
     
  2. jcsd
  3. Jul 19, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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