I am using Galerkin's method for solving a generalized eigenvalue problem numerically, which results from a linear stability analysis. Due to the physics of the problem the following boundaries of the ansatz have to be fulfilled:
$f(0) = f(1) = f'(0) = f''(1) = 0$
So the hard question is now, if anyone knows a complete function space $f$, which satisfies these boundaries. I know that in general this kind of boundary value problem can only be solved if a differential equation is given, but unfortunately the physics of linear stability doesn't tell us that. Thanks in advance for ideas on how to create such a function space or ideas if such space even exists in finite dimensions?
Of course, such function spaces exist. For example, the space of all polynomials $f$ of degree at most $n$ (where $n\ge 4$) that satisfy the conditions $f(0) = f(1) = f'(0) = f''(1) = 0$. To get a more explicit description of this space, note that $f$ can be written as $$f(x)=x^2(1-x)p(x) \tag1$$ where $p$ is a polynomial of degree at most $n-3$. The form (1) takes care of all conditions except $f''(1)=0$. To satisfy the latter, differentiate (1) twice to obtain $f''(1)=-2p'(1)-4p(1)$. Since $p'(1)=-2p(1)$, it follows that $p$ can be written as $p(x)=c(2x-1)+(1-x)^2q(x)$ for constant $c$ and some polynomial $q$ of degree at most $n-5$. Thus, the space consists of the functions $$f(x)=x^2(1-x)\left(c(2x-1)+(1-x)^2q(x)\right),\quad c\in\mathbb R, \ \deg q\le n-5 \tag2$$