A Basis functions and spanning a solution space

joshmccraney

Hi PF

Given some linear differential operator $L$, I'm trying to solve the eigenvalue problem $L(u) = \lambda u$. Given basis functions, call them $\phi_i$, I use a variational procedure and the Ritz method to approximate $\lambda$ via the associated weak formulation
$$\langle L(\phi_i),\phi_j\rangle = \lambda \langle \phi_i,\phi_j\rangle.$$

As you can see, this expression is now a matrix equation, solutions to which are straightforward. For my particular problem, the basis functions are $$\phi_j = \cos\left( \frac{\pi j}{2}(x+1) \right) \cosh\left( \frac{\pi j}{2}(y+h) \right).$$

However, this solution, when inputted into the weak formulation equation, does not output correct eigenvalues. However, $\phi_j$ can be split into even and odd components:
$$\phi_j^o = \sin \left( \pi(j-1/2)x \right)\cosh\left( \pi(p-1/2)(y+h) \right)\\ \phi_j^e = \cos \left( \pi j x \right)\cosh\left( \pi j(y+h) \right)$$

Now to obtain eigenvalues I solve two separate equations, one for even eigenvalues and one for odd:
$$\langle L(\phi_i^e),\phi_j^e\rangle = \lambda \langle \phi_i^e,\phi_j^e\rangle\\ \langle L(\phi_i^o),\phi_j^o\rangle = \lambda \langle \phi_i^o,\phi_j^o\rangle.$$

This latter approach gives correct solutions: why? Any insight or direction is greatly appreciated.

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Orodruin

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Please be more specific. Your post seems to suggest that you have some particular linear differential operator in mind. Also, please show what you have done more explicitly and be more explicit in what you obtain.

joshmccraney

Please be more specific. Your post seems to suggest that you have some particular linear differential operator in mind. Also, please show what you have done more explicitly and be more explicit in what you obtain.
The real problem I'm worried is kind of long, tough to explain, and could scare off anyone who could potentially help me (posted it in a previous thread and everyone ran , and I don't blame them). I tried finding the post but couldn't.

So I'll briefly summarize: fluid rests in a 2D channel of domain $\Omega$, the gas-fluid interface $\Gamma$, fluid contact line $\gamma$, and channel wall and bottom $\Sigma$. Center depth is $h$. The governing hydrodynamic equations of motion are then
$$\nabla^2 \phi = 0 \,\,\,\,\,[\Omega]\\ \phi_n = 0 \,\,\,\,\,[\Sigma]\\ \pm\phi' + \cos a \cot a \phi = 0 \,\,\,\,\, [\gamma]\\ \int_\Gamma \phi_n = 0 \,\,\,\,\, [\Gamma]\\ -d_s^2\phi_n - c^2 \phi_n = \lambda \phi \,\,\,\,\,[\Gamma].$$

My basis functions in my first post solve the first 2 equations analytically. Equation 5 is the eigenvalue problem, where $L \phi_n \equiv -d_s^2\phi_n - c^2 \phi_n$, and I solve this using inverse differential operators. I build equations 3 and 4 into those operators. Subscripted $n$ denotes a normal derivative to the given geometric structure.

To relate all this to my initial post, equation 5 is the eigenvalue problem I approximately solve. The basis functions in the first post exactly solve the first 2 equations (easy to verify). It's confusing that this problem only admits solutions when working with even and odd functions separately, which is my question.

"Basis functions and spanning a solution space"

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