# Solving higher order PDE's

1. Jun 19, 2015

Hello there!
So here's my problem, while you solve the Euler Bernoulli beam Equation by separation of variables, how do I have to prove the separated function of space are orthogonal? If so, are hyperbolic sines and cosines orthogonal when you have a product or a linear combination of them?

The pde is- [; u_{tt}+\alpha^{2} u_{xxxx} = o ;] where [; \alpha ;] is a constant that is material dependent. The separated function of space is [; F(x) = \sum_{n=1}^{\infty} [cosh(\beta_{n}x)-cos(\beta_{n}x)]-[sinh(\beta_{n}x)-sin(\beta_{n}x)] ;] where [; \beta_{n} ;] is some constant.

2. Jun 20, 2015

### pasmith

You are looking for solutions of $X_\lambda^{(4)} = \lambda X_\lambda$ for $\lambda \in \mathbb{R}$. Now for fixed $\lambda$ there is a four-dimensional subspace of solutions, and given an arbitrary inner product on that space there will exist a basis which is orthogonal with respect to that inner product. (The constraint you haven't mentioned is that $X_\lambda$ needs to satisfy the boundary conditions at each end of the beam. That constraint may cause you to reject some of these solutions.) So the difficulty is to find an inner product with respect to which $X_\lambda$ and $X_\mu$ are necessarily orthogonal when $\lambda \neq \mu$.

A sufficient condition for this is that the operator $f \mapsto f^{(4)}$ should be self-adjoint with respect to the inner product, ie. $$(f^{(4)},g) = (f,g^{(4)})$$ for every $f$ and $g$. Now if you take (somewhat arbitrarily) the inner product $$(f,g) = \int_{-L}^L f(x)(g(x))^{*}\,dx$$ where ${}^{*}$ denotes the complex conjugate, then repeatedly integrating $(f^{(4)},g)$ by parts yields $$(f^{(4)},g) = (f,g^{(4)}) + \left[ f'''g^{*} - f''(g^*)' + f'(g^*)'' - f(g^*)''' \right]_{-L}^{L}.$$

3. Jun 21, 2015