- #1

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- Homework Statement
- I need help with finding the approximate solution of a second order differential equation.

- Relevant Equations
- I am given the differential equation as in

$$

\lambda \frac{d^2u}{dx^2} + q = 0

$$

which is defined over $$x \in [-L/2,L/2]$$ and where $$q = a+bx$$. I am supposed to find an approximative solution using a spectral method, i.e. using cosine and sine terms that fulfills the essential boundary conditions given by

$$

u \bigg ( -\frac{L}{2} \bigg ) = 0, u \bigg (\frac{L}{2} \bigg ) = 0

$$

The appoximative solution should be represented by a Fourier series and by noticing that the terms in the integral are orthogonal, the matrix of the system of equations should be diagonal.

We choose an approximative solution given by

$$

u_N(x) = \frac{a_0}{2} + \sum_{n=1}^N a_n \cos nx + b_n \sin nx

$$

Comparing this approximative solution with the differential equation yields that

$$

\frac{a_0}{2} = a

$$

and the boundary conditions yields the equation system

$$

a + \sum_{n=1}^N a_n \cos \bigg ( \frac{nL}{2} \bigg ) + b_n \sin \bigg ( \frac{nL}{2} \bigg ) = 0 \\

a + \sum_{n=1}^N a_n \cos \bigg ( -\frac{nL}{2} \bigg ) + b_n \sin \bigg ( -\frac{nL}{2} \bigg ) = 0

$$

So at least I got a system of two equations but I do not know where to go from here. How should I use the orthogonality to set up a matrix from these two equations?

$$

u_N(x) = \frac{a_0}{2} + \sum_{n=1}^N a_n \cos nx + b_n \sin nx

$$

Comparing this approximative solution with the differential equation yields that

$$

\frac{a_0}{2} = a

$$

and the boundary conditions yields the equation system

$$

a + \sum_{n=1}^N a_n \cos \bigg ( \frac{nL}{2} \bigg ) + b_n \sin \bigg ( \frac{nL}{2} \bigg ) = 0 \\

a + \sum_{n=1}^N a_n \cos \bigg ( -\frac{nL}{2} \bigg ) + b_n \sin \bigg ( -\frac{nL}{2} \bigg ) = 0

$$

So at least I got a system of two equations but I do not know where to go from here. How should I use the orthogonality to set up a matrix from these two equations?