Second order differential equation

[Linder88]

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?

 

[BvU]

Comparing this approximative solution with the differential equation yields that
${a_0\over 2}=a$

How ?

and the boundary conditions yields the equation system

I miss the $+bx$ from
$$
\lambda \frac{d^2u}{dx^2} + a + bx = 0
$$

(and I find the use of $a_n$ and $b_n$ in your notation confusing, since $a$ and $b$ are allready used).

 

[pasmith]

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
$$
\frac{a_0}{2} = a
$$

No. It's the second derivative of $u$ that is equal to $\frac{-q(x)}{\lambda}$, not $u$ itself. When you differentiate the constant term you get zero, and in any case a non-zero constant does not satisfy the boundary condition.

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?

You have $u(x) \approx \sum_n u_n\phi_n(x)$ where $$\phi_n(-L/2) = \phi_n(L/2) = 0.$$ Therefore $$\lambda \sum_n u_n \phi_n''(x) \approx -q(x)$$ and we determine the $u_n$ by requiring that the inner product with $\phi_m$ should be exact, ie. $$\lambda \sum_n u_n \int_{-L/2}^{L/2} \phi_n''(x) \phi_m(x)\,dx = -\int_{-L/2}^{L/2} q(x)\phi_m(x)\,dx.$$ The matrix you are looking for is the matrix of coefficients $$M_{mn} = \int_{-L/2}^{L/2} \phi_n''(x) \phi_m(x)\,dx.$$

Now the choice of a Fourier basis means that $$\phi_n(x) = A_n\cos(k_n x) + B_n\sin(k_n x)$$ and $k_n$ is such that the boundary conditions are satisfied by something other than $A_n = B_n = 0$.