# PDE - Review for Exam II

Here are some problems to play with for the exam on Wednesday.

1. Find the full Fourier series of $$f(x)=x$$ over the interval $$[-1,1]$$.

2. Solve the heat problem $$u_t = u_{xx}$$ on the interval $$[0,\pi]$$ subject to $$u(0,t)=u(\pi,t)=0$$ and $$u(x,0)=x^2 \sin(x)$$.
Note: You may assume that $\int_0^{\pi} x^2 \sin(x) \sin(n\pi x) \, dx = \begin{cases} - \frac{\pi}{4} + \frac{\pi^{3}}{6} & \text{for}\: n = 1 \\- \frac{4 \left(-1\right)^{n} \pi n}{n^{4} - 2 n^{2} + 1} & \text{otherwise} \end{cases}.$

3. Given two functions $$f$$ and $$g$$ continuous on the unit interval, define their inner product by $\langle f,g\rangle =\int _0^{1}f(x)g(x)dx$ Given any three such functions $$f$$, $$g$$, and $$h$$, prove that $\langle f-g,h\rangle =\langle f,h\rangle -\langle g,h\rangle.$

4. Suppose you are asked to estimate the solution to the boundary value problem $y'' + x^3 y = e^x \text{ subject to } y(0) = 0, y(2) = 1.$ You do so by breaking the interval $$[0,2]$$ into 4 equal pieces allowing you to approximate $$y''$$ with the symmetric difference quotient for the second derivative. Write down the resulting linear system of equations that the $$y_i \approx y(x_i)$$ must satisfy.

1. The figure below shows the graph of two functions on the unit square. Which of these can be a solution of of Laplace’s equation $$\Delta u=0$$. You must clearly state a property that $$u$$ must satisfy that one graph clearly does not have in order to get full credit.