\documentclass[11pt]{article}
\begin{document}

\title{HW 5}
\author{}
\date{}
\maketitle
\begin{enumerate}
\item Find conjugate $f^*(x^*)$ to the function 
$$ f(x)= \frac{1}{2} a \, x^2 + b \, x +  \frac{1}{2} c $$
Compare $\min f(x)$ and $\max f^*(x^*)$. 

Compute $x\, x^* $

\item Find conjugate $f^*(x^*)$ to the function 
$$ f(x)= \max\left \{ \frac{(x-1)^2}{2} , \,  \frac{(x-1)^2}{2} \right \} $$
Using this result, compute conjugate $g^*(x^*)$ to the function 
$$ g(x)= \min \left \{ \frac{(x-1)^2}{2} , \,  \frac{(x-1)^2}{2} \right\} $$
Compute the convex envelope of $g(x)$. 

\item Using Ritz method, approximate solution to the problem
$$ I=\min_{u(x)} \int_0^1 \left(x (u' )^2 + (1-x^2) u^2 \right) dx, \quad u(0)=1, ~u(1)=0 $$
Represent the solution as a sum of polynoms and trigonometric functions and account for boundary conditions:
$$ u(x)= (1-x) + x(1-x) \left( \sum_{k=0}^N \left(a_{2k} \cos(\pi k x) + a_{2k+1}\sin(\pi k x),,\right) \right) 
$$
 find optimal valuers of parameters, graph the solution. How the graph varies when $N$ increases? 
\item Using Galerkin method, approximate the solution to the ODE 
$$
u''' - x u'' + 1-x)^2 u' - x(1-x) u + x^5 =0, \quad u(0)=1, u(1)=u'(1)=0 $$
Use polynomials up to degree $N$ with unknown coefficients to approximate the solution and to account for boundary conditions, use polynomials as the trial functions $\psi$. Graph the approximates of the solution for different values $N$. 

Refs:

1. http://fischerp.cs.illinois.edu/tam470/refs/galerkin2.pdf

2. http://www.sd.ruhr-uni-bochum.de/downloads/Galerkin\_method.pdf

3. https://en.wikipedia.org/wiki/Galerkin\_method
\end{enumerate}


\end{document}
