Numerical Schemes in the Context of Differential Equations

The goal of this project is to introduce standard solution schemes in the realm of numerical methods. We start with the following damped harmonic oscillator,

\[\ddot x + 2 \zeta \dot x + x = 0.\]

$\textbf{1.}$ We give a comprehensive treatment of the implicit Euler scheme, as well as the explicit Euler schemes for the model above. notebook, pdf

$\textbf{2.}$ A detailed construction of the finite differencing & shooting methods is given, before solving intermediate and boundary value problems for the case of $\zeta = 2.$ notebook, pdf

$\textbf{3.}$ Finally, we numerically solve the heat equation in two dimensions

\[\frac{\partial u}{\partial t} = \alpha \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right)\]

for the following initial conditions;

\[u(t, x, y) = \begin{cases}0, \quad \quad ~~~~~t = 0; ~~~~~x,y \in (0, 100)\\ 0, \quad \quad ~~~~x = 0; ~~~~~~~~~y \in (0, 100)\\0, \quad \quad ~~~~x=100; ~~~~~y \in (0, 100)\\ 200, \quad \quad y=0; ~~~~~~~~~x \in [0,100]\\200, \quad \quad y=100; ~~~~~x \in [0, 100]\\ \end{cases}\]

to conclude this nice tutorial. Many thanks to Mohammad Bennani for his contributions. $\fbox{$\xi$}$ notebook, pdf