Differential Equations 11

Partial Differential Equations and its definition and properties. Some important PDEs are given and an example given of how to solve one. It goes through how to make a well-posed PDE question and the modelling framework. An example is given of modelling car traffic going through the steps of the modelling framework.

Example of Near constant car density continued with interpretation. D'Alembert's solutions with an example of some cars being able to pass other cars. Heat equation and solving it with separation of variables coming to the full general solution.

Square Wave example with matlab script to plot solution. Theorem of functions being orthogonal and proving this is the case for different periods. Deriving the value of the coefficients for the fourier series. Square wave Redux.

Separation of variables to solve the Heat equation and represent as a summation solution. Example of copper bar and solving for its solution to transfer of heat over time. Includes matlab code of graphing of solution. Different Boundary conditions: Dirichlet and Neumann boundary conditions, motivation for Fourier series.

Theorem for Fourier series, the fourier series and the integrals to solve the coefficients. Forced oscillation example and graphing of solution. Generalising Fourier series to an Arbitrary period and derivation.

Graph and solve fourier series for square wave with arbitrary period. Fourier Cosine and Sine series with matlab example. Half-range expansion with examples.

Matlab Script graphing Wave Equation solution of separation of variables with Fourier series. Heat Flow equation example solving and graphing with Matlab. PDEs and waves, shows part way through deriving the PDE for the wave equation.

Linearising wave equation for a water column. Using separation of variables to create a general solution to the wave equation.

Laplace's equation for a steady state equation. Steady State heat equation finding a fourier series solution. Computer algebra to find integrals. Matlab script for graphing fourier solutions. Solving Laplace's equation with Dirichlet and/or Neumann boundary conditions with an example.

Water Waves in 2D. Using Dirichlet Boundary conditions in an example to solve the wave equation for a rectangular region. Procedure how to solve Wave equation in this way using Fourier series.