Advanced Engineering Mathematics notes

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.

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.

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

Singular solutions for ODEs with examples and a graph. Existence and uniqueness theorem example. Picard's Iteration Method theory and an example.

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.

Example of solving the time it takes for water to flow out of a tank and its Matlab code. Solving Exact ODEs with example.

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

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.

Looking at the steps to Modelling Ordinary Differential Equations and the different types of forms: first order ODEs in explicit form, implicit form. How to solve Separable ODEs using a Chauvet Cave example. Lastly looking at the Concepts of a particular solution and a general solution.

How to make ODEs exact with a integration factor, finding a particular solution for Exact ODEs. Show why integrating factor works. Example finding an integrating factor, finding an exact solution and graphing the solution in Matlab.