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24. PDEs and waves
- Class notes • 5 pages • 2019
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D'Alembert's solution and derivation of the form of solutions. Classification of PDEs with an example using the wave equation.
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4. Second and higher order ODEs
- Class notes • 4 pages • 2019
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Solving Linear ODEs: definition of linear/nonlinear and homogeneous/nonhomogeneous. General solution to first order ODE with integrating factor. Example of a first order linear IVP. Explanation to why the General solution formula works. Example of Hormone levels and their rates of changes. Using Matlab to solve ODEs.
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21. Representing periodic functions by Fourier series & PDEs and waves
- Class notes • 4 pages • 2019
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Graph and solve fourier series for square wave with arbitrary period. Fourier Cosine and Sine series with matlab example. Half-range expansion with examples.
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18. Representing periodic functions by Fourier series
- Class notes • 3 pages • 2019
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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.
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23. PDEs and waves
- Class notes • 4 pages • 2019
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Linearising wave equation for a water column. Using separation of variables to create a general solution to the wave equation.
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20. Representing periodic functions by Fourier series
- Class notes • 3 pages • 2019
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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.
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1. First order ordinary differential equations
- Class notes • 3 pages • 2019
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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.
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6. Second and higher order ODEs
- Class notes • 3 pages • 2019
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Singular solutions for ODEs with examples and a graph. Existence and uniqueness theorem example. Picard's Iteration Method theory and an example.
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2. First order ordinary differential equations
- Class notes • 3 pages • 2019
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Example of solving the time it takes for water to flow out of a tank and its Matlab code. Solving Exact ODEs with example.
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3. First order ordinary differential equations & Second and higher order ODEs
- Class notes • 2 pages • 2019
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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.