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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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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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22. PDEs and waves
- Class notes • 5 pages • 2019
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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.
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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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26. PDEs and waves
- Class notes • 5 pages • 2019
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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.
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Series solutions of ODEs create special functions
- Class notes • 7 pages • 2019
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27. Power Series Method process of solving and example. Picard iteration method for second order ODE using Matlab. Power series solution for nonlinear ODEs. Power series properties (Radius of convergence). Existence of power series.
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28. PDEs and waves & Series solutions of ODEs create special functions
- Class notes • 7 pages • 2019
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Series Solution and how they can be found. Modelling of shallow water and finding solutions. Variation of Parameters and an example. Variation of parameters general solution and deriving it. Power Series method and using is to solve ODEs creating a recurrence relationship.
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29. Series solutions of ODEs create special functions
- Class notes • 5 pages • 2019
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Legrendre polynomials and ODE, finding a power series to solve creating a recurrence relation for the coefficients. Convergence for power series. Method of Frobenius for circular and cylindrical geometries. Singular points and an example. Bessel's equation with singular points. Frobenius method and its solutions.
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30. More partial differential equations
- Class notes • 5 pages • 2019
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Bessel equation and looking at Frobenius solution when using cylindrical coordinates. Show how nu is a solution a Frobenius solution to the Bessel Equation. Finding a recurrence relation for the coefficients of the Bessel functions. Plotting Bessel functions of the first kind. Generalising the Bessel series solution of first kind of order nu where half integers can used.
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31. More partial differential equations
- Class notes • 4 pages • 2019
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Gamma Function and solving the integral. Gamma Function for half integers and Matlab Script. Bessel Function series and plot with different values for nu. Bessel functions, special cases and the Bessel equation General Solution. Indicial Equation and its derivation. Frobenius Theorem and the different cases for different solutions.