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10. Second and higher order ODEs
- Class notes • 5 pages • 2019
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Proof for multiple roots for solutions. Theory for the general solution for nonhomogeneous linear ODEs and an example. Method of undetermined guessing and an example which is plotted with matlab code. An example of an IVP for nonhomogeneous solutions.
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13. Second and higher order ODEs
- Class notes • 5 pages • 2019
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Reduction of order method to find all terms of the homogeneous solution. Euler Cauchy Equations and the types of solution for the roots of the characteristic equation. Example using Reduction of order finding the general solution. Modelling Mathematical ecology, by looking at a fish example, finding solutions to a system of equations of how different factors affect fish populations.
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15. Second and higher order ODEs
- Class notes • 5 pages • 2019
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Solving systems of equations using matlab and creating a general solution. Shows example of a repeated root problem and how to solve. Goes through the theorem for system of equations and a proof. Gives an example of modelling a system of equations.
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19. Representing periodic functions by Fourier series
- Class notes • 5 pages • 2019
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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.
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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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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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Exam Revision
- Summary • 14 pages • 2019
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7. Second and higher order ODEs
- Class notes • 5 pages • 2019
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Example finished from the end of the last notes. Definitions of linearly independent, linear dependence and linear combination with an example. Existence and uniqueness theorem. Example of solving a third order IVP. Definition of the Wronskian, the Wronskian test with examples and a proof.
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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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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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