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29. Series solutions of ODEs create special functions
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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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8. Second and higher order ODEs
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Picard's Iteration method with example and using matlab to solve. Classification of ODEs: order and linearity. Initial value problem theory: autonomy, deterministism, homogeneous linear ODEs and superposition with proof. Different cases of equation giving examples of the different types and features.
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12. Second and higher order ODEs
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Damped system and the different scenarios and their solutions. Solutions plotted in matlab and graphed. Looked at Resonance in forced oscillations and the particular solution for this equation. Graph the solution of types of solution for the particular solution for different constant values and interpreted the results. Looked at more general methods of solving ODEs: finding a solution that is a factor multiple of a known solution.
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17. Partial differential equations
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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.
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26. PDEs and waves
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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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32. More partial differential equations
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Frobenius ODEs, and looking at special cases. Using the indicial equation to find what type of solution the Frobenius ODEs. Edge Waves redux finding a power series solution.
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16. Partial differential equations
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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.
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14. Second and higher order ODEs
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Looking at steady state solution for modelling the populations of predators and prey with graphs. Constructing systems of equations for different stages of life to model the population of a species.
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5. Second and higher order ODEs
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Example of a separable solution. Existence and Uniqueness Theorem (no proof) with two example questions. Showing how to work out the integrating factor to make exact ODEs exact.
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31. More partial differential equations
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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.
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