Differential Equations 11

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.

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.

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.

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.

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.

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.

Looking at the wave equation and solving using polar coordinates for a wave propagating on the surface of a circular pool. Special case with no angular dependence and finding solutions.

D'Alembert's solution and derivation of the form of solutions. Classification of PDEs with an example using the wave equation.

Further Fourier Series looking at a Maclaurin Series, inner product, weight function, orthogonal and norm. These have definitions equations and examples. Define an orthogonal set and examples of these. Legrendre Polynomials, Bessel functions and generalised fourier series. Definition for completeness of functions. Complex Fourier series and example.

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.