Eleonoraagostinelli

The package starts showing the solution of important PDEs, then it focuses on ODEs and how to find series solutions, to then go back to PDEs and focus on how to solve PDEs using separation of variables and reducing them to ODEs that can be easily solved. Finally it show how integral transforms can be used to solved PDEs

This chapter introduces important PDEs such as the heat equation, wave equation, Laplace equation and Helmholtz equation and gives examples of how to solve them using separation of variables. Then it introduces the 3 coordinate systems of polars, cylindrical polars and spherical, showing how to go from one system to another. Finally it shows how separation of variables can be used to to solve some important types of equation such as the Euler-type equation, the Bessel's equation and the Legendr...

This chapter focuses on ODEs, which are essential when solving PDEs. In particular it show how to solve ordinary differential equation using the Frobenius method, i.e. finding a series solution to the ODE.

This document introduces to Sturm-Liouville eigenvalue problems, regular, singular and periodic, to find eigenvalues and eigenfunctions of partial differential equations. In introduces the concept of orthogonality of eigenfunctions. Then it focuses on how to generate the generalised Fourier series expansion of a function, and in particular the Fourier-Bessel and Fourier-Legendre series. Finally, it introduces inhomogeneous problems and how to solve them.

Providing a general approach to solve problems involving partial differential equations using the separation of variables method

Introduction to Fourier and Laplace transforms and their reverse transform and their use to solve partial differential equations

This package gives an introduction to applied stochastic methods: starting with the Brownian process, stochastic calculus and SDEs, to exploring the connection from PDEs and SDEs and finding methods to approximate complex systems of SDEs to simpler and solvable ones. Finally it introduces linear response theory, to predict the response of a system

Providing methods to use the known past behaviour of a stochastic process in order to predict its response to a perturbation

Investigating the connection between stochastic differential equations and partial differential equations in order to use PDEs to solve or find characteristics of the SDE and viceversa

Providing methods to approximate complex systems of stochastic differential equations to simpler ones that can be solved