Partial Differential Equations Done Right
Camrud - Partial Differential Equations Done Right.pdfThis book should serve as a first course in PDEs for a traditional undergraduate path. Most notably, a full sequence of calculus, linear algebra, and ODEs are necessary prerequisites.
Topics covered include:
Construction of PDEs from physical principles
Domains and boundaries in arbitrary dimension
A review of ODEs solution methods
Outer products and tensor fields
The Taylor series for scalar fields
The Fourier series and transform for scalar fields
Flows, Hamiltonians, and the method of characteristics
Construction of the Laplace, heat, and wave equations
Separation of variables
Fourier methods for the solution to the Laplace, heat, and wave equations
Classification of elliptic, parabolic, hyperbolic PDEs
Integral transforms, the Dirac delta function, and Green's functions
Eigenvalues and eigenfunctions
The spectral theorem for Hermitian PDE operators
The Cauchy-Kovalevskaya theorem and power series methods
Finite difference and Galerkin methods
Video lectures (each a guided read-through) corresponding to the entire book may be found on my YouTube channel (see below).