This course unit surveys analytical methods for linear and nonlinear first and second order PDEs. We will discuss exact solutions, series solutions, Fourier transforms and nonlinear transforms, with a view to developing, applying and analysing a broad toolbox of methods to solve problems in applied mathematics.
No prior knowledge of PDEs is required, but experience with vector calculus and general undergraduate methods courses would be very helpful.
1. Introduction
Basic notation. Classification of PDEs, examples of common PDEs.
2. First order PDEs
Construction of solutions to linear and nonlinear first order PDEs via method of characteristics. Application of Cauchy data. Examples of shock formation.
3. Linear second order PDEs
Characteristics of second order PDEs, classification, reduction to normal form. Well-posedness of boundary conditions.
4. Fourier series
Properties of full and half range Fourier series, and discussion of orthogonality. Use of separable solutions in constructing series solutions for appropriate BVPs and IVPs.
5. Sturm-Liouville systems
Definition of Sturm-Liouville systems, and proofs of main properties for regular S-L systems. Further discussion of applicability of series solutions.
6. Fourier transforms
Connection to Fourier series. Summary of main properties of Fourier transforms, and examples of calculation. Inversion via contour integration, and relation to convolution properties. Examples of solution of linear PDEs in infinite domains, and use of sine and cosine transforms in semi-infinite domains.
7. Nonlinear PDEs
Failure of superposition principle. Cole-Hopf transform for Burgers' equation. Examples of Backlund transforms. Inverse scattering methods for the KdV equation.