Elementary Partial Differential Equations II Topics
Biomathematics and Applied and Computational Mathematics at Florida State University
References:
- Partial Differential Equations: an introduction, by Walter A. Strauss, John Wiley, 1992
- Partial Differential Equations, by L. C. Evans, American Mathematical Society, 1998
- Functional Analysis, Walter Rudin, McGraw-Hill Science/Engineering/Math, 1991, Second Edition
Topics:
- Background
- Classification of 2nd order linear PDEs as elliptic, hyperbolic, or parabolic, and their associated properties.
- Well-posedness of PDEs: existence, uniqueness, and continuous dependence on data (a.k.a. stability)
- Similarity solutions for linear/nonlinear PDEs with examples
- The diffusion equation in ℝ1 and ℝd
- The maximum principle
- The heat kernel (a similarity solution)
- Well-posendess results
- Use of reflections with the heat kernel
- The wave equation in ℝ1 and ℝd
- D'Alambert's solution revisited (with reflections), and characteristics
- Well-posedness results
- Advanced solution techniques
- Background: distributions and weak convergence
- Green's functions
- Fourier transform methods
- Nonlinear PDEs
- Similarity solutions
- The method of characteristics
- The Burgers-Hopf equation: shock and rarefaction waves
- The (viscous) Burgers equation and the Cole-Hopf transform
- The KdV equation (brief)
- Linear stability analysis of PDEs
- Linearization about an equilibrium solution; stable and unstable modes
- Examples from biology (Turing mechanism and diffusive predator-prey models)
