Partial Differential Equations

Classification & First-Order

1. Classification of PDEselliptic, parabolic, hyperbolic; well-posedness (Hadamard)5.5–10 h
2. First-Order PDEsmethod of characteristics, conservation laws intro5.5–10 h

Classical Equations

1. The Wave Equationd'Alembert solution, domain of dependence, reflections5.5–10 h
2. The Heat Equationmaximum principle, smoothing, fundamental solution5.5–10 h
3. Laplace & Poisson Equationsharmonic functions, mean value property, maximum principle5.5–10 h
4. Separation of Variablesspatial and temporal dependence, eigenfunction expansion5.5–10 h

Methods & Theory

1. Fourier Series & Transformsorthogonality, convergence, solving via transforms5.5–10 h
2. Green's Functionsfundamental solutions, boundary value problems5.5–10 h
3. Sturm-Liouville Theoryeigenvalue problems, completeness, applications5.5–10 h
4. Introduction to Nonlinear PDEsshock waves, solitons, reaction-diffusion5.5–10 h