Derivation of classical equations of applied math including quasilinear hyperbolic equations, Laplace, Poisson and Helmholtz equations, advection-diffusion equation, wave equation, etc., in various orthogonal curvilinear coordinate systems. The gradient, divergence, curl and Laplacian operators in Cartesian, cylindrical and spherical coordinates, acting on scalar or vector fields. Methods of solution on bounded domains using eigenfunction expansions and the associated Sturm-Liouville eigenvalue problems, including Bessel functions and Legendre polynomials. Method of characteristics for systems of hyperbolic equations, including nonlinear waves and shocks. Solution on unbounded domains using Laplace and Fourier transforms. Calculation of the transforms and their inverses using integration in the complex plane, including residue theory and integrals involving branch cuts. Green's functions. Fredholm's solvability criteria for linear systems. Calculus of variations and the Euler-Lagrange equations. Analysis of simple nonlinear dynamical systems in the phrase plane. Applications of linear algebra to the study of differential equations. Matrix exponentials. Prerequisite: Advanced Calculus, Differential Equations, Linear Algebra, Complex Variables. |