ACM 201b: Advanced Partial Differential Equations, Dr. Bruno
Content:  Heat equation, maximum principle, solutions in bounded domains; waves, one-dimensional wave equation, D'Alambert formula, domain of dependence and of influence; wave equation in N dimensions, Darboux equation, Hadamard's method of descent, Duhamel principle; water waves, linearization, dispersion relation, phase and group velocities, shallow-water equations, gas dynamics, hyperbolic systems, Riemann invariants, shock waves, existence of solutions to hyperbolic systems; integral equations, single- and double-layer potentials, Fredholm's theorem,
Books: John, F, "Partial Differential Equations", Springer-Verlag 1982


 

ACM 201a: Advanced Partial Differential Equations, Dr. Bruno
Content: Existence and uniqueness for ordinary Ordinary Differential Equations (ODEs); first-order Partial Differential Equations (PDEs), quasi-linear equations, characteristics, shocks, shock condtiions, Burger's equation, nonlinear equations, geometrical optics; Cauchy-Kowalevsky; secon-order equations, hyperbolic, parabolic, and elliptic; Laplace's equation, Green's formulae, weak solutions, maximum principle, existence of solutions of Dirichlet problem; Hilbert space methods, general Sobolev spaces, distributions, method of partition of unity; eigenvalue problems.
Books: John, F, "Partial Differential Equations", Spring-Verlag 1982

ACM 210b: Advanced Numerical PDEs, Dr. Hou
Content:  Multiple-scales problems, homogenization theory, finite-element approximation, Cea's lemma; two-phase flow problems, homogenization for time-dependent problems, semi-linear hyperbolic systems, particle methods, convergence analysis for particle methods, wavelet-based homogenization, variational multi-scale method, incompressible Euler equations,
Books: Class Notes
 

ACM 210a: Advanced Numerical PDEs, Dr. Hou
Content:  Stability for linear hyperbolic systems, local truncation error, dynamic and numerical stability, phase error, dissipation schemes; nonlinear hyperbolic PDEs, entropy condition, Godunov's scheme, Riemann problems, Lax-Wendroff theorem; high-resolution schemes, flux limiters, Roe's scheme, semi-discrete ENO methods, Runge-Kutta total-variation-diminishing schemes; level sets methods,
Books: Class Notes

AE 160a: Continuum Mechanics of Solid and Liquids, Dr. Ortiz
Content: Kinematic of deformable bodies, Lagrangian picture (solids), analysis of local deformation, orientation, finite spatial rotations, three-dimensional rotations, vector and matrix representations, Euler's theorem, polar decomposition, spectral decomposition, exponential representation of rotations, algebra of quaternions, compatibility equations, discontinuous deformations, dislocations, integrability of systems of PDEs, Christoffel symbols, properties of the Riemann-Christoffel tensor, linearized kinematics, Eulerian picture (fluids).
Books: Class Notes