Partial Differential Equations  

Hyperbolic
* Properties: They admit an initial-value formulation.
* Examples: The wave equation and klein-gordon equation.
@ General references: Leray 52; Kundt & Newman JMP(68) [and characteristic propagation]; Beyer gq/05-ln [linear and quasi-linear, semigroup methods].
@ Non-linear: T Li 94 [quasi-linear]; Shatah & Sogge in Chrusciel 97; Claudel & Newman PRS(98) [quasi-linear, with singularity in t].
@ Characteristic: Frittelli JPA(04)mp, mp/04 [first-order, stability]; Nicolas m.AP/05 [second-order wave equations].
@ Singularities: Geroch JMP(83) [non-singularity theorem]; Witt 95 [with conical points]; Vickers & Wilson gq/01 [hypersurface singularities].

Elliptic
* Examples: The Poisson equation (> s.a. gravitating matter).
* Steady state equation: The general stationary limit of both the general wave and the diffusion equations,

– · (p u) + qu = F(x) .

Special cases are the Poisson equation 2u = –f (p = const, q = 0, and f:= F/p) and the laplace equation.
@ Non-linear: in Nirenberg 74; Kiessling PhyA(00)mp [Poisson-Boltzmann, Paneitz]; López-Fontán et al PhyA(07) [Poisson-Boltzmann].

Parabolic
@ References: A Friedman 64.

Integrability Conditions for a System of PDE's
* Idea: Conditions to be satisfied by the (known) functions appearing in a system of coupled pde's, in order for it to admit solutions.
* Example: If u, x = F(x, y) and u, y = G(x, y), need F, y = G, x .

Types and Solution Methods > s.a. Cauchy, Dirichlet, Neumann, Robin Problem; green functions; laplace equation; symmetries.
* History: The symmetry reduction method of finding group-invariant solutions was proposed by S Lie in the XIX cy.
* Separation of variables: Leads to ode's in eigenvalue form.
@ Superposition of solutions: Zhdanov JPA(94) [non-linear separation of variables]; Cariñena & Ramos mp/01/AAM.
@ Symmetries and reduction: Baumann et al JPA(94) [non-classical]; Anderson et al CMP(00)mp/99, CMP(00)mp/99, mp/01-in [generalization to non-transverse actions]; Nucci TMP(05) [Lie group analysis]; Gaeta & Mancinelli IJGMP(05)mp/06 [asymptotic symmetries]; Cicogna & Laino RVMP(06)mp [conditional symmetries].
@ Non-symmetric solutions: Martina et al mp/01/JPA [-dimensional symmetry group].
@ First-order: Holcman & Kupka QJM(05)mp/03 [on compact manifolds]; Bogoyavlenskij CMP(96) [existence of Hamiltonian structures].
@ Second-order: LaChapelle AP(04)mp, AP(04)mp [linear, path integral method].
@ Linear: Hörmander 85; Fokas PRS(04)m.AP [boundary-value, variable coefficients].
@ Non-linear: Adomian 94; Kong & Hu PLA(98) [solutions, geometric]; Ramm mp/00; Ludu et al mp/02/IJCMS [multiscale analysis]; Peng PLA(03) [including sine-Gordon]; Fairlie JPA(04)mp, JNMP(05)mp/04 [implicit solutions]; Debnath 05; Lü PLA(06) [Burgers equation-based solutions]; Khater et al IJTP(06) [conservation laws]; Torres-Córdoba a0709 [Monge equation, solution]; Sals & Gómez a0805 [coupled systems]; > s.a. Riemann Equation, wave equation.
@ Stochastic: Hochberg et al PRE(99)cm [stochastic noise]; Hairer mp/01 [reaction-diffusion]; > s.a. effective potential, stochastic processes.
@ Other types: Visser & Yunes IJMPA(03)gq/02 [scale-invariant]; Barnaby & Kamran a0709 [infinitely many derivatives, initial value].
@ Spectral methods: Bonazzola et al JCAM(99)gq/98 [in general relativity].

In Mathematical Physics, Examples > s.a. diffusion; einstein's equation; initial value formulation of general relativity; numerical relativity.
@ General references: Rubinstein 94; Geroch gq/96; Gràcia et al IJGMP(04)mp [geometrical aspects]; Calin & Chang 04 [on Riemannian manifolds].
@ Hyperbolicity: Gundlach & Martín-García PRD(04)gq [symmetric]; Reula gq/04 [strong]; Beig gq/04-in [rev].

Other References > s.a. differential equations.
@ General: Webster 47; Sommerfeld 49; Ayres 52; Petrovsky 54; Bers et al 64; Garabedian 64; Meis & Marcowitz 81; John 82; Bellman & Adomian 85; Zachmanoglou & Thoe 86; Stephani 89; Hubbard & West 90; Cronin 94.
@ Books, III: Stephenson 85; Edelen & Wang 92; Folland 95; Christodoulou 00.
@ And Lie groups: Olver 86; Dresner 98.
@ Initial value formulation: Bers et al 64 [non-second-order diagonal].
@ Conservation laws: Anco & Bluman EJAM(02)mp/01, EJAM(02)mp/01.
@ Computational: Wolf EJAM(02)cs.SC/03 [conservation laws]; Hawley & Matzner CQG(04)gq/03 [elliptic equations and holes]; Valiquette & Winternitz JPA(05)mp [discretization and symmetries]; > s.a. computational physics, numerical relativity.
@ With Mathematica: Vvedensky 92; Ross 95; Stavroulakis & Tersian 99.
@ Handbook: Zwillinger 89.
@ Related topics: Werschulz 91 [complexity]; Medvedev PRS(99) [Poincaré normal form]; Evans BAMS(04) [entropy methods].

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 26 jun 2008