Partial Differential Equations| Partial Differential Equations |
Hyperbolic
> s.a. types of wave equations; klein-gordon equation.
* Properties: They admit an initial-value formulation.
@ General references: Leray 52;
Kundt & Newman JMP(68) [and characteristic propagation];
Beyer gq/05-ln [linear and quasi-linear, semigroup methods];
Hattori 13 [II/III].
@ Non-linear: Li 94 [quasi-linear];
Shatah & Sogge in Chruściel 97;
Claudel & Newman PRS(98) [quasi-linear, with singularity in time].
@ Characteristic: Frittelli JPA(04)mp,
JPA(05)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];
Chen 10;
Ghergu & Taliaferro 16.
Elliptic > s.a. Poisson Equation.
* Steady-state equation: The general
stationary limit of both the general wave and
the diffusion equations,
\[ -\nabla\cdot(p\nabla u) + qu = F(x)\;; \]
Special cases are the Poisson equation \(\nabla^2u = -f\) (the case with \(p\) = const,
\(q\) = 0, and \(f:= F/p\)) and the laplace equation.
@ Numerical methods: Okawa IJMPA(13)-a1308-ln [and constraints in numerical relativity].
@ Non-linear: in Nirenberg 74;
Kiessling PhyA(00)mp [Poisson-Boltzmann, Paneitz];
López-Fontán et al PhyA(07) [Poisson-Boltzmann].
Parabolic
@ References: Friedman 64;
Lieberman 96.
Integrability Conditions for a System of PDEs
* 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. Boundary-Value
Problems; green functions; Inverse
Scattering; laplace equation; symmetries.
* History: The
symmetry reduction method of finding group-invariant solutions
was proposed by S Lie in the XIX century.
* Symmetries and
reduction: Any symmetry reduces a second-order differential equation
to a first-order equation; Variational symmetries of the action
(exemplified by central field dynamics) lead to conservation laws,
symmetries of only the equations of motion (exemplified by scale-invariant
hydrostatics) yield first-order non-conservation laws between invariants.
* Separation of variables:
Leads to ordinary differential equations in eigenvalue form.
@ Superposition of solutions: Zhdanov JPA(94) [non-linear separation of variables];
Cariñena & Ramos AAM-mp/01;
> s.a. Ermakov System.
@ Symmetries and reduction: Baumann et al JPA(94) [non-classical];
Anderson et al CMP(00)mp/99,
CMP(00)mp/99,
mp/01-proc
[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];
Bludman a1106 [non-variational symmetries];
Steinberg & de Melo Marinho a1409-ln [computational approach].
@ Non-symmetric solutions: Martina et al JPA(01)mp [infinite-dimensional symmetry group].
@ First-order: Holcman & Kupka QJM(05)mp/03 [on compact manifolds];
Bogoyavlenskij CMP(96) [existence of Hamiltonian structures];
López 12.
@ Second-order: LaChapelle AP(04)mp,
AP(04)mp [linear, path-integral method];
Cioranescu et al 18.
@ Linear: Hörmander 85;
Polyanin 01 [handbook].
@ Non-linear: Adomian 94;
Kong & Hu PLA(98) [solutions, geometric];
Ramm MMMAA-mp/00;
Ludu et al IJCMS-mp/02 [multiscale analysis];
Peng PLA(03) [including sine-Gordon];
Fairlie JPA(04)mp,
JNMP(05)mp/04 [implicit solutions];
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];
Polyanin & Zaitsev 11 [handbook];
Anco et al Sigma(11)-a1105 [method of group foliation reduction];
Maheswari & Sahadevan JPA(11) [conservation laws];
Debnath 12;
Tadmor BAMS(12) [numerical methods];
Li & Song 16 [variational methods];
> s.a. Riemann Equation; types of wave equations.
@ Stochastic: Hochberg et al PRE(99)cm [stochastic noise];
Hairer Nonlin(02)-mp/01 [reaction-diffusion];
> s.a. effective potential; stochastic processes.
@ Other types: Visser & Yunes IJMPA(03)gq/02 [scale-invariant];
Barnaby & Kamran JHEP(08)-a0709,
JHEP(08)-a0809 [infinitely many derivatives, initial-value problem];
Tehseen & Prince JPA(13)-a1302 [using differential geometric methods];
Ablinger et al a1608 [coupled systems, in terms of power series];
> s.a. differential equations [fractional], Combinatorial PDEs.
@ Spectral methods: Bonazzola et al JCAM(99)gq/98 [in general relativity];
Piotrowska et al a1712 [non-smooth problems].
In Mathematical Physics
> s.a. chaotic systems; diffusion.
@ General references: Rubinstein 94;
Geroch gq/96;
Calin & Chang 04 [on Riemannian manifolds];
Kirkwood 12;
Lein a1508-ln.
@ Geometrical aspects:
Zharinov 92;
Gràcia et al IJGMP(04)mp.
@ Hyperbolicity: Gundlach & Martín-García PRD(04)gq [symmetric];
Reula gq/04 [strong];
Beig LNP(06)gq/04 [rev].
> Gravity-related examples:
see einstein's equation; initial-value
formulation of general relativity; numerical relativity.
Other References > s.a. differential equations.
@ General: Webster 47;
Sommerfeld 49; Ayres 52;
Petrovsky 54
(reprint 91);
Bers et al 64; Garabedian 64;
Meis & Marcowitz 81; John 82;
Bellman & Adomian 85;
Zachmanoglou & Thoe 86;
Stephani 89;
Hubbard & West 90;
Cronin 94;
Xu a1205 [algebraic approaches].
@ Books, III: Edelen & Wang 92;
Folland 95;
Stephenson 96;
Christodoulou 00.
@ And Lie groups: Olver 93;
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;
Zhang a1409 [new technique];
Anco & Kara EJAM-a1510 [symmetry invariance].
@ 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];
Formaggia et al 12;
Bartels 15 [non-linear];
> s.a. computational physics; Finite-Element Method;
numerical relativity; Courant-Friedrichs-Lewy
Condition.
@ With Mathematica: Vvedensky 92;
Ross 04; Stavroulakis & Tersian
04.
@ Handbook: Zwillinger 89;
Polyanin et al 01 [first-order].
@ Related topics: Werschulz 91 [complexity];
Medvedev PRS(99) [Poincaré normal form];
Evans BAMS(04) [entropy methods].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 19 nov 2018