Differential Equations |
In General
* History: Originated with the
invention of calculus (Newton and Leibniz, 1670–1680); They were first used to
solve geometrical problems, then dynamical ones starting with Euler around 1730.
* Order: The highest derivative
appearing in the equation.
* Degree: The highest power to
which the highest derivative is raised, if the differential equation is polynomial.
* Varying the equation: "Close
differential equations give close solutions", but this closeness is not uniform
over t; The solutions almost never stay close, and the trajectories, as sets,
may or may not be close; > s.a. chaos.
@ General references:
Vrabie 16 [intro text];
Székelyhidi 16 [intro text].
@ Existence, uniqueness, regularity: in Gallavotti 83.
@ Symmetries: Krasil'shchik & Vinogradov
98 [and conservation laws];
Duarte & da Mota a1007 [semi-algorithmic method for finding Lie symmetries];
Peng a2009
[modified formal Lagrangian formulation for general differential equations].
@ And topology:
Filippov 98.
@ Related topics: Gitman & Kupriyanov JPA(07)-a0710 [action principle];
> s.a. Variational Principles.
Types and Solution Methods > s.a. Boundary-Value
Problems; Darboux Transformation; ordinary
differential equations; partial differential equations.
* Numerical: A robust one
is the Runge-Kutta method (errors can only grow at a polynomial rate).
@ General references: Titchmarsh 62 [eigenfunction expansion];
Ablinger et al a1601 [coupled systems];
Loustau 16 [numerical].
@ Integrable:
Bracken a0903 [and theory of surfaces].
@ Stochastic:
Burrage et al PRS(04) [numerical];
Williams PhyA(08)
[stochastic Ansatz for solution];
> s.a. partial differential equations.
@ Discretization: Murata et al JPA(10) [systematic, for first-order equations];
Bihlo JPA(13)-a1210 [invariant meshless discretization schemes];
Campoamor-Stursberg et al JPA(16)-a1507 [for ODEs, symmetry preserving];
> s.a. Difference Equations.
@ Runge-Kutta method:
Kalogiratou et al PRP(14);
O'Sullivan JPCS(17)-a1702 [Factorized Runge-Kutta-Chebyshev methods];
Tapley a2105 [preservation of second integrals];
> s.a. computational physics.
@ Other types: de Gosson et al mp/00-proc,
Kedlaya 10 [p-adic];
Maj mp/04-proc [pseudo-differential wave equations];
Carlsson et al a1403
[linear differential equations with infinitely many derivatives];
Grabowski & de Lucas a1810 [superdifferential equations];
> functional analysis; p-Adic Numbers;
quaternions.
Integro-Differential Equations
* Idea:
Typical examples include time-dependent diffusion equations containing a parameter
(e.g., the temperature) that depends on integrals of the unknown distribution function;
The standard approach to solving the resulting non-linear pde involves the use of
iterative predictor-corrector algorithms.
@ General references:
Becker JMP(99) [perturbative approach];
Dattoli et al NCB(04) [operator method].
@ Applications:
Tarasov TMP(09)-a1107 [fractional, electromagnetic waves in a dielectric].
Fractional Order Differential Equations > s.a. diffusion;
fractional calculus; fractals.
@ Books:
Kilbas et al 06;
Guo et al 15 [and numerical solutions];
Zhou et al 16 [basic theory].
@ General references:
Gorenflo & Mainardi in(97)-a0805;
Ciesielski & Leszczynski mp/03-conf [and anomalous diffusion];
Duan JMP(05) [time and space];
Kochubei FCAA-a0806 [solutions and singularities];
Podlubny et al JCP(09) [matrix approach];
Wu a1006 [variational approach];
Saxena et al a1109 [computable solutions];
Leo et al CRM(14)-a1402, a1405 [symmetries].
@ Specific types of equations: Saxena et al EJPAM-a1406
[analytical solutions of fractional order Laplace, Poisson and Helmholtz equations in two variables].
@ And brownian motion:
Mainardi & Pironi EM(96)-a0806;
Lim & Muniandy PLA(00);
Hochberg & Pérez-Mercader PLA(02) [and renormalization];
McCauley et al PhyA(07) [vs Gaussian Markov processes, and Hurst exponents];
Duarte & Guimarães PLA(08) [and fractional derivatives];
> s.a. brownian motion.
@ Fractional quantum mechanics / Schrödinger equation:
Laskin PRE(02)qp;
Naber JMP(04)mp [time-fractional];
Herrmann mp/05;
Guo & Xu JMP(06) [examples];
Iomin PRE(09)-a0909;
Jeng et al JMP(10) [non-locality];
Laskin a1009 [rev].
@ Other applications:
Hilfer ed-00 [in physics];
Klafter et al ed-11;
Tarasov TMP(09)-a1107 [electromagnetic fields in dielectric media].
Specific types of equations:
see fokker-planck equation;
poisson equation.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 25 may 2021