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.

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