Differential Equations

In General
* History: Originated with the invention of calculus (Newton and Leibniz, 1670–1680); 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 de 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.
@ Existence, uniqueness, regularity: in Gallavotti 83.
@ Symmetries: Krasil'shchik & Vinogradov 98 [and conservation laws].
@ And topology: Filippov 98_UM.
@ Related topics: Gitman & Kupriyanov a0710 [action principle].

Types and Solution Methods > s.a. Boundary Value Problems; Darboux Transformation; ordinary 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]; Polianin et al 02 [handbook].
@ Stochastic: Burrage et al PRS(04) [numerical]; > s.a. partial de's.

Other Types of Equations > s.a. Difference Equations; functional analysis; partial differential equations; Quaternions.
* Integro-differential: 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.
@ Fractional order: Gorenflo & Mainardi in(97)-a0805; Ciesielski & Leszczynski mp/03-in [and anomalous diffusion]; Duan JMP(05) [time and space]; > s.a. analysis, diffusion.
@ Integro-differential: Becker JMP(99) [perturbative approach]; Dattoli et al NCB(04) [operator method].
@ Other types: de Gosson et al mp/00-in [p-adic]; Maj mp/04-in [pseudodifferential wave equations].

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