Ordinary Differential Equations| Ordinary Differential Equations |
First-Order Equations
* Existence theorems: A differential
equation y' = g(x, y), with g continuously
differentiable in a region R ⊂ \(\mathbb R^2\), admits an infinity of
of solutions f(x, y, C) = 0, such that for all
(x, y) ∈ \(\mathbb R^2\) there passes 1! solution.
* Example: The Langevin equation for brownian motion.
* Non-linear: May have more arbitrary constants
than one expects; For example, y' = y2/3/3,
or the Riccati equation below, have 2.
* Riccati equation: The non-linear
equation y' = a y2
+ b y + c; It can be reduced wlog to y'
= y2 + c, whose solution is y
= −ψ−1ψ',
where ψ'' = −c ψ; The latter is to be
solved, a linear second-order equation; (Note: c may not be constant!).
- Example: y' = ±
(y2−c), with c
constant; The 2-parameter family of solutions reduces to 1-parameter; The solution
of ψ'' = cψ with c a constant is ψ
= A exp{−c1/2x}
+ B exp{c1/2x}, so
\[ \def\ee{{\rm e}}
y = -{\psi'\over\psi} = \sqrt{c}\,{A-B\,\ee^{\pm2\sqrt{c}x}_{\phantom o}\over A+B\,\ee^{\pm2\sqrt{c}x}_{\phantom o}}
= \sqrt{c}\,{1-D\,\ee^{\pm2\sqrt{c}x}_{\phantom o}\over 1+D\,\ee^{\pm2\sqrt{c}x}_{\phantom o}} \;.\]
- Blow-up in finite time: The
solutions of dx/dt = x1+ε,
for ε > 0.
@ Riccati:
Cariñena & Ramos IJMPA(99) [and groups];
Rosu et al JPA(03)mp/01 [generalization];
Cariñena et al EJDE(07)-a0810 [integrable, geometric approach];
> s.a. quaternions.
@ Other types, solutions:
Kosovtsov mp/02 [operator method],
mp/02 [rational],
mp/02 [integrating factors].
Second-Order Equations
> s.a. integrable systems; Special Functions;
Sturm-Liouville Theory; WKB Method.
* Methods for solution:
- u''(x)
+ p(x) u'(x)
= r(x), substitute v(x):= u'(x);
- u''(x)
+ p(x) u'(x)
+ q(x) u(x) = , {see #581};
- u''(x)
+ p(x) u'(x) + q(x) u(x)
= r(x), can be reduced to the form without the q(x)
term by u(x) =: v(x) h(x),
where h(x) solves the homogeneous equation.
* Eigenvalue problems: −y''(x)
+ x2N+2y(x)
= xNE y(x),
for −∞ < x < ∞, can be solved in closed form
[@ Bender & Wang JPA(01)mp].
* Non-linear example:
u'' = ± u2, one solution is
u = ± 6/(x+c)2.
@ General references:
Crampin & Saunders JGP(05) [Cartan theory, duality];
Rafiq et al PLA(08) [homotopy perturbation method].
@ Books: Ayres 52;
Coddington & Levinson 55;
Nemytskii & Stepanov 60;
Pontrjagin 70;
Arnold 73, 83;
Braun 83;
Stroud 74.
@ Eigenvalue problems: Ciftci et al JPA(05)mp/04 [asymptotic iteration method].
@ Delay-differential equations: in Kaplan & Glass 95 [II];
Simmendinger et al PRE(99)mp/01.
@ Non-linear equations: Asch et al mp/01 [h3(h''+h')
= 1 as t → ∞];
Cornejo-Pérez & Rosu PTP(05)mp;
Chandrasekar et al JPA(06) [linearization];
Ying & Candès JCP(06) [phase flow method for constructing phase maps].
@ Rational equations: Avellar et al AMC(07)mp/05,
mp/05 [elementary first-order integrals].
Different Types and Other Topics > s.a. differential equations.
* Order reduction: Linear ode's with
non-constant coefficients can be reduced in order if one knows any single solution;
If u(x) = f(x) is a particular solution, use the
ansatz u(x) = f(x) v(x), and the
ode becomes an equation of one order less for v'(x).
* Linear homogeneous:
It can be reduced to an eigenvalue problem.
@ General references: Sakovich PLA(03) [third-order, non-linear];
Górka et al CQG(12)-a1208 [with infinitely many derivatives, initial-value problem];
Gurappa et al a1205 [linear].
@ Systems:
Gaeta LMP(97) [normal form];
Mennicken & Möller
03 [boundary eigenvalue problems];
Cariñena & de Lucas DissM(11)-a1103 [Lie systems].
References
@ General:
Polyanin & Zaitsev 02 [exact solutions, handbook];
Schroers 11 [practical guide];
Deng 14 [lectures, problems and solutions];
Nandakumaran et al 17.
@ Symmetries:
Govinder & Leach JPA(95) [non-local];
Abraham-Shrauner et al JPA(95) [hidden contact symmetries];
Athorne JPA(97) [linear, homogeneous equations].
@ Approaches: Diver JPA(93) [genetic algorithm];
Bagarello IJTP(04),
same as IJTP(05)?? [non-commutative strategy];
Bervillier JPA(09)-a0812 [conformal mappings and other methods];
White 10 [asymptotic analysis].
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send feedback and suggestions to bombelli at olemiss.edu – modified 19 nov 2018