nLab differential equation

Idea

A differential equation is an equation involving terms that are derivatives (or differentials). One sometimes distinguishes partial differential equations (which involve partial derivatives) from ordinary differential equations (which don't).

As sub-$\infty$-groupoids of tangent and jet bundle

Analogous to how ordinary equations determine and are determined by their spaces of solutions – the corresponding schemes – accordingly differential equations $F(\partial^i x)$ on sections of a bundle $E \to X$ determine and are determined by their solution spaces, which are sub-D-schemes of the jet bundle D-scheme:

$$\array{ Solutions(F) &&\hookrightarrow&& Jet(E) \\ & \searrow && \swarrow \\ && \Im(X) }$$

In fact there is an equivalence of categories between the Eilenberg-Moore category of the jet comonad over $X$ and the category of partial differential equations with variables in $X$ (Marvan 86).

At least in nice cases for differential equations on functions on $X$ this is equivalently modeled by sub-Lie algebroids of tangent Lie algebroids.

$$\array{ Solutions(F) &&\hookrightarrow&& T X }$$

see exterior differential system for details

In terms of synthetic differential geometry

A perspective on differential equations from the nPOV of synthetic differential geometry is given in

• William Lawvere, Toposes of laws of motion , transcript of a talk in Montreal, Sept. 1997 (pdf)

  (on the description of differential equations in terms of synthetic differential geometry)

See also the appendix of

• Outline of synthetic differential geometry , seminar notes (1998) (pdf)

First order homogeneous ODEs as extension problems in a smooth topos

In a smooth topos with $X$ and $A$ any two objects and $D = \{x \in R | x^2 = 0\}$ the abstract tangent vector, a diagram, of the form

$$\array{ {*}&\to& D &\stackrel{v}{\to}& X \\ &{}_{\mathllap{v(f)(x)}}\searrow& {}^{\mathllap{v(f)}}\downarrow & \swarrow_{\mathrlap{f}} \\ && A }$$

may be read as

• $f$ is a smooth function on $X$ with values in $A$;

• whose derivative along the tangent vector $v \in T_x X \subset T X = X^D$…

• …is the tangent vector $v(f) \in T_{f(x)} A \subset T A = A^D$.

Accordingly, the diagram

$$\array{ D &\stackrel{v}{\to}& X \\ {}_{\mathllap{\alpha}}\downarrow \\ A }$$

may be read as encoding the differential equation $v(f)_x = \alpha$ (at one point) whose solutions $f \in A^X$ are the extensions that complete this diagram.

To get differential equations in the more common sense that they impose a condition on the derivative of $f$ at each single point of $X$ and varying smoothly with $X$, we think of $f : X \to A$ in terms of the exponential map

$$f^X : X^X \to A^X$$

and consider

$$\array{ &&{*} \\ && \downarrow & \searrow^{\mathrlap{Id_X}} \\ {*}&\to& D &\stackrel{v}{\to}& X^X \\ &{}_{\mathllap{f}}\searrow&{}^{\mathllap{\alpha}}\downarrow & \swarrow_{\mathrlap{f^X}} \\ &&A^X } \,.$$

In this diagram now

• $v : D \to X^X$ is the adjunct of a vector field $X \to T X = X^D$ on $X$;

  (the commutativity of the top right triangle ensures that indeed this $X \to T X$ is a section of the tangent bundle projection $T X \to X$);

• $\alpha : D \to A^X$ is the adjunct of a map $X \to T A = A^D$ that sends each point $x \in X$ to a tangent vector in $T_{f(x)} A$

  (enforced by the commutativity of the left bottom triangle)

• and the commutativity of the right lower triangle is the differential equation

  $$v(f) = \alpha \,.$$

Examples

• heat equation

• wave equation, Klein-Gordon equation, sine-Gordon equation

• Navier-Stokes equation

• Schrödinger equation

• Dirac equation

• Green hyperbolic partial differential equation

Related concepts

• differentiation, differential calculus

• diffiety

• ordinary differential equation, partial differential equation

• linear differential equation

• integrable differential equation, formally integrable differential equation

• generalized solution of a PDE

• geometric analysis

• Green's function

• Sturm–Liouville theory

• initial value problem

• differential Galois theory

• D-module, D-geometry

• homotopy analysis method

References

General

General accounts:

• Jean-François Pommaret, Partial Differential Equations and Group Theory, Springer (1994) [doi:10.1007/978-94-017-2539-2]

• Sergiu Klainerman, PDE as a unified subject 2000 [pdf]

• Boris Kruglikov, Valentin Lychagin, Geometry of differential equations [pdf]

• Arthemy Kiselev, The twelve lectures in the (non)commutative geometry of differential equations, preprint IHES M/12/13 pdf

• Yves Andre, Solution algebras of differential equations and quasi-homogeneous varieties, arXiv:1107.1179

• Mikhail Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-Verlag, Berlin, 1986. x+363 pp.

• Peter Olver, Applications of Lie groups to differential equations, Springer; Equivalence, invariants, and symmetry, Cambridge Univ. Press 1995.

On the partial differential equations appearing in mathematical physics:

• Cliff P. Burgess, Primer on Partial Differential Equations for Physicists, lecture notes (1990) [pdf, pdf]

• Robert Geroch, Partial Differential Equations of Physics, in: General Relativity: Proceedings Edited by G.S. Hall and J.R. Pulham. Edinburgh, IOP Publishing, 1996. p. 19 (arXiv:gr-qc/9602055)

• Bijan Kumar Bagchi, Partial Differential Equations for Mathematical Physicists, Routledge (2020) [ISBN:9780367227029]

Via jet bundles and D-modules

Discussion of the spaces of solutions is in

• Batu Güneysu, Markus Pflaum, The profinite dimensional manifold structure of formal solution spaces of formally integrable PDE’s (arXiv:1308.1005)

Characterization of the category of partial differential equations as the Eilenberg-Moore category of coalgebras over the jet comonad:

• Michal Marvan, A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno (pdf)

Discussion in differential cohesion:

• Igor Khavkine, Urs Schreiber, Synthetic geometry of differential equations: I. Jets and comonad structure (arXiv:1701.06238)

Discussion in differentially cohesive modal homotopy type theory:

• Felix Wellen, Formalizing Cartan Geometry in Modal Homotopy Type Theory, 2017

In the language of D-modules and hence for the special case of linear differential equations, this appears as prop. 3.4.1.1 in

• Dennis Gaitsgory, Nick Rozenblyum, Crystals and D-modules (arXiv:1111.2087)

Other

A domain specific programming language for differential equations is presented in

• Martin S. Alnaes, Anders Logg, Kristian B. Oelgaard, Marie E. Rognes, Garth N. Wells, Unified Form Language: A domain-specific language for weak formulations of partial differential equations, arXiv:1211.4047

See also

• Dispersive Wiki