nLab
differential equation

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

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          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Variational calculus

          Equality and Equivalence

          Contents

          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( ix)F(\partial^i x) on sections of a bundle EXE \to X determine and are determined by their solution spaces, which are sub-D-schemes of the jet bundle D-scheme:

          Solutions(F) Jet(E) (X) \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 XX and the category of partial differential equations with variables in XX (Marvan 86).

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

          Solutions(F) TX \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

          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 XX and AA any two objects and D={xR|x 2=0}D = \{x \in R | x^2 = 0\} the abstract tangent vector, a diagram, of the form

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

          may be read as

          • ff is a smooth function on XX with values in AA;

          • whose derivative along the tangent vector vT xXTX=X Dv \in T_x X \subset T X = X^D

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

          Accordingly, the diagram

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

          may be read as encoding the differential equation v(f) x=αv(f)_x = \alpha (at one point) whose solutions fA Xf \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 ff at each single point of XX and varying smoothly with XX, we think of f:XAf : X \to A in terms of the exponential map

          f X:X XA X f^X : X^X \to A^X

          and consider

          * Id X * D v X X f α f X A X. \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:DX Xv : D \to X^X is the adjunct of a vector field XTX=X DX \to T X = X^D on XX;

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

          • α:DA X\alpha : D \to A^X is the adjunct of a map XTA=A DX \to T A = A^D that sends each point xXx \in X to a tangent vector in T f(x)AT_{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)=α. v(f) = \alpha \,.

          Examples

          References

          General accounts include

          • 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.

          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 is due to

          • 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)

          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

          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

          Last revised on December 11, 2017 at 06:28:04. See the history of this page for a list of all contributions to it.