nLab linear differential equation

Redirected from "linear partial differential equations".
Contents

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

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

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& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

A differential equation (ordinary or partial) is called linear if the linear combination of any of its solutions is still a solution, hence if its space of solutions is a vector space.

Equivalently this means that the differential operator that corresponds to the differential equation is a linear operator.

Linear differential equations may be analyzed via harmonic analysis by applying Fourier transform to decompose solutions as superpositions of plane wave “harmonics” (e.g. Hörmander 90).

In physics a field theory whose equations of motion is a linear partial differential equation is called a free field theory.

Where a general (possibly non-linear) differential equation is equivalently an object in the slice category over the de Rham shape of the space of its free variables, a linear differential equation is more specifically a linear object in this slice. In the context of algebraic geometry these are the D-modules.

Examples

References

  • Lars Hörmander, The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

See also

Last revised on May 23, 2022 at 06:34:13. See the history of this page for a list of all contributions to it.