System of differential equations

In mathematics, a system of differential equations is a finite set of differential equations. A system can be either linear or non-linear. Also, a system can be either a system of ordinary differential equations or a system of partial differential equations.

Linear system of differential equations

Main article: Linear differential equation

Like any system of equations, a system of linear differential equations is said to be overdetermined if there are more equations than the unknowns. A system of Cauchy–Riemann equations is an example of an overdetermined system.

For an overdetermined system to have a solution, it needs to satisfy the compatibility conditions. For example, consider the system:

${\frac {\partial u}{\partial x_{i}}}=f_{i},1\leq i\leq m.$

Then the necessary conditions for the system to have a solution are:

${\frac {\partial f_{i}}{\partial x_{k}}}-{\frac {\partial f_{k}}{\partial x_{i}}}=0,1\leq i,k\leq m.$

See also: Cauchy problem and Ehrenpreis's fundamental principle.

Non-linear system of differential equations

Main article: Nonlinear differential equations

Perhaps the most famous example of a non-linear system of differential equations is the Navier–Stokes equations. Unlike the linear case, the existence of a solution of a non-system is a difficult problem (cf. Navier–Stokes existence and smoothness.)

See also: h-principle.

Differential system

A differential system is a means of studying a system of partial differential equations using geometric ideas such as differential forms and vector fields.

See also: Category:differential systems.

See also

• integral geometry

References

• L. Ehrenpreis, The Universality of the Radon Transform, Oxford Univ. Press, 2003.
• Gromov, M. (1986), Partial differential relations, Springer, ISBN 3-540-12177-3