In mathematics, a system of differential equations is a finite set of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. Such a system can be either
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
or
non-linear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
. Also, such a system can be either a system of
ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
or a system of
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
.
Linear system of differential equations
Like any system of equations, a system of linear differential equations is said to be
overdetermined if there are more equations than the unknowns.
For an overdetermined system to have a solution, it needs to satisfy the
compatibility conditions
Compatibility may refer to:
Computing
* Backward compatibility, in which newer devices can understand data generated by older devices
* Compatibility card, an expansion card for hardware emulation of another device
* Compatibility layer, compon ...
.
For example, consider the system:
:
Then the necessary conditions for the system to have a solution are:
:
See also:
Cauchy problem
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. A Cauchy problem can be an initial value problem or a boundary value proble ...
and
Ehrenpreis's fundamental principle.
Non-linear system of differential equations
Perhaps the most famous example of a non-linear system of differential equations is the
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
. Unlike the linear case, the existence of a solution of a non-linear 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
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
using geometric ideas such as differential forms and vector fields.
For example, the compatibility conditions of an overdetermined system of differential equations can be succinctly stated in terms of differential forms (i.e., a form to be exact, it needs to be closed). See
integrability conditions for differential systems In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
for more.
See also:
:differential systems.
Notes
See also
*
Integral geometry In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformation ...
*
Cartan–Kuranishi prolongation theorem Given an exterior differential system defined on a manifold ''M'', the Cartan–Kuranishi prolongation theorem says that after a finite number of ''prolongations'' the system is either ''in involution'' (admits at least one 'large' integral mani ...
References
*L. Ehrenpreis, ''The Universality of the Radon Transform'', Oxford Univ. Press, 2003.
*Gromov, M. (1986), Partial differential relations, Springer,
*M. Kuranishi, "Lectures on involutive systems of partial differential equations" , Publ. Soc. Mat. São Paulo (1967)
*Pierre Schapira, ''Microdifferential systems in the complex domain,'' Grundlehren der Math- ematischen Wissenschaften, vol. 269, Springer-Verlag, 1985.
Further reading
*https://mathoverflow.net/questions/273235/a-very-basic-question-about-projections-in-formal-pde-theory
*https://www.encyclopediaofmath.org/index.php/Involutional_system
*https://www.encyclopediaofmath.org/index.php/Complete_system
*https://www.encyclopediaofmath.org/index.php/Partial_differential_equations_on_a_manifold
{{math-stub
Differential equations
Differential systems
Multivariable calculus