''in the theory of partial differential equations''
101 bytes (13 words) - 16:57, 7 February 2011
...m concerns a [[Hyperbolic partial differential equation|hyperbolic partial differential equation]], or a second-order hyperbolic system, in two independent variabl
...y coincide with the sides of this angle. This problem has been studied for equations of the form (1).
5 KB (813 words) - 19:42, 5 June 2020
...$D$ of the variable $x = (x_1,\ldots,x_n)$ the solution $u(x)$ of a linear differential equation
...y differential equations]]; [[Boundary value problem, partial differential equations]].
471 bytes (78 words) - 21:19, 2 May 2012
...rface of the characteristic cone can be converted with the aid of interior differential operators to an especially simple form which can be used for the solution
\frac{\partial u }{\partial n }
2 KB (242 words) - 05:53, 30 May 2023
...Differential equation, partial, oblique derivatives|Differential equation, partial, oblique derivatives]].
$$\frac{df}{dl}=\sum_{i=1}^nl_i(P)\frac{\partial f}{\partial x_i},\quad P=(x_1,\ldots,x_n),$$
1 KB (187 words) - 17:57, 30 July 2014
The homogeneous partial differential equation
$$\frac{\partial u}{\partial t}-a^2\sum_{k=1}^n\frac{\partial^2u}{\partial x_k^2}=0.$$
2 KB (322 words) - 19:05, 30 July 2014
''bicharacteristic strip, of a linear partial differential operator''
...=1,\dots,n$, are defined by solving a system of $2n$ ordinary differential equations
3 KB (419 words) - 22:32, 10 December 2018
The partial differential equation
\frac{\partial ^ {2} u }{\partial t ^ {2} }
2 KB (306 words) - 08:25, 6 June 2020
...(x,a)$, $x=(x_1,\dots,x_n)$, $a=(a_1,\dots,a_n)$, of a first-order partial differential equation
...t(x_1,\dots,x_n,u,\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\right)=0,\label{1}\tag{1}$$
2 KB (389 words) - 12:23, 17 February 2024
''of a linear partial differential equation''
A solution of a partial differential equation $ Lu ( x) = 0 $,
3 KB (487 words) - 08:49, 13 May 2022
...olving a partial differential equation or a system of partial differential equations with given conditions on the characteristic manifolds (cf. [[Characteristic
...e set arbitrarily. They must satisfy certain conditions, determined by the differential equation. Accordingly, the characteristic Cauchy problem will usually be no
3 KB (535 words) - 22:45, 10 December 2018
\frac{\partial ^ {2} u ( t, x) }{\partial t ^ {2} }
\frac{\partial ^ {2} u ( t, x) }{\partial x ^ {2} }
3 KB (438 words) - 13:28, 26 March 2023
...equations of mathematical physics such as the [[Maxwell equations|Maxwell equations]] and the [[Wave equation|wave equation]]) have solutions in the form (*).
.... John, "Plane waves and spherical means applied to partial differential equations" , Interscience (1955)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign=
1 KB (210 words) - 08:28, 6 June 2020
A differential equation of the form
which is a simple model of a second-order partial differential equation of mixed elliptic-hyperbolic type with two independent variables $
2 KB (300 words) - 06:00, 30 May 2023
A partial differential equation of the form
$$\sum_{i=1}^m\left(\frac{\partial\tau}{\partial x^i}\right)^2=\frac{1}{c^2(x^1,\dots,x^m)}.$$
3 KB (444 words) - 12:04, 24 March 2024
* For geometric structures and partial differential equations - conditions guaranteeing existence of solutions ([[Frobenius integrability
* For differential equations (both ordinary and partial) and dynamical systems:
1 KB (149 words) - 07:42, 3 May 2012
...idiary (initial, boundary and other) conditions. Approximating an original differential problem by a difference scheme is one of the methods of approximating the o
the grid, and the derivatives entering into the differential equation are replaced by difference relations on the grid $ G _ {h} $.
5 KB (788 words) - 11:03, 27 April 2024
...class of evolution equations includes, first of all, ordinary differential equations and systems of the form
...rresponding difference equations, usually belong to the class of evolution equations.
3 KB (443 words) - 15:35, 14 February 2020
...ich it is defined. For example, if $L(\partial/\partial x_k)$ is a partial differential operator written out in some coordinate system $(x_1,\ldots,x_n)$, if $x_k=
\phi^* L\left( \frac{\partial}{\partial x} \right)u
3 KB (392 words) - 21:35, 24 July 2012
...ue problem of special form, in which it is required to find solutions to a differential equation
\frac \partial {\partial n }
7 KB (954 words) - 20:18, 10 January 2024