$#C+1 = 17 : ~/encyclopedia/old_files/data/E035/E.0305520 Elliptic partial differential equation
A partial differential equation of order $ m $,
2 KB (314 words) - 20:22, 17 January 2024
\frac{\partial u }{\partial t }
and $ \Delta ^ {2} u = \sum _ {j=1} ^ {n} \partial ^ {2} u / \partial x _ {j} ^ {2} $
3 KB (400 words) - 20:22, 10 January 2024
...y|coerciveness inequality]]. Coercive boundary value problems for elliptic equations are sometimes called elliptic boundary value problems [[#References|[4]]].
\frac \partial {\partial x _ 1}
8 KB (1,200 words) - 09:02, 1 February 2020
...imes in the theory of [[Partial differential equation|partial differential equations]].
...domain. Since holomorphic functions are solutions of the [[Cauchy-Riemann equations]], this concept is indeed a generalization of the one above.
2 KB (344 words) - 08:39, 15 January 2014
The partial differential equation that can be written in complex form as follows:
...valign="top"> L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964)</TD></TR></table>
2 KB (311 words) - 10:59, 29 May 2020
...al{A} ( \Omega ) : \Omega \text { open } \}$ provides a [[Sheaf|sheaf]] of differential algebras on ${\bf R} ^ { n }$. Convolution with a sequence of mollifiers $(
...lution equations. In the latter case the spatial derivative $( \partial / \partial x _ { k } ) u ( x )$ may be replaced by the difference operator
2 KB (342 words) - 16:45, 1 July 2020
The representation of solutions of differential equations by algebraic formulas, using an a priori given supply of functions and pre-
...The [[Bernoulli equation|Bernoulli equation]] is an example of an ordinary differential equation that is integrable by quadrature. J. Liouville (cf. [[#References|
4 KB (607 words) - 15:57, 14 February 2020
''system in involution, involutive system of partial differential equations''
A system of first-order partial differential equations
6 KB (940 words) - 19:10, 11 January 2024
A partial differential equation
\sum_{\abs{\alpha} = m} \frac{\partial F(x,Du)}{\partial q_\alpha} \xi^\alpha \geq 0
2 KB (296 words) - 22:17, 2 May 2012
...ytic function]]; [[Elliptic partial differential equation|Elliptic partial differential equation]]). To apply the transform, complex (independent) variables $z=x+i
...partial x^2}+\frac{\partial^2}{\partial y^2}=4\frac{\partial^2}{\partial z\partial z^*}$$
4 KB (623 words) - 15:18, 14 February 2020
...$, $u$, $p=(p_1,\dots,p_n)$ is a first integral of the first-order partial differential equation
$$u=u(x_1,\dots,x_n),\quad p_i=\frac{\partial u}{\partial x_i},\quad1\leq i\leq n,$$
2 KB (314 words) - 06:56, 23 March 2024
...lution of the other system (equation), and conversely, where both systems (equations) are considered in one and the same domain.
The most thoroughly studied are the algebraic equations (cf. [[Algebraic equation|Algebraic equation]]). Solving them was a princip
4 KB (692 words) - 13:38, 31 July 2014
...n)$, where the function $u=\phi(x_1,\dots,x_n)$ is a solution of a partial differential equation. For example, consider the linear homogeneous first-order equation
$$X_1\frac{\partial u}{\partial x_1}+\dotsb+X_n\frac{\partial u}{\partial x_n}=0.\label{*}\tag{*}$$
2 KB (314 words) - 17:39, 14 February 2020
...ntegral equations and differential equations are also integro-differential equations.
==Linear integro-differential equations.==
5 KB (685 words) - 19:38, 12 January 2024
...partial^2u}{\partial x_i\partial x_k}+\sum_{i=1}^nb_i(x)\frac{\partial u}{\partial x_i}+c(x)u=0\tag{*}$$
$$G(x,y)=o(e^{-a|x-y|}),\quad\frac{\partial G}{\partial x_i}=o(e^{-a|x-y|})$$
1 KB (190 words) - 08:35, 31 October 2014
..., pressure, and density. Due to thermal expansion the convection/diffusion equations describing the reaction dynamics are intrinsically coupled with the fluid d
...inearity due to the reaction kinetics. The systems of partial differential equations describing combustion can be studied using numerical or analytical techniqu
3 KB (400 words) - 17:25, 7 February 2011
...= 22 : ~/encyclopedia/old_files/data/D032/D.0302040 Differential equation, partial, with singular coefficients
A partial differential equation the coefficients and/or the free term of which have discontinuitie
5 KB (714 words) - 17:33, 5 June 2020
...ence region of the difference equation covers the dependence region of the differential equation, in the sense that
...he convergence and stability of explicit difference schemes for hyperbolic equations cf. [[#References|[a1]]]–[[#References|[a5]]]. Reference [[#References|[a
2 KB (345 words) - 07:04, 27 May 2023
...stem in the domain $U$ is described by the system of ordinary differential equations
$$\frac{dx^i}{dt}=\pm\sum_jg^{ij}\frac{\partial f}{\partial x^j},\quad i=1,\dots,n,$$
2 KB (250 words) - 10:12, 24 August 2014
$#C+1 = 42 : ~/encyclopedia/old_files/data/D031/D.0301900 Differential equation, abstract
...a differential equation with operator coefficients. The classical abstract differential equation which is most frequently encountered is the equation
6 KB (905 words) - 17:33, 5 June 2020