Ernst denklemi

**Ernst denklemi**, matematik'te doğrusal-olmayan bir kısmi diferansiyel denklem'dir.

## Adı
Ünlü fizikçi Frederick J. Ernst tarafından bulunmuş olduğundan, *"Ernst denklemi"* olarak adlandırılmıştır.

## Ernst denklemi
Sağ tarafında Birinci dereceden kısmî türevler içeren ve doğrusal olmayan terimleri olan  bir denklemdir. Çözümü aranan *u* karmaşık fonksiyonunun gerçel kısmı **R**(*u*), denklemin sol tarafındaki İkinci dereceden kısmî türevlerin çarpımı halinde belirdiğinden, denklemin her iki tarafı da doğrusal-olmayan *(non-linear)* terimler ihtivâ etmektedir. Denklem aşağıdaki şekilde verilmektedir:

:\displaystyle \Re(u)(u_{rr}+u_r/r+u_{zz}) = (u_r)^2+(u_z)^2.

### Kullanım amacı
Einstein alan denklemlerinin noksansız çözümlerini elde etmek için kullanılan doğrusal olmayan bir kısmi türevsel denklemdir.

## Bibliyografya
*
### İlgili yayınlar
#### '' '' mecmuasında

*1971: Frederick J. Ernst, *Exterior-Algebraic Derivation of Einstein Field Equations Employing a Generalized Basis*
*1974: Frederick J. Ernst, *Complex potential formulation of the axially symmetric gravitational field problem*
*1974: Frederick J. Ernst, *Weyl conform tensor for stationary gravitational fields*
*1975: Frederick J. Ernst, *Black holes in a magnetic universe*
*1975: Frederick J. Ernst, *Erratum: Complex potential formulation of the axially symmetric gravitational field problem*
*1975: John E. Economou & Frederick J. Ernst, *Weyl conform tensor of =2 Tomimatsu–Sato spinning mass gravitational field*
*1976: Frederick J. Ernst & Walter J. Wild, *Kerr black holes in a magnetic universe*
*1976: Frederick J. Ernst, *New representation of the Tomimatsu–Sato solution*
*1976: Frederick J. Ernst, *Removal of the nodal singularity of the C-metric*
*1977: Frederick J. Ernst, *A new family of solutions of the Einstein field equations*
*1978: Frederick J. Ernst, *Coping with different languages in the null tetrad formulation of general relativity*
*1978: Frederick J. Ernst & I. Hauser, *Field equations and integrability conditions for special type N twisting gravitational fields*
*1978: Frederick J. Ernst, *Generalized C-metric*
*1978: Isidore Hauser & Frederick J. Ernst, *On the generation of new solutions of the Einstein–Maxwell field equations*
*1979: I. Hauser & Frederick J. Ernst, *SU(2,1) generation of electrovacs from Minkowski space
*1979: (Erratum) Coping with different languages in the null tetrad formulation of general relativity
*1979: (Erratum) Generalized C metric*
*1980: Isidore Hauser & Frederick J. Ernst, *A homogeneous Hilbert problem for the Kinnersley–Chitre transformations of electrovac space-times*
*1980: Isidore Hauser & Frederick J. Ernst, *A homogeneous Hilbert problem for the Kinnersley–Chitre transformations*
*1981: Isidore Hauser & Frederick J. Ernst, *Proof of a Geroch conjecture*
*1982: Dong-sheng Guo & Frederick J. Ernst, *Electrovac generalization of Neugebauer's N = 2 solution of the Einstein vacuum field equations*
*1983: Y. Chen, Dong-sheng Guo & Frederick J. Ernst, *Charged spinning mass field involving rational functions*
*1983: Cornelius Hoenselares & Frederick J. Ernst, *Remarks on the Tomimatsu–Sato metrics*
*1987: Frederick J. Ernst, Alberto Garcia D & Isidore Hauser, *Colliding gravitational plane waves with noncollinear polarization. I*
*1987: Frederick J. Ernst, Alberto Garcia D & Isidore Hauser, *Colliding gravitational plane waves with noncollinear polarization. II*
*1988: Frederick J. Ernst, Alberto Garcia D & Isidore Hauser, *Colliding gravitational plane waves with noncollinear polarization. III*
*1989: Wei Li & Frederick J. Ernst, *A family of electrovac colliding wave solutions of Einstein's equations*
*1989: Isidore Hauser & Frederick J. Ernst, *Initial value problem for colliding gravitational plane waves. I*
*1989: Isidore Hauser & Frederick J. Ernst, *Initial value problem for colliding gravitational plane waves. II*
*1990: Isidore Hauser & Frederick J. Ernst, *Initial value problem for colliding gravitational plane waves. III*
*1990: Cornelius Hoenselares & Frederick J. Ernst, *Matching pp waves to the Kerr metric*
*1991: Wei Li, Isidore Hauser & Frederick J. Ernst, *Colliding gravitational plane waves with noncollinear polarizations*
*1991: Wei Li, Isidore Hauser & Frederick J. Ernst, *Colliding gravitational waves with Killing–Cauchy horizons*
*1991: Wei Li, Isidore Hauser & Frederick J. Ernst, *Colliding wave solutions of the Einstein–Maxwell field* equations
*1991: Isidore Hauser & Frederick J. Ernst, *Initial value problem for colliding gravitational plane waves. IV*
*1991: Wei Li, Isidore Hauser & Frederick J. Ernst, *Nonimpulsive colliding gravitational waves with noncollinear polarizations*
*1993: Frederick J. Ernst & Isidore Hauser, *On [[Metin Gürses|Gürses]]'s symmetries of the [[Einstein alan denklemleri|Einstein equations]]*

## Kaynakça