Laplaceova jednačina

Laplasova jednačina' je eliptička parcijalna diferencijalna jednačina drugoga reda oblika:

${\displaystyle \qquad \nabla ^{2}\varphi =0}$

Rešenja Laplasove jednačine su harmoničke funkcije. Laplasova jednačina je značajna u matematici, elektromagnetizmu, astronomiji i dinamici fluida.

U tri demenzije Laplasiva jednačina može da se prikaže u različitim koordinatnim sistemima. U kartezijevom koordinatnom sistemu je oblika:

${\displaystyle \Delta f={\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}=0.}$

U cilindričnom koordinatnom sistemu je:

${\displaystyle \Delta f={1 \over r}{\partial \over \partial r}\left(r{\partial f \over \partial r}\right)+{1 \over r^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}=0}$

U sfernom koordinatnom sistemu je:

${\displaystyle \Delta f={1 \over \rho ^{2}}{\partial \over \partial \rho }\!\left(\rho ^{2}{\partial f \over \partial \rho }\right)\!+\!{1 \over \rho ^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over \rho ^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}=0.}$

U zakrivljenom koordinatnom sistemu je:

${\displaystyle \Delta f={\partial \over \partial \xi ^{i}}\!\left({\partial f \over \partial \xi ^{k}}g^{ki}\right)\!+\!{\partial f \over \partial \xi ^{j}}g^{jm}\Gamma _{mn}^{n}=0,}$

ilir

${\displaystyle \Delta f={1 \over {\sqrt {|g|}}}{\partial \over \partial \xi ^{i}}\!\left({\sqrt {|g|}}g^{ij}{\partial f \over \partial \xi ^{j}}\right)=0,\quad (g=\mathrm {det} \{g_{ij}\}).}$

U polarnom koordinatnom dvodimenzionalnom sistemu je oblika:

${\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial u}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}u}{\partial \phi ^{2}}}=0}$

U dvodimenzionalnom kartezijevom sistemu je:

${\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0}$

Laplasova jednačina se često rešava uz pomoć Grinove funkcije i Grinova teorema:

${\displaystyle \int _{V}(\phi \nabla ^{2}\psi -\psi \nabla ^{2}\phi )dV=\int _{S}(\phi \nabla \psi -\psi \nabla \phi )\cdot d{\hat {\sigma }}.}$

Definicija Grinove funkcije je:

${\displaystyle \nabla ^{2}G(x,x')=\delta (x-x').}$

Uvrstimo u Grinov teorem ${\displaystyle \psi =G}$ pa dobijamo:

${\displaystyle {\begin{aligned}&{}\quad \int _{V}\left[\phi (x')\delta (x-x')-G(x,x')\nabla ^{2}\phi (x')\right]\ d^{3}x'\\[6pt]&=\int _{S}\left[\phi (x')\nabla 'G(x,x')-G(x,x')\nabla '\phi (x')\right]\cdot d{\hat {\sigma }}'.\end{aligned}}}$

Sada možemo da rešimo Laplasovu jednačinu ${\displaystyle \nabla ^{2}\phi (x)=0}$ u slučaju Nojmanovih ili Dirihleovih rubnih uslova. Uzimajući u obzir:

${\displaystyle \int \limits _{V}{\phi (x')\delta (x-x')\ d^{3}x'}=\phi (x)}$

pa se jednačina svodi na:

${\displaystyle \phi (x)=\int _{V}G(x,x')\rho (x')\ d^{3}x'+\int _{S}\left[\phi (x')\nabla 'G(x,x')-G(x,x')\nabla '\phi (x')\right]\cdot d{\hat {\sigma }}'.}$

Kada nema rubnih uslova Grinova funkcija je:

${\displaystyle G(x,x')={\dfrac {1}{|x-x'|}}.}$

• Sommerfeld A, Partial Differential Equations in Physics, New York: Academic Press (1949)
• Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.
• Morse PM, Feshbach H . Methods of Theoretical Physics, Part I. New York:. Šablon:Page1
• Laplasova jednačina