Laplaceova jednačina

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

${\mathrm{\nabla }}^2\phi =0$

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

Definicija

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

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

U cilindričnom koordinatnom sistemu je:

$\mathrm{\Delta }f=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial f}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}f}{\partial {\varphi }^2}+\frac{{\partial }^{2}f}{\partial z^2}=0$

U sfernom koordinatnom sistemu je:

$\mathrm{\Delta }f=\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial f}{\partial \theta })+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}f}{\partial {\phi }^2}=0.$

U zakrivljenom koordinatnom sistemu je:

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

ilir

$\mathrm{\Delta }f=\frac{1}{\sqrt{|g|}}\frac{\partial }{\partial {\xi }^i}\left(\sqrt{|g|}{g}^{ij}\frac{\partial f}{\partial {\xi }^j}\right)=0,(g=\mathrm{d}\mathrm{e}\mathrm{t}\{g_{ij}\}).$

Dvodimenzionalni sistem

U polarnom koordinatnom dvodimenzionalnom sistemu je oblika:

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

U dvodimenzionalnom kartezijevom sistemu je:

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

Grinova funkcija

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

$\underset{V}{\int }(\varphi {\mathrm{\nabla }}^2\psi -\psi {\mathrm{\nabla }}^2\varphi )dV=\underset{S}{\int }(\varphi \mathrm{\nabla }\psi -\psi \mathrm{\nabla }\varphi )\cdot d\hat{\sigma }.$

Definicija Grinove funkcije je:

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

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

$\begin{array}{cc}& \underset{V}{\int }[\varphi (x^{\prime })\delta (x-x^{\prime })-G(x,x^{\prime }){\mathrm{\nabla }}^2\varphi (x^{\prime })]d^3{x}^{\prime }\\ & =\underset{S}{\int }[\varphi (x^{\prime }){\mathrm{\nabla }}^{\prime }G(x,x^{\prime })-G(x,x^{\prime }){\mathrm{\nabla }}^{\prime }\varphi (x^{\prime })]\cdot d{\hat{\sigma }}^{\prime }.\end{array}$

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

$\underset{V}{\int }\varphi (x^{\prime })\delta (x-x^{\prime })d^3{x}^{\prime }=\varphi (x)$

pa se jednačina svodi na:

$\varphi (x)=\underset{V}{\int }G(x,x^{\prime })\rho (x^{\prime })d^3{x}^{\prime }+\underset{S}{\int }[\varphi (x^{\prime }){\mathrm{\nabla }}^{\prime }G(x,x^{\prime })-G(x,x^{\prime }){\mathrm{\nabla }}^{\prime }\varphi (x^{\prime })]\cdot d{\hat{\sigma }}^{\prime }.$

Kada nema rubnih uslova Grinova funkcija je:

$G(x,x^{\prime })=\frac{1}{|x-x^{\prime }|}.$

Literatura

• 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