Laplasova jednačina' je eliptička parcijalna diferencijalna jednačina drugoga reda oblika:
![{\displaystyle \qquad \nabla ^{2}\varphi =0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EytaNntKPoNs0o2vAoNoPajmPnjdEa2i0zjaQzgdEajrDajvAz2wO)
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.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EaAwPyta1nDsNaqrBatJAytG4yqnEztoQzDmNzAwNntG5a2iNnjmP)
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}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FyjKNyjFCzAa3yqrFyja0nDs5njnDzqoQoqa0athDaAvAoNG2aNm5)
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.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NzDw1oDzAotvCnjJEzDnBotiPoDw2otm2aNnCzgo1yjnEaNmOzjC2)
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,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DotiQnqvBotnDnjhFotFAzjs5zqwOoDKQoDlDoDa1zgw1aghAoAa2)
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}\}).}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EothEaNoQnDK2nqdEoNo3oDCQzAaOyje4ntBBoNGQyjzDajaPntFD)
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}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Fz2wNa2zDnjvBaDGNzNeNnjw1zqhFngo2aAs2ajwPoqrBo2s2zDrB)
U dvodimenzionalnom kartezijevom sistemu je:
![{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Nots1zjo3otCNatG5aje0ajwOntw3ztaPaNo3oNC3nAi0ags3zjBE)
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 }}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BaNvDz2i5age4aDiPntFDz2o2ztdAago5otm1zjCQajC2yghBzqzA)
Definicija Grinove funkcije je:
![{\displaystyle \nabla ^{2}G(x,x')=\delta (x-x').}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EoDK1nto1oqs1aNlCotoOothFyjJEzgrCnji0aAdBztBCytnEzNFC)
Uvrstimo u Grinov teorem
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}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zta5yjw5zAoNote3ajJFagoQytdCaArBaAiNzgs2yji5ago2ngdF)
Sada možemo da rešimo Laplasovu jednačinu
u slučaju Nojmanovih ili Dirihleovih rubnih uslova. Uzimajući u obzir:
![{\displaystyle \int \limits _{V}{\phi (x')\delta (x-x')\ d^{3}x'}=\phi (x)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OnDvDzje3zts5o2zBaNK1yta1aDCQo2a5nts5o2oNoNs1ajCPzjsP)
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 }}'.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CaDrEzqwPnDhFytzEzDG2a2w4zNw3ztw2otnBoDiOajeQote1zqw0)
Kada nema rubnih uslova Grinova funkcija je:
![{\displaystyle G(x,x')={\dfrac {1}{|x-x'|}}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zjCNatrBajG5njhEzgoNaDGNoAzFnqeNoAnBoNC4yqw1ygdBoNzE)
- 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