Fungsi Green
Menemukan fungsi Green > Tabel fungsi Green

Tabel berikut memberikan gambaran umum tentang fungsi Green dari operator diferensial yang sering muncul, di mana  \textstyle r={\sqrt {x^{2}+y^{2}+z^{2}}},  \textstyle \rho ={\sqrt {x^{2}+y^{2}}},  \textstyle \Theta (t) adalah fungsi langkah Heaviside,  \textstyle J_{\nu }(z) adalah fungsi Bessel,  \textstyle I_{\nu }(z) adalah fungsi Bessel yang dimodifikasi dari jenis pertama, dan  \textstyle K_{\nu }(z) adalah  Fungsi Bessel yang dimodifikasi dari jenis kedua. Di mana waktu (  t ) muncul di kolom pertama, fungsi Green lanjutan (kausal) terdaftar.
| Operator diferensial  L | Fungsi Green  G | Contoh aplikasi |
| --- | --- | --- |
|  \partial _{t}^{n+1} |  {\frac {t^{n}}{n!}}\Theta (t) |  |
|  \partial _{t}+\gamma  |  \Theta (t)\mathrm {e} ^{-\gamma t} |  |
|  \left(\partial _{t}+\gamma \right)^{2} |  \Theta (t)t\mathrm {e} ^{-\gamma t} |  |
|  \partial _{t}^{2}+2\gamma \partial _{t}+\omega _{0}^{2} |  \Theta (t)\mathrm {e} ^{-\gamma t}~{\frac {\sin(\omega t)}{\omega }}    with     \omega ={\sqrt {\omega _{0}^{2}-\gamma ^{2}}} |  Osilator harmonik teredam 1D |
|  Operator Laplace 2D  \Delta _{\text{2D}}=\partial _{x}^{2}+\partial _{y}^{2} |  {\frac {1}{2\pi }}\ln \rho     with     \rho ={\sqrt {x^{2}+y^{2}}} | Persamaan 2D Poisson |
|  Operator Laplace 3D  \Delta _{\text{3D}}=\partial _{x}^{2}+\partial _{y}^{2}+\partial _{z}^{2} |  {\frac {-1}{4\pi r}}    with     r={\sqrt {x^{2}+y^{2}+z^{2}}} | Persamaan Poisson |
|  Operator Helmholtz  \Delta _{\text{3D}}+k^{2} |  {\frac {-\mathrm {e} ^{-ikr}}{4\pi r}}=i{\sqrt {\frac {k}{32\pi r}}} H_{1/2}^{(2)}(kr) =i{\frac {k}{4\pi }}\, h_{0}^{(2)}(kr) | 3D stasioner persamaan Schrödinger untuk partikel bebas |
|  \Delta -k^{2} in  n dimensions |  -(2\pi )^{-n/2}\left({\frac {k}{r}}\right)^{n/2-1}K_{n/2-1}(kr) | Potensi Yukawa,  Penyebar Feynman |
|  \partial _{t}^{2}-c^{2}\partial _{x}^{2} |  {\frac {1}{2c}}\Theta (t-|x/c|) | 1D wave equation |
|  \partial _{t}^{2}-c^{2}\,\Delta _{\text{2D}} |  {\frac {1}{2\pi c{\sqrt {c^{2}t^{2}-\rho ^{2}}}}}\Theta (t-\rho /c) | 2D persamaan gelombang |
| D'Alembert operator  \square ={\frac {1}{c^{2}}}\partial _{t}^{2}-\Delta _{\text{3D}} |  {\frac {\delta (t-{\frac {r}{c}})}{4\pi r}} | 3D persamaan gelombang |
|  \partial _{t}-k\partial _{x}^{2} |  \Theta (t)\left({\frac {1}{4\pi kt}}\right)^{1/2}\mathrm {e} ^{-x^{2}/4kt} | 1D difusi |
|  \partial _{t}-k\,\Delta _{\text{2D}} |  \Theta (t)\left({\frac {1}{4\pi kt}}\right)\mathrm {e} ^{-\rho ^{2}/4kt} | 2D difusi |
|  \partial _{t}-k\,\Delta _{\text{3D}} |  \Theta (t)\left({\frac {1}{4\pi kt}}\right)^{3/2}\mathrm {e} ^{-r^{2}/4kt} | 3D difusi |
|  {\frac {1}{c^{2}}}\partial _{t}^{2}-\partial _{x}^{2}+\mu ^{2} |  {\frac {1}{2}}\left[\left(1-\sin {\mu ct}\right)(\delta (ct-x)+\delta (ct+x))+\mu \Theta (ct-|x|)J_{0}(\mu u)\right],\,u={\sqrt {c^{2}t^{2}-x^{2}}} | 1D Persamaan Klein–Gordon |
|  {\frac {1}{c^{2}}}\partial _{t}^{2}-\Delta _{\text{2D}}+\mu ^{2} |  {\frac {1}{4\pi }}\left[(1+\cos(\mu ct)){\frac {\delta (ct-\rho )}{\rho }}+\mu ^{2}\Theta (ct-\rho )\operatorname {sinc} (\mu u)\right],\,u={\sqrt {c^{2}t^{2}-\rho ^{2}}} | 2D Persamaan Klein–Gordon |
|  \square +\mu ^{2} |  {\frac {1}{4\pi }}\left[{\frac {\delta \left(t-{\frac {r}{c}}\right)}{r}}+\mu c\Theta (ct-r){\frac {J_{1}\left(\mu u\right)}{u}}\right],\,u={\sqrt {c^{2}t^{2}-r^{2}}} | 3D Persamaan Klein–Gordon |
|  \partial _{t}^{2}+2\gamma \partial _{t}-c^{2}\partial _{x}^{2} |  {\frac {1}{2}}e^{-\gamma t}\left[\delta (ct-x)+\delta (ct+x)+\Theta (ct-|x|)\left({\frac {\gamma }{c}}I_{0}\left({\frac {\gamma u}{c}}\right)+{\frac {\gamma t}{u}}I_{1}\left({\frac {\gamma u}{c}}\right)\right)\right],\,u={\sqrt {c^{2}t^{2}-x^{2}}} | telegrapher's equation |
|  \partial _{t}^{2}+2\gamma \partial _{t}-c^{2}\,\Delta _{\text{2D}} |  {\frac {e^{-\gamma t}}{4\pi }}\left[(1+e^{-\gamma t}+3\gamma t){\frac {\delta (ct-\rho )}{\rho }}+\Theta (ct-\rho )\left({\frac {\gamma \sinh \left({\frac {\gamma u}{c}}\right)}{cu}}+{\frac {3\gamma t\cosh \left({\frac {\gamma u}{c}}\right)}{u^{2}}}-{\frac {3ct\sinh \left({\frac {\gamma u}{c}}\right)}{u^{3}}}\right)\right],\,u={\sqrt {c^{2}t^{2}-\rho ^{2}}} | 2D konduksi panas relativistik |
|  \partial _{t}^{2}+2\gamma \partial _{t}-c^{2}\,\Delta _{\text{3D}} |  {\frac {e^{-\gamma t}}{20\pi }}\left[\left(8-3e^{-\gamma t}+2\gamma t+4\gamma ^{2}t^{2}\right){\frac {\delta (ct-r)}{r^{2}}}+{\frac {\gamma ^{2}}{c}}\Theta (ct-r)\left({\frac {1}{cu}}I_{1}\left({\frac {\gamma u}{c}}\right)+{\frac {4t}{u^{2}}}I_{2}\left({\frac {\gamma u}{c}}\right)\right)\right],\,u={\sqrt {c^{2}t^{2}-r^{2}}} | 3D konduksi panas relativistik |