Green's function number

As an example, number X11 denotes the Green's function that satisfies the heat equation in the domain (0 < x < L) for boundary conditions of type 1 (Dirichlet) at both boundaries x = 0 and x = L. Here X denotes the Cartesian coordinate and 11 denotes the type 1 boundary condition at both sides of the body. The boundary value problem for the X11 Green's function is given by

${\displaystyle {\begin{aligned}{\dfrac {\partial ^{2}G}{\partial x^{2}}}+{\dfrac {1}{\alpha }}\delta (t-\tau )\delta (x-x')={\dfrac {1}{\alpha }}{\dfrac {\partial G}{\partial t}};&\;\;0<x<L;\;\;\;t>0\\G|_{x=0}=0;\;\;G|_{x=L}=0;\;\;G|_{t<\tau }=0.&\end{aligned}}}$

Here ${\displaystyle \alpha }$ is the thermal diffusivity (m²/s) and ${\displaystyle \delta }$ is the Dirac delta function. This GF is developed elsewhere. [13] [14]

As another Cartesian example, number X20 denotes the Green's function in the semi-infinite body (${\displaystyle 0<x<\infty }$) with a Neumann (type 2) boundary at x = 0. Here X denotes the Cartesian coordinate, 2 denotes the type 2 boundary condition at x = 0 and 0 denotes the zeroth type boundary condition (boundedness) at ${\displaystyle x=\infty }$. The boundary value problem for the X20 Green's function is given by

${\displaystyle {\begin{aligned}{\dfrac {\partial ^{2}G}{\partial x^{2}}}+{\dfrac {1}{\alpha }}\delta (t-\tau )\delta (x-x')={\dfrac {1}{\alpha }}{\dfrac {\partial G}{\partial t}};&\;\;0<x<\infty$ ;\;\;\;t>0\\{\dfrac {\partial G}{\partial n}}|_{x=0}=0;\;\;G|_{x\rightarrow \infty }{\mbox{ is bounded}};\;\;G|_{t<\tau }=0.&\end{aligned}}}

This GF is published elsewhere. [15] [16]

As a two-dimensional example, number X10Y20 denotes the Green's function in the quarter-infinite body (${\displaystyle 0<x<\infty }$, ${\displaystyle 0<y<\infty }$) with a Dirichlet (type 1) boundary at x = 0 and a Neumann (type 2) boundary at y = 0. The boundary value problem for the X10Y20 Green's function is given by

${\displaystyle {\begin{aligned}{\dfrac {\partial ^{2}G}{\partial x^{2}}}+{\dfrac {\partial ^{2}G}{\partial y^{2}}}+{\dfrac {1}{\alpha }}\delta (t-\tau )\delta (x-x')\delta (y-y')={\dfrac {1}{\alpha }}{\dfrac {\partial G}{\partial t}};&\;\;0<x<\infty$ ;\;\;0<y<\infty ;\;\;\;t>0\\G|_{x=0}=0;\;\;{\dfrac {\partial G}{\partial n}}|_{y=0}=0;\;\;&\\G|_{x\rightarrow \infty }{\mbox{ is bounded}};\;\;G|_{y\rightarrow \infty }{\mbox{ is bounded}};\;\;G|_{t<\tau }=0.&\end{aligned}}}

Applications of related half-space and quarter-space GF are available. [17]

As an example in the cylindrical coordinate system, number R03 denotes the Green's function that satisfies the heat equation in the solid cylinder (0 < r < a) with a boundary condition of type 3 (Robin) at r = a. Here letter R denotes the cylindrical coordinate system, number 0 denotes the zeroth boundary condition (boundedness) at the center of the cylinder (r = 0), and number 3 denotes the type 3 (Robin) boundary condition at r = a. The boundary value problem for R03 Green's function is given by

${\displaystyle {\dfrac {1}{r}}{\dfrac {\partial }{\partial r}}\left(r{\dfrac {\partial G}{\partial r}}\right)+{\dfrac {1}{\alpha }}\delta (t-\tau ){\dfrac {\delta (r-r')}{2\pi r'}}={\dfrac {1}{\alpha }}{\dfrac {\partial G}{\partial t}};\;\;0<r<a;\;\;\;t>0}$

${\displaystyle G|_{r=0}{\mbox{ is bounded}};\;\;k{\dfrac {\partial G}{\partial n}}|_{r=a}+hG|_{r=a}=0;\;\;G|_{t<\tau }=0.}$

Here ${\displaystyle k}$ is thermal conductivity (W/(m K)) and ${\displaystyle h}$ is the heat transfer coefficient (W/(m² K)). See [18] [19] for this GF.

As another example, number R10 denotes the Green's function in a large body containing a cylindrical void (a < r < ${\displaystyle \infty }$) with a type 1 (Dirichlet) boundary condition at r = a. Again letter R denotes the cylindrical coordinate system, number 1 denotes the type 1 boundary at r = a, and number 0 denotes the type zero boundary (boundedness) at large values of r. The boundary value problem for the R10 Green's function is given by

${\displaystyle {\begin{aligned}{\dfrac {1}{r}}{\dfrac {\partial }{\partial r}}\left(r{\dfrac {\partial G}{\partial r}}\right)+{\dfrac {1}{\alpha }}\delta (t-\tau ){\dfrac {\delta (r-r')}{2\pi r'}}={\dfrac {1}{\alpha }}{\dfrac {\partial G}{\partial t}};&\;\;a<r<\infty$ ;\;\;\;t>0\\G|_{r=a}=0;\;\;G|_{r\rightarrow \infty }{\mbox{ is bounded}};\;\;G|_{t<\tau }=0.&\end{aligned}}}

This GF is available elsewhere. [20] [21]

As a two dimensional example, number R01${\displaystyle \phi }$00 denotes the Green's function in a solid cylinder with angular dependence, with a type 1 (Dirichlet) boundary condition at r = a. Here letter ${\displaystyle \phi }$ denotes the angular coordinate, and numbers 00 denote the type zero boundaries for angle; here no physical boundary takes the form of the periodic boundary condition. The boundary value problem for the R01${\displaystyle \phi }$00 Green's function is given by

${\displaystyle {\begin{aligned}{\dfrac {1}{r}}{\dfrac {\partial }{\partial r}}\left(r{\dfrac {\partial G}{\partial r}}\right)+{\dfrac {1}{r^{2}}}{\dfrac {\partial ^{2}G}{\partial \phi ^{2}}}+{\dfrac {1}{\alpha }}\delta (t-\tau ){\dfrac {\delta (r-r')}{2\pi r'}}\delta (\phi -\phi ')={\dfrac {1}{\alpha }}{\dfrac {\partial G}{\partial t}};&\;\;0<r<a;\;\;\;0<\phi <2\pi$ ;\;\;\;t>0\\G|_{r=0}{\mbox{ is bounded,}}\;\;G|_{r=a}=0;&\\G|_{\phi =0}=G|_{\phi =2\pi };{\dfrac {\partial G}{\partial \phi }}|_{\phi =0}={\dfrac {\partial G}{\partial \phi }}|_{\phi =2\pi };\;\;G|_{t<\tau }=0.&\end{aligned}}}

Both a transient [22] and steady form [23] of this GF are available.

As an example in the spherical coordinate system, number RS02 denotes the Green's function for a solid sphere (0 < r < b ) with a type 2 (Neumann) boundary condition at r = b. Here letters RS denote the radial-spherical coordinate system, number 0 denotes the zeroth boundary condition (boundedness) at r=0, and number 2 denotes the type 2 boundary at r = b. The boundary value problem for the RS02 Green's function is given by

${\displaystyle {\dfrac {1}{r^{2}}}{\dfrac {\partial }{\partial r}}\left(r^{2}{\dfrac {\partial G}{\partial r}}\right)+{\dfrac {1}{\alpha }}\delta (t-\tau ){\dfrac {\delta (r-r')}{4\pi r^{2}}}={\dfrac {1}{\alpha }}{\dfrac {\partial G}{\partial t}};\;\;0<r<b;\;\;\;t>0}$

${\displaystyle G|_{r=0}{\mbox{ is bounded}};\;\;{\dfrac {\partial G}{\partial n}}|_{r=b}=0;\;\;G|_{t<\tau }=0.}$

This GF is available elsewhere. [24]