Plate, steady GF, polynomial and series form.

For the 1D plate, the steady GF satisfies:

$${\frac{d^{2}G}{dx^{2}}} \delta \prime$$ = (4)

$${\frac{dG}{dn_{i}}}$$ = 0;  i = 1 or 2 (5)

There are two alternative forms of the 1D GF, polynomial and series.


Polynomial Form, 1-D Plate
The GF for the 1-D plate are given in Table 1 below. Note that these functions are piecewise linear with a jump in slope at x = x^$\prime$ with the exception of special case X22 which is discussed later.

Table 1. Steady 1D Green's Function for cases XIJ.
Note B₁ = h₁L/k;  B₂ = h₂L/k.

Case	Range	1D steady GF; units are meters.
X11	x > x$\prime$	x$\prime$(1 - x/L)
	x < x$\prime$	x(1 - x$\prime$/L)
X12	x > x$\prime$	x$\prime$
	x < x$\prime$	x
X13	x > x$\prime$	x$\prime$[1 - B2(x/L)/(1 + B2)]
	x < x$\prime$	x[1 - B2(x$\prime$/L)/(1 + B2)]
X21	x > x$\prime$	L - x
	x < x$\prime$	L - x$\prime$
X22 *	x > x$\prime$	( x$\prime$2 + (x)2)/(2L) - x + L/3
	x < x$\prime$	( x2 + (x$\prime$2)2)/(2L) - x$\prime$ + L/3
X23	x > x$\prime$	L(1 + 1/B2 - x/L)
	x < x$\prime$	L(1 + 1/B2 - x$\prime$/L)
X31	x > x$\prime$	(B1x$\prime$ - B1x$\prime$x/L + L - x)/(1 + B1)
	x < x$\prime$	(B1x - B1xx$\prime$/L + L - x$\prime$)/(1 + B1)
X32	x > x$\prime$	L(1/B1 + x$\prime$/L)
	x < x$\prime$	L(1/B1 + x/L)
X33	x > x$\prime$	$\left(\vphantom{ B_{1}B_{2}x^{\prime }+B_{1}x^{\prime }-B_{1}B_{2}x^{\prime }x/L-B_{2}x+B_{2}L+L}\right.$B1B2x$\prime$ + B1x$\prime$ - B1B2x$\prime$x/L - B2x + B2L + L$\left.\vphantom{ B_{1}B_{2}x^{\prime }+B_{1}x^{\prime }-B_{1}B_{2}x^{\prime }x/L-B_{2}x+B_{2}L+L}\right)$
		÷ (B1B2 + B1 + B2)
	x < x$\prime$	$\left(\vphantom{ B_{1}B_{2}x+B_{1}x-B_{1}B_{2}x^{\prime }x/L-B_{2}x^{\prime } +B_{2}L+L}\right.$B1B2x + B1x - B1B2x$\prime$x/L - B2x$\prime$ + B2L + L$\left.\vphantom{ B_{1}B_{2}x+B_{1}x-B_{1}B_{2}x^{\prime }x/L-B_{2}x^{\prime } +B_{2}L+L}\right)$
		÷ (B1B2 + B1 + B2)

^* Special temperature solution needed with this pseudo GF.

Series Form, 1-D Plate.
The GF in the 1-D plate, cases XIJ (I and J = 1,2,3) may be expressed in series form as follows:

$$\prime \sum_{n=1}^{\infty } {\frac{1}{\lambda _{n}^{2}}} {\frac{X_{n}(x)X_{n}(x^{\prime })}{N_{x}(\lambda_n)}}$$ (6)

Here N_x($\lambda_{n}^{}$) denotes the norm of the n^th eigenfunction. (Strictly speaking, these are the squares of the norms of the eigenfunctions.) Eigenfunctions X_n(x) satisfy the following differential equation:

$$\lambda_{n}^{2}$$ (7)

Table 2 contains eigenfunctions X_n(x) and norms and Table 3 contains the associated eigenconditions (and eigenvalues for simple cases). The 1D series form is important because it arises naturally as part of steady 2D and 3D temperature solutions. Whenever it appears, if you can recognize the 1D series form and replace it with the polynomial form, this will improve the numerical convergence.

Table 2. Eigenfunctions and inverse norm.
Index n = 1, 2,... for all cases except X22 with n = 0, 1, 2,... .

Case

X_n(x)

N_x^-1

X11

sin($\lambda_{n}^{}$x)

2/L

X12

sin($\lambda_{n}^{}$x)

2/L

X13

sin($\lambda_{n}^{}$x)

2$\phi_{2n}^{}$/L

X21

cos($\lambda_{n}^{}$x)

2/L

X22

cos($\lambda_{n}^{}$x);	$\lambda_{n}^{}$ $\neq$ 0
1;	$\lambda_{n}^{}$ = 0

2/L	for $\lambda_{n}^{}$ $\neq$ 0
1/L	for $\lambda_{n}^{}$ = 0

X23

cos($\lambda_{n}^{}$x)

2$\phi_{2n}^{}$/L

X31

sin($\lambda_{n}^{}$(L - x))

2$\phi_{1n}^{}$/L

X32

cos($\lambda_{n}^{}$(L - x))

2$\phi_{1n}^{}$/L

X33

$\lambda_{n}^{}$Lcos($\lambda_{n}^{}$x) + (h₁L/k)sin($\lambda_{n}^{}$x)

2$\Phi_{n}^{}$/L

note:

$\phi_{in}^{}$ = $\left[\vphantom{ (\lambda _{n}L)^{2}+(h_{i}L/k)^{2}}\right.$($\lambda_{n}^{}$L)² + (h_iL/k)²$\left.\vphantom{ (\lambda _{n}L)^{2}+(h_{i}L/k)^{2}}\right]$ ÷ $\left[\vphantom{ (\lambda _{n}L)^{2}+(h_{i}L/k)^{2}+h_{i}L/k}\right.$($\lambda_{n}^{}$L)² + (h_iL/k)² + h_iL/k$\left.\vphantom{ (\lambda _{n}L)^{2}+(h_{i}L/k)^{2}+h_{i}L/k}\right]$

$\Phi_{n}^{}$ = $\phi_{2n}^{}$ ÷ $\left[\vphantom{ (\lambda _{n}L)^{2}+(h_{1}L/k)^{2}+(h_{1}L/k)\phi _{2n}}\right.$($\lambda_{n}^{}$L)² + (h₁L/k)² + (h₁L/k)$\phi_{2n}^{}$ $\left.\vphantom{ (\lambda _{n}L)^{2}+(h_{1}L/k)^{2}+(h_{1}L/k)\phi _{2n}}\right]$

Table 3. Eigenconditions and eigenvalues for X_n(x).
Index n = 1, 2,... for all cases except X22 with n = 0, 1, 2,....

Case	Eigencondition	Eigenvalues
X11	sin($\lambda_{n}^{}$L) = 0	${\frac{n\pi }{L}}$, n = 1, 2,...
X12	cos($\lambda_{n}^{}$L) = 0	${\frac{(2n-1)\pi }{2L}}$, n = 1, 2,...
X13	$\lambda_{n}^{}$Lcot($\lambda_{n}^{}$L) = - h2L/k
X21	cos($\lambda_{n}^{}$L) = 0	${\frac{(2n-1)\pi }{2L}}$, n = 1, 2,...
X22	sin($\lambda_{n}^{}$L) = 0	${\frac{n\pi }{L}}$, n = 0, 1, 2,...
X23	$\lambda_{n}^{}$Ltan($\lambda_{n}^{}$L) = h2L/k
X31	$\lambda_{n}^{}$Lcot($\lambda_{n}^{}$L) = - h1L/k
X32	$\lambda_{n}^{}$Ltan($\lambda_{n}^{}$L) = h1L/k
X33	tan($\lambda_{n}^{}$L) = [$\lambda_{n}^{}$(h1 + h2)/k]/[$\lambda_{n}^{2}$ - h1h2k-2]