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Plate, transient 1-D.

X11 Plate, with G = 0 (Dirichlet) at x = 0 and at x = L.
a. Best convergence for (t - $ \tau$) small:
GX11(x, t | x$\scriptstyle \prime$,$\displaystyle \tau$) = [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2  
    x $\displaystyle \sum_{n=-\infty }^{\infty }$$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{% (2nL+x-x^{\prime })^{2}}{4... ...exp \left[ -\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{% (2nL+x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{% (2nL+x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{% (2nL+x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ - exp$\displaystyle \left[\vphantom{ -\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{% (2nL+x-x^{\prime })^{2}}{4\... ...xp \left[ -\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  

b. Best convergence for (t - $ \tau$) large:

GX11(x, t | x$\scriptstyle \prime$,$\displaystyle \tau$) = $\displaystyle {\frac{2}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{m^{2}\pi ^{2}\alpha (t-\tau )}{L^{2}}}\right.$ - $\displaystyle {\frac{m^{2}\pi ^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{m^{2}\pi ^{2}\alpha (t-\tau )}{L^{2}}}\right]$sin(m$\displaystyle \pi$$\displaystyle {\frac{x}{L}}$)sin(m$\displaystyle \pi$$\displaystyle {\frac{x^{\prime }}{L}}$)    

X12 Plate, with G = 0 (Dirichlet) at x = 0 and $ \partial$G/$ \partial$x = 0 (Neumann) at x = L.
a. Best convergence for (t - $ \tau$) small:
GX12(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2  
    x $\displaystyle \sum_{n=-\infty }^{\infty }$(- 1)n$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{% (2nL+x-x^{\prime })^{2}}{4... ...exp \left[ -\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{% (2nL+x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{% (2nL+x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{% (2nL+x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ - exp$\displaystyle \left[\vphantom{ -\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{% (2nL+x-x^{\prime })^{2}}{4\... ...xp \left[ -\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  

b. Best convergence for (t - $ \tau$) large:
GX12(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{2}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right]$sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{x}{L}}$)sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{x^{\prime }}{L}}$)  
where $\displaystyle \beta_{m}^{}$ = (2m - 1)($\displaystyle \pi$/2), for m = 1, 2,....  

X13 Plate with G = 0 (Dirichlet) at x = 0 and k$ \partial$G/$ \partial$x + hG = 0 (convection) at x = L.
a. For $ \alpha$(t - $ \tau$)/L2 $ \leq$ 0.022 use the following approximation:
GX13(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) $\displaystyle \approx$ [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}% \right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}% \right] }\right.$  
    $\displaystyle \left.\vphantom{ -\exp \left[ -\frac{(x+x^{\prime })^{2}}{4\alpha... ... +\exp \left[ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$ -exp$\displaystyle \left[\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ + exp$\displaystyle \left[\vphantom{ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ -\exp \left[ -\frac{(x+x^{\prime })^{2}}{4\alpha... ...+\exp \left[ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  
    - $\displaystyle {\frac{h}{k}}$exp$\displaystyle \left[\vphantom{ \frac{h(2L-x-x^{\prime })}{k}+h^{2}\frac{\alpha (t-\tau )}{k^{2}}}\right.$$\displaystyle {\frac{h(2L-x-x^{\prime })}{k}}$ + h2$\displaystyle {\frac{\alpha (t-\tau )}{k^{2}}}$ $\displaystyle \left.\vphantom{ \frac{h(2L-x-x^{\prime })}{k}+h^{2}\frac{\alpha (t-\tau )}{k^{2}}}\right]$  
    x erfc$\displaystyle \left[\vphantom{ \frac{2L-x-x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}+h\frac{\left[ \alpha (t-\tau )\right] ^{1/2}}{k}}\right.$$\displaystyle {\frac{2L-x-x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}}$ + h$\displaystyle {\frac{\left[ \alpha (t-\tau )\right] ^{1/2}}{k}}$ $\displaystyle \left.\vphantom{ \frac{2L-x-x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}+h\frac{\left[ \alpha (t-\tau )\right] ^{1/2}}{k}}\right]$  

b. For all values, but best for $ \alpha$(t - $ \tau$)/L2 large:
GX13(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{2}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right]$$\displaystyle \left(\vphantom{ \frac{\beta _{m}^{2}+B^{2}}{\beta _{m}^{2}+B^{2}+B}}\right.$$\displaystyle {\frac{\beta _{m}^{2}+B^{2}}{\beta _{m}^{2}+B^{2}+B}}$ $\displaystyle \left.\vphantom{ \frac{\beta _{m}^{2}+B^{2}}{\beta _{m}^{2}+B^{2}+B}}\right)$  
    x sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{x}{L}}$)sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{x^{\prime }}{L}}$)  
    with $\displaystyle \beta_{m}^{}$cot$\displaystyle \beta_{m}^{}$ = - B and where B = hL/k.  

X14 Plate with G = 0 (Dirichlet) at x = 0 and k$ \partial$G/$ \partial$x + ($ \rho$cb)2$ \partial$G/$ \partial$t = 0 (thin film) at x = L.
GX14(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) =   $\displaystyle {\frac{2}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right]$$\displaystyle \left(\vphantom{ \frac{C_{2}^{2}\beta _{m}^{2}+1}{C_{2}^{2}\beta _{m}^{2}+C_{2}+1}}\right.$$\displaystyle {\frac{C_{2}^{2}\beta _{m}^{2}+1}{C_{2}^{2}\beta _{m}^{2}+C_{2}+1}}$ $\displaystyle \left.\vphantom{ \frac{C_{2}^{2}\beta _{m}^{2}+1}{C_{2}^{2}\beta _{m}^{2}+C_{2}+1}}\right)$  
    x sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{x}{L}}$)sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{x^{\prime }}{L}}$)  
with $\displaystyle \beta_{m}^{}$tan$\displaystyle \beta_{m}^{}$ = $\displaystyle {\frac{1}{C_{2}}}$ , $\displaystyle \beta_{m}^{}$ > 0 for m = 0, 1, 2...  
and where C2 = $\displaystyle {\frac{(\rho cb)_{2}}{\rho cL}}$.  

X15 Plate with G = 0 (Dirichlet) at x = 0 and k$ \partial$G/$ \partial$x + h2G + ($ \rho$cb)2$ \partial$G/$ \partial$t = 0 (thin film with convection) at x = L.

GX15(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) & = & $\displaystyle {\frac{2}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}\right]$$\displaystyle {\frac{1}{N_{m}}}$sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{x}{L}}$)sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{x^{\prime }}{L}}$)    


    with Nm = $\displaystyle {\frac{(B_{2}-C_{2}\beta _{m}^{2})^{2}+\beta _{m}^{2}+B_{2}+C_{2}\beta _{m}^{2}}{(B_{2}-C_{2}\beta _{m}^{2})^{2}+\beta _{m}^{2}}}$,  
    eigencondition (B2 - C2$\displaystyle \beta_{m}^{2}$)tan$\displaystyle \beta_{m}^{}$ = - $\displaystyle \beta_{m}^{}$,    $\displaystyle \beta_{m}^{}$ > 0 for m = 0, 1, 2...  
    and where C2 = $\displaystyle {\frac{(\rho cb)_{2}}{\rho cL}}$ and B2 = $\displaystyle {\frac{h_{2}L}{k}}$.  

X21 Plate, with $ \partial$G/$ \partial$x = 0 (Neumann) at x = 0 and G = 0 (Dirichlet) at x = L.
a. Best convergence for (t - $ \tau$) small:
GX21(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2  
    x $\displaystyle \sum_{n=-\infty }^{\infty }$(- 1)n$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{% (2nL+x-x^{\prime })^{2}}{4... ...exp \left[ -\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{% (2nL+x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{% (2nL+x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{% (2nL+x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ + exp$\displaystyle \left[\vphantom{ -\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{% (2nL+x-x^{\prime })^{2}}{4\... ...xp \left[ -\frac{% (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  

b. Best convergence for (t - $ \tau$) large:
GX21(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{2}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}\right]$cos($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{x}{L}}$)cos($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{x^{\prime }}{L}}$)  
where $\displaystyle \beta_{m}^{}$ = (2m - 1)($\displaystyle {\frac{\pi }{2}}$), for m = 1, 2,....  

X22 Plate, with $ \partial$G/$ \partial$x = 0 (Neumann) at x = 0 and at x = L.
a. Best convergence for (t - $ \tau$) small:
GX22(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2  
    x $\displaystyle \sum_{n=-\infty }^{\infty }$$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{ (2nL+x-x^{\prime })^{2}}{4\... ...\exp \left[ -\frac{ (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{ (2nL+x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{ (2nL+x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{ (2nL+x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ + exp$\displaystyle \left[\vphantom{ -\frac{ (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{ (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{ (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{ (2nL+x-x^{\prime })^{2}}{4\a... ...exp \left[ -\frac{ (2nL+x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  

b. Best convergence for (t - $ \tau$) large:

GX22(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{L}}$$\displaystyle \left[\vphantom{ 1+2\sum_{m=1}^{\infty }\exp \left( -\frac{m^{2}\... ... L^{2}}\right) \cos (m\pi \frac{x}{L})\cos (m\pi \frac{x^{\prime }}{L})}\right.$1 + 2$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left(\vphantom{ -\frac{m^{2}\pi ^{2}\alpha (t-\tau )}{ L^{2}}}\right.$ - $\displaystyle {\frac{m^{2}\pi ^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{m^{2}\pi ^{2}\alpha (t-\tau )}{ L^{2}}}\right)$cos(m$\displaystyle \pi$$\displaystyle {\frac{x}{L}}$)cos(m$\displaystyle \pi$$\displaystyle {\frac{x^{\prime }}{L}}$)$\displaystyle \left.\vphantom{ 1+2\sum_{m=1}^{\infty }\exp \left( -\frac{m^{2}\... ... L^{2}}\right) \cos (m\pi \frac{x}{L})\cos (m\pi \frac{x^{\prime }}{L})}\right]$    

X23 Plate, with $ \partial$G/$ \partial$x = 0 (Neumann) at x = 0 and k$ \partial$G/$ \partial$x + h2G = 0 (convection) at x = L.
a. For small values of $ \alpha$(t - $ \tau$)/L2 $ \leq$ 0.022 use the following approximation:
GX23(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) $\displaystyle \approx$ [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )} \right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )} \right] }\right.$  
    $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{(x+x^{\prime })^{2}}{4\alpha... ... +\exp \left[ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$ +exp$\displaystyle \left[\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ + exp$\displaystyle \left[\vphantom{ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{(x+x^{\prime })^{2}}{4\alpha... ...+\exp \left[ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  
    - $\displaystyle {\frac{B_{2}}{L}}$exp$\displaystyle \left[\vphantom{ B_{2}\frac{2l-x-x^{\prime }}{L}+B_{2}^{2}\frac{ \alpha (t-\tau )}{L^{2}}}\right.$B2$\displaystyle {\frac{2l-x-x^{\prime }}{L}}$ + B22$\displaystyle {\frac{ \alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ B_{2}\frac{2l-x-x^{\prime }}{L}+B_{2}^{2}\frac{ \alpha (t-\tau )}{L^{2}}}\right]$  
    x erfc$\displaystyle \left[\vphantom{ \frac{2L-x-x^{\prime }}{\left[ 4\alpha (t-\tau ) \right] ^{1/2}}+B_{2}\frac{\left[ \alpha (t-\tau )\right] ^{1/2}}{L}}\right.$$\displaystyle {\frac{2L-x-x^{\prime }}{\left[ 4\alpha (t-\tau ) \right] ^{1/2}}}$ + B2$\displaystyle {\frac{\left[ \alpha (t-\tau )\right] ^{1/2}}{L}}$ $\displaystyle \left.\vphantom{ \frac{2L-x-x^{\prime }}{\left[ 4\alpha (t-\tau ) \right] ^{1/2}}+B_{2}\frac{\left[ \alpha (t-\tau )\right] ^{1/2}}{L}}\right]$  

b. For any value of $ \alpha$(t - $ \tau$)/L2 but best for $ \alpha$(t - $ \tau$)/L2 > 0.022:
GX23(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{2}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}\right]$  
    x $\displaystyle \left(\vphantom{ \frac{\beta _{m}^{2}+B_{2}^{2}}{\beta _{m}^{2}+B_{2}^{2}+B_{2}}}\right.$$\displaystyle {\frac{\beta _{m}^{2}+B_{2}^{2}}{\beta _{m}^{2}+B_{2}^{2}+B_{2}}}$ $\displaystyle \left.\vphantom{ \frac{\beta _{m}^{2}+B_{2}^{2}}{\beta _{m}^{2}+B_{2}^{2}+B_{2}}}\right)$cos($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{x}{L}}$)cos($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{x^{\prime }}{L}}$)  
with $\displaystyle \beta_{m}^{}$tan$\displaystyle \beta_{m}^{}$ = B2 and where B2 = h2L/k.  

X24 Plate with $ \partial$G/$ \partial$x = 0 (Neumann) at x = 0 and k$ \partial$G/$ \partial$x + ($ \rho$cb)2$ \partial$G/$ \partial$t = 0 (thin film) at x = L.
GX24(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{1+C_{2}}}$  
    + $\displaystyle {\frac{2}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}\right]$$\displaystyle {\frac{\cos (\beta _{m}x/L)\cos (\beta _{m}x^{\prime }/L)}{N_{m}}}$  
Nm = $\displaystyle {\frac{\left( 1+C_2^2\beta _{m}^{2}+C_{2}\right) }{1+C_2^2\beta _{m}^{2}}}$,  
with tan$\displaystyle \beta_{m}^{}$ = - C2$\displaystyle \beta_{m}^{}$ and where C2 = $\displaystyle {\frac{(\rho cb)_{2}}{\rho cL}}$.  

X25 Plate with $ \partial$G/$ \partial$x = 0 (Neumann) at x = 0 and k$ \partial$G/$ \partial$x + h2G + ($ \rho$cb)2$ \partial$G/$ \partial$t = 0 (thin film with convection) at x = L.

GX25(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{2}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}\right]$$\displaystyle {\frac{\cos (\beta _{m}x/L)\cos (\beta _{m}x^{\prime }/L)}{N_{m}}}$    


    with Nm = $\displaystyle {\frac{\left[ \beta _{m}^{2}+D^{2}\right] (1+2C_{2})+D \left[ 1-2C_{2}D\right] }{\beta _{m}^{2}+D^{2}}}$,  
    eigencondition $\displaystyle \beta_{m}^{}$tan$\displaystyle \beta_{m}^{}$ = B2 - C2$\displaystyle \beta_{m}^{2}$  
    and where B2 = $\displaystyle {\frac{h_{2}L}{k}}$, C2 = $\displaystyle {\frac{(\rho cb)_{2}}{\rho cL}}$ and D = (B2 - C2$\displaystyle \beta_{m}^{2}$).  

X31 Plate, with - k$ \partial$G/$ \partial$x + hG = 0 (convection) at x = 0 and G = 0 (Dirichlet) at x = L.
a. For small values of $ \alpha$(t - $ \tau$)/L2 $ \leq$ 0.022 use the following approximation:
GX31(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) $\displaystyle \approx$ [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )} \right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )} \right] }\right.$  
    $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{(x+x^{\prime })^{2}}{4\alpha... ... +\exp \left[ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$ +exp$\displaystyle \left[\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ - exp$\displaystyle \left[\vphantom{ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{(x+x^{\prime })^{2}}{4\alpha... ...+\exp \left[ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  
    - $\displaystyle {\frac{h}{k}}$exp$\displaystyle \left[\vphantom{ \frac{h(x+x^{\prime })}{k}+\frac{h^{2}\alpha (t-\tau )}{k^{2}}}\right.$$\displaystyle {\frac{h(x+x^{\prime })}{k}}$ + $\displaystyle {\frac{h^{2}\alpha (t-\tau )}{k^{2}}}$ $\displaystyle \left.\vphantom{ \frac{h(x+x^{\prime })}{k}+\frac{h^{2}\alpha (t-\tau )}{k^{2}}}\right]$  
    x erfc$\displaystyle \left[\vphantom{ \frac{x+x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}+\frac{h}{k}\left[ \alpha (t-\tau )\right] ^{1/2}}\right.$$\displaystyle {\frac{x+x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}}$ + $\displaystyle {\frac{h}{k}}$$\displaystyle \left[\vphantom{ \alpha (t-\tau )}\right.$$\displaystyle \alpha$(t - $\displaystyle \tau$)$\displaystyle \left.\vphantom{ \alpha (t-\tau )}\right]^{1/2}_{}$ $\displaystyle \left.\vphantom{ \frac{x+x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}+\frac{h}{k}\left[ \alpha (t-\tau )\right] ^{1/2}}\right]$  

b. For any value of $ \alpha$(t - $ \tau$)/L2 but best for $ \alpha$(t - $ \tau$)/L2 > 0.022:
GX31(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{2}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right]$$\displaystyle \left(\vphantom{ \frac{\beta _{m}^{2}+B^{2}}{\beta _{m}^{2}+B^{2}+B}}\right.$$\displaystyle {\frac{\beta _{m}^{2}+B^{2}}{\beta _{m}^{2}+B^{2}+B}}$ $\displaystyle \left.\vphantom{ \frac{\beta _{m}^{2}+B^{2}}{\beta _{m}^{2}+B^{2}+B}}\right)$  
    x sin$\displaystyle \left[\vphantom{ \beta _{m}(1-\frac{x}{L})}\right.$$\displaystyle \beta_{m}^{}$(1 - $\displaystyle {\frac{x}{L}}$)$\displaystyle \left.\vphantom{ \beta _{m}(1-\frac{x}{L})}\right]$sin$\displaystyle \left[\vphantom{ \beta _{m}(1-\frac{x^{\prime }}{L})}\right.$$\displaystyle \beta_{m}^{}$(1 - $\displaystyle {\frac{x^{\prime }}{L}}$)$\displaystyle \left.\vphantom{ \beta _{m}(1-\frac{x^{\prime }}{L})}\right]$  
where $\displaystyle \beta_{m}^{}$cot$\displaystyle \beta_{m}^{}$ = - B and B = hL/k.  

X32 Plate, with - k$ \partial$G/$ \partial$x + hG = 0 (convection) at x = 0 and $ \partial$G/$ \partial$x = 0 (Neumann) at x = L.
a. For small values of $ \alpha$(t - $ \tau$)/L2 $ \leq$ 0.022 use the following approximation:
GX32(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) $\displaystyle \approx$ [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}% \right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}% \right] }\right.$  
    $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{(x+x^{\prime })^{2}}{4\alpha... ... +\exp \left[ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$ +exp$\displaystyle \left[\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ + exp$\displaystyle \left[\vphantom{ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{(x+x^{\prime })^{2}}{4\alpha... ...+\exp \left[ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  
    - $\displaystyle {\frac{h}{k}}$exp$\displaystyle \left[\vphantom{ \frac{h(x+x^{\prime })}{k}+\frac{h^{2}\alpha (t-\tau )}{k^{2}}}\right.$$\displaystyle {\frac{h(x+x^{\prime })}{k}}$ + $\displaystyle {\frac{h^{2}\alpha (t-\tau )}{k^{2}}}$ $\displaystyle \left.\vphantom{ \frac{h(x+x^{\prime })}{k}+\frac{h^{2}\alpha (t-\tau )}{k^{2}}}\right]$  
    x erfc$\displaystyle \left[\vphantom{ \frac{x+x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}+\frac{h}{k}\left[ \alpha (t-\tau )\right] ^{1/2}}\right.$$\displaystyle {\frac{x+x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}}$ + $\displaystyle {\frac{h}{k}}$$\displaystyle \left[\vphantom{ \alpha (t-\tau )}\right.$$\displaystyle \alpha$(t - $\displaystyle \tau$)$\displaystyle \left.\vphantom{ \alpha (t-\tau )}\right]^{1/2}_{}$ $\displaystyle \left.\vphantom{ \frac{x+x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}+\frac{h}{k}\left[ \alpha (t-\tau )\right] ^{1/2}}\right]$  

b. For any value of $ \alpha$(t - $ \tau$)/L2 but best for $ \alpha$(t - $ \tau$)/L2 > 0.022:
GX32(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{2}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right]$$\displaystyle \left(\vphantom{ \frac{\beta _{m}^{2}+B^{2}}{\beta _{m}^{2}+B^{2}+B}}\right.$$\displaystyle {\frac{\beta _{m}^{2}+B^{2}}{\beta _{m}^{2}+B^{2}+B}}$ $\displaystyle \left.\vphantom{ \frac{\beta _{m}^{2}+B^{2}}{\beta _{m}^{2}+B^{2}+B}}\right)$  
    x cos$\displaystyle \left[\vphantom{ \beta _{m}(1-\frac{x}{L})}\right.$$\displaystyle \beta_{m}^{}$(1 - $\displaystyle {\frac{x}{L}}$)$\displaystyle \left.\vphantom{ \beta _{m}(1-\frac{x}{L})}\right]$cos$\displaystyle \left[\vphantom{ \beta _{m}(1-\frac{x^{\prime }}{L})}\right.$$\displaystyle \beta_{m}^{}$(1 - $\displaystyle {\frac{x^{\prime }}{L}}$)$\displaystyle \left.\vphantom{ \beta _{m}(1-\frac{x^{\prime }}{L})}\right]$  
with $\displaystyle \beta_{m}^{}$tan$\displaystyle \beta_{m}^{}$ = B and where B = hL/k.  

X33 Plate, with - k$ \partial$G/$ \partial$x + h1G = 0 (convection) at x = 0 and k$ \partial$G/$ \partial$x + h2G = 0 (convection) at x = L.
a. For small values of $ \alpha$(t - $ \tau$)/L2 $ \leq$ 0.022 use the following approximation:
GX33(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) $\displaystyle \approx$ [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}% \right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}% \right] }\right.$  
    $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{(x+x^{\prime })^{2}}{4\alpha... ... +\exp \left[ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$ +exp$\displaystyle \left[\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ + exp$\displaystyle \left[\vphantom{ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{(x+x^{\prime })^{2}}{4\alpha... ...+\exp \left[ -\frac{(2L-x-x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  
    - $\displaystyle {\frac{h_{1}}{k}}$exp$\displaystyle \left[\vphantom{ \frac{h_{1}(x+x^{\prime })}{k}+\frac{% h_{1}^{2}\alpha (t-\tau )}{k^{2}}}\right.$$\displaystyle {\frac{h_{1}(x+x^{\prime })}{k}}$ + $\displaystyle {\frac{% h_{1}^{2}\alpha (t-\tau )}{k^{2}}}$ $\displaystyle \left.\vphantom{ \frac{h_{1}(x+x^{\prime })}{k}+\frac{% h_{1}^{2}\alpha (t-\tau )}{k^{2}}}\right]$  
    x erfc$\displaystyle \left[\vphantom{ \frac{x+x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}+\frac{h_{1}}{k}\left[ \alpha (t-\tau )\right] ^{1/2}}\right.$$\displaystyle {\frac{x+x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}}$ + $\displaystyle {\frac{h_{1}}{k}}$$\displaystyle \left[\vphantom{ \alpha (t-\tau )}\right.$$\displaystyle \alpha$(t - $\displaystyle \tau$)$\displaystyle \left.\vphantom{ \alpha (t-\tau )}\right]^{1/2}_{}$ $\displaystyle \left.\vphantom{ \frac{x+x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}+\frac{h_{1}}{k}\left[ \alpha (t-\tau )\right] ^{1/2}}\right]$  
    - $\displaystyle {\frac{h_{2}}{k}}$exp$\displaystyle \left[\vphantom{ \frac{h_{2}(2L-x-x^{\prime })}{k}+\frac{% h_{2}^{2}\alpha (t-\tau )}{k^{2}}}\right.$$\displaystyle {\frac{h_{2}(2L-x-x^{\prime })}{k}}$ + $\displaystyle {\frac{% h_{2}^{2}\alpha (t-\tau )}{k^{2}}}$ $\displaystyle \left.\vphantom{ \frac{h_{2}(2L-x-x^{\prime })}{k}+\frac{% h_{2}^{2}\alpha (t-\tau )}{k^{2}}}\right]$  
    x erfc$\displaystyle \left[\vphantom{ \frac{2L-x-x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}+\frac{h_{2}}{k}\left[ \alpha (t-\tau )\right] ^{1/2}}\right.$$\displaystyle {\frac{2L-x-x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}}$ + $\displaystyle {\frac{h_{2}}{k}}$$\displaystyle \left[\vphantom{ \alpha (t-\tau )}\right.$$\displaystyle \alpha$(t - $\displaystyle \tau$)$\displaystyle \left.\vphantom{ \alpha (t-\tau )}\right]^{1/2}_{}$ $\displaystyle \left.\vphantom{ \frac{2L-x-x^{\prime }}{\left[ 4\alpha (t-\tau )% \right] ^{1/2}}+\frac{h_{2}}{k}\left[ \alpha (t-\tau )\right] ^{1/2}}\right]$  

b. For any value of $ \alpha$(t - $ \tau$)/L2 but best for $ \alpha$(t - $ \tau$)/L2 > 0.022:
GX33(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{2}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right]$  
    x $\displaystyle \left[\vphantom{ \beta _{m}\cos (\beta _{m}x/L)+B_{1}\sin (\beta _{m}x/L)% }\right.$$\displaystyle \beta_{m}^{}$cos($\displaystyle \beta_{m}^{}$x/L) + B1sin($\displaystyle \beta_{m}^{}$x/L)$\displaystyle \left.\vphantom{ \beta _{m}\cos (\beta _{m}x/L)+B_{1}\sin (\beta _{m}x/L)% }\right]$  
    x $\displaystyle {\frac{\left[ \beta _{m}\cos (\beta _{m}x^{\prime }/L)+B_{1}\sin ... ...ta _{m}^{2}+B_{1}^{2})\left[ 1+B_{2}/(\beta _{m}^{2}+B_{2}^{2})\right] +B_{1}}}$  
with tan$\displaystyle \beta_{m}^{}$ = $\displaystyle {\frac{\beta _{m}(B_{1}+B_{2})}{\beta _{m}^{2}-B_{1}B_{2}}}$ and where B1 = $\displaystyle {\frac{h_{1}L}{k}}$, B2 = $\displaystyle {\frac{h_{2}L}{k}}$.  

X34 Plate, with - k$ \partial$G/$ \partial$x + h2G = 0 (convection) at x = 0 and k$ \partial$G/$ \partial$x + ($ \rho$cb)2 $ \partial$G/$ \partial$t = 0 (thin film) at x = L.
GX34(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right]$$\displaystyle {\frac{X_{m}(x)X_{m}(x^{\prime })}{N_{m}}}$  
where   Xm(x) = B1sin($\displaystyle \beta_{m}^{}$x/L) + $\displaystyle \beta_{m}^{}$cos($\displaystyle \beta_{m}^{}$x/L)  
Nm = $\displaystyle \left\{\vphantom{ \frac{(\beta _{m}^{2}+B_{1}^{2})}{2}+\beta _{m}^{2}C_{2}+% \frac{\tan \beta _{m}}{1+\tan ^{2}\beta _{m}}}\right.$$\displaystyle {\frac{(\beta _{m}^{2}+B_{1}^{2})}{2}}$ + $\displaystyle \beta_{m}^{2}$C2 + $\displaystyle {\frac{\tan \beta _{m}}{1+\tan ^{2}\beta _{m}}}$ $\displaystyle \left.\vphantom{ \frac{(\beta _{m}^{2}+B_{1}^{2})}{2}+\beta _{m}^{2}C_{2}+% \frac{\tan \beta _{m}}{1+\tan ^{2}\beta _{m}}}\right.$  
    x $\displaystyle \left.\vphantom{ \left( \frac{(\beta _{m}^{2}-B_{1}^{2})}{2\beta ... ...eft[ C_{2}\left( B_{1}^{2}-\beta _{m}^{2}\right) +B_{1}\right] \right) }\right.$$\displaystyle \left(\vphantom{ \frac{(\beta _{m}^{2}-B_{1}^{2})}{2\beta _{m}}% ... ...ight) \left[ C_{2}\left( B_{1}^{2}-\beta _{m}^{2}\right) +B_{1}\right] }\right.$$\displaystyle {\frac{(\beta _{m}^{2}-B_{1}^{2})}{2\beta _{m}}}$ + 2C2B1$\displaystyle \beta_{m}^{}$ + tan$\displaystyle \left(\vphantom{ \beta _{m}}\right.$$\displaystyle \beta_{m}^{}$ $\displaystyle \left.\vphantom{ \beta _{m}}\right)$$\displaystyle \left[\vphantom{ C_{2}\left( B_{1}^{2}-\beta _{m}^{2}\right) +B_{1}}\right.$C2$\displaystyle \left(\vphantom{ B_{1}^{2}-\beta _{m}^{2}}\right.$B12 - $\displaystyle \beta_{m}^{2}$ $\displaystyle \left.\vphantom{ B_{1}^{2}-\beta _{m}^{2}}\right)$ + B1$\displaystyle \left.\vphantom{ C_{2}\left( B_{1}^{2}-\beta _{m}^{2}\right) +B_{1}}\right]$ $\displaystyle \left.\vphantom{ \frac{(\beta _{m}^{2}-B_{1}^{2})}{2\beta _{m}}% ... ...ight) \left[ C_{2}\left( B_{1}^{2}-\beta _{m}^{2}\right) +B_{1}\right] }\right)$ $\displaystyle \left.\vphantom{ \left( \frac{(\beta _{m}^{2}-B_{1}^{2})}{2\beta ... ...ft[ C_{2}\left( B_{1}^{2}-\beta _{m}^{2}\right) +B_{1}\right] \right) }\right\}$  
with tan$\displaystyle \beta_{m}^{}$ = $\displaystyle {\frac{B_{1}-C_{2}\beta _{m}^{2}}{\beta _{m}(1+-B_{1}C_{2})}}$ and where   B1 = $\displaystyle {\frac{h_{1}L}{k}}$, C2 = $\displaystyle {\frac{(\rho cb)_{2}}{\rho cL}}$.  

X35 Plate, with - k$ \partial$G/$ \partial$x + h1G = 0 (convection) at x = 0 and k$ \partial$G/$ \partial$x + h2G + ($ \rho$cb)2 $ \partial$G/$ \partial$t = 0 (thin film with convection) at x = L.
GX35(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right]$$\displaystyle {\frac{X_{m}(x)X_{m}(x^{\prime })}{N_{m}}}$  
where   Xm(x) = B1sin($\displaystyle \beta_{m}^{}$x/L) + $\displaystyle \beta_{m}^{}$cos($\displaystyle \beta_{m}^{}$x/L)  
Nm = $\displaystyle {\frac{(\beta _{m}^{2}+B_{1}^{2})}{2}}$ + $\displaystyle \beta_{m}^{2}$C2 + $\displaystyle {\frac{\tan \beta _{m}}{1+\tan ^{2}\beta _{m}}}$  
    x $\displaystyle \left(\vphantom{ \frac{(\beta _{m}^{2}-B_{1}^{2})}{2\beta _{m}}% ... ...ight) \left[ C_{2}\left( B_{1}^{2}-\beta _{m}^{2}\right) +B_{1}\right] }\right.$$\displaystyle {\frac{(\beta _{m}^{2}-B_{1}^{2})}{2\beta _{m}}}$ + 2C2B1$\displaystyle \beta_{m}^{}$ + tan$\displaystyle \left(\vphantom{ \beta _{m}}\right.$$\displaystyle \beta_{m}^{}$ $\displaystyle \left.\vphantom{ \beta _{m}}\right)$$\displaystyle \left[\vphantom{ C_{2}\left( B_{1}^{2}-\beta _{m}^{2}\right) +B_{1}}\right.$C2$\displaystyle \left(\vphantom{ B_{1}^{2}-\beta _{m}^{2}}\right.$B12 - $\displaystyle \beta_{m}^{2}$ $\displaystyle \left.\vphantom{ B_{1}^{2}-\beta _{m}^{2}}\right)$ + B1$\displaystyle \left.\vphantom{ C_{2}\left( B_{1}^{2}-\beta _{m}^{2}\right) +B_{1}}\right]$ $\displaystyle \left.\vphantom{ \frac{(\beta _{m}^{2}-B_{1}^{2})}{2\beta _{m}}% ... ...ight) \left[ C_{2}\left( B_{1}^{2}-\beta _{m}^{2}\right) +B_{1}\right] }\right)$  
with tan$\displaystyle \beta_{m}^{}$ = $\displaystyle {\frac{\beta _{m}(B_{1}+B_{2}-C_{2}\beta _{m}^{2})}{\beta _{m}^{2}-B_{1}(B_{2}-C_{2}\beta _{m}^{2})}}$   
and   B1 = $\displaystyle {\frac{h_{1}L}{k}}$, B2 = $\displaystyle {\frac{h_{2}L}{k}}$ , C2 = $\displaystyle {\frac{(\rho cb)_{2}}{\rho cL}}$.  

X41 Plate, with - k$ \partial$G/$ \partial$x + ($ \rho$cb)1$ \partial$G/$ \partial$t = 0 (thin film) at x = 0 and G = 0 (Dirichlet) at x = L.
GX41(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{L}}$$\displaystyle \sum_{m=1}^{\infty }$$\displaystyle {\frac{1}{N_{m}}}$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}\right]$  
    x sin$\displaystyle \left[\vphantom{ \beta _{m}(1-\frac{x}{L})}\right.$$\displaystyle \beta_{m}^{}$(1 - $\displaystyle {\frac{x}{L}}$)$\displaystyle \left.\vphantom{ \beta _{m}(1-\frac{x}{L})}\right]$sin$\displaystyle \left[\vphantom{ \beta _{m}(1-\frac{x^{\prime }}{L})}\right.$$\displaystyle \beta_{m}^{}$(1 - $\displaystyle {\frac{x^{\prime }}{L}}$)$\displaystyle \left.\vphantom{ \beta _{m}(1-\frac{x^{\prime }}{L})}\right]$  
    Nm = $\displaystyle \left[\vphantom{ \left( C_{1}\beta _{m}\right) ^{2}+C_{1}+1}\right.$$\displaystyle \left(\vphantom{ C_{1}\beta _{m}}\right.$C1$\displaystyle \beta_{m}^{}$ $\displaystyle \left.\vphantom{ C_{1}\beta _{m}}\right)^{2}_{}$ + C1 + 1$\displaystyle \left.\vphantom{ \left( C_{1}\beta _{m}\right) ^{2}+C_{1}+1}\right]$/2  
where $\displaystyle \beta_{m}^{}$tan$\displaystyle \beta_{m}^{}$ = 1/C1, C1 = $\displaystyle {\frac{% \left( \rho cb\right) _{1}}{\rho cL}}$.  

X42 Plate, with - k$ \partial$G/$ \partial$x + ($ \rho$cb)1$ \partial$G/$ \partial$t = 0 (thin film) at x = 0 and $ \partial$G/$ \partial$x = 0 (Neumann) at x = L.
GX42(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{L}}$$\displaystyle \left\{\vphantom{ \frac{1}{% N_{0}}+\sum_{m=1}^{\infty }\exp \lef... ...pha (t-\tau )% }{L^{2}}\right] \frac{X_{m}(x)X_{m}(x^{\prime })}{N_{m}}}\right.$$\displaystyle {\frac{1}{% N_{0}}}$ + $\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )% }{L^{2}}}\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )% }{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )% }{L^{2}}}\right]$$\displaystyle {\frac{X_{m}(x)X_{m}(x^{\prime })}{N_{m}}}$ $\displaystyle \left.\vphantom{ \frac{1}{% N_{0}}+\sum_{m=1}^{\infty }\exp \left... ...ha (t-\tau )% }{L^{2}}\right] \frac{X_{m}(x)X_{m}(x^{\prime })}{N_{m}}}\right\}$  
where   Xm(x) = cos($\displaystyle \beta_{m}^{}$x/L) - C1$\displaystyle \beta_{m}^{}$sin($\displaystyle \beta_{m}^{}$x/L)  
N0 = 1 + C1,  Nm = $\displaystyle \left[\vphantom{ \left( C_{1}\beta _{m}\right) ^{2}+C_{1}+1% }\right.$$\displaystyle \left(\vphantom{ C_{1}\beta _{m}}\right.$C1$\displaystyle \beta_{m}^{}$ $\displaystyle \left.\vphantom{ C_{1}\beta _{m}}\right)^{2}_{}$ + C1 + 1$\displaystyle \left.\vphantom{ \left( C_{1}\beta _{m}\right) ^{2}+C_{1}+1% }\right]$/2  
where $\displaystyle \beta_{m}^{}$cot$\displaystyle \beta_{m}^{}$ = -1/C1, C1 = $\displaystyle {\frac{% \left( \rho cb\right) _{1}}{\rho cL}}$.  

X51 Plate, with - k$ \partial$G/$ \partial$x + hG + ($ \rho$cb)1$ \partial$G/$ \partial$t = 0 (thin film) at x = 0 and G = 0 (Dirichlet) at x = L.
GX51(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right]$$\displaystyle {\frac{X_{m}(x)X_{m}(x^{\prime })}{N_{m}}}$  
where   Xm(x) = Dmsin($\displaystyle \beta_{m}^{}$x/L) + cos($\displaystyle \beta_{m}^{}$x/L)  
Nm = $\displaystyle {\textstyle\frac{1}{2}}$$\displaystyle \left[\vphantom{ D_{m}^{2}+D_{m}/\beta _{m}+2C_{1}+1}\right.$Dm2 + Dm/$\displaystyle \beta_{m}^{}$ + 2C1 + 1$\displaystyle \left.\vphantom{ D_{m}^{2}+D_{m}/\beta _{m}+2C_{1}+1}\right]$ , Dm =   $\displaystyle {\frac{B}{\beta _{m}}}$ - C1$\displaystyle \beta_{m}^{}$  
where tan$\displaystyle \beta_{m}^{}$ = $\displaystyle {\frac{\beta _{m}}{C_{1}\beta _{m}^{2}-B}}$, C1 = $\displaystyle {\frac{\left( \rho cb\right) _{1}}{\rho cL}}$, B = $\displaystyle {\frac{hL% }{k}}$  

X52 Plate, with - k$ \partial$G/$ \partial$x + hG + ($ \rho$cb)1$ \partial$G/$ \partial$t = 0 (thin film) at x = 0 and $ \partial$G/$ \partial$x = 0 (Neumann) at x = L.
GX52(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{L}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{L^{2}}% }\right]$$\displaystyle {\frac{X_{m}(x)X_{m}(x^{\prime })}{N_{m}}}$  
where   Xm(x) = Dmsin($\displaystyle \beta_{m}^{}$x/L) + cos($\displaystyle \beta_{m}^{}$x/L)  
Nm = $\displaystyle {\textstyle\frac{1}{2}}$$\displaystyle \left[\vphantom{ D_{m}^{2}+\frac{D_{m}}{\beta _{m}}+2C_{1}+1}\right.$Dm2 + $\displaystyle {\frac{D_{m}}{\beta _{m}}}$ + 2C1 + 1$\displaystyle \left.\vphantom{ D_{m}^{2}+\frac{D_{m}}{\beta _{m}}+2C_{1}+1}\right]$, Dm =   $\displaystyle {\frac{B}{\beta _{m}}}$ - C1$\displaystyle \beta_{m}^{}$  
where tan$\displaystyle \beta_{m}^{}$ = Dm, C1 = $\displaystyle {\frac{\left( \rho cb\right) _{1}}{\rho cL}}$, B = $\displaystyle {\frac{hL}{k}}$  

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Next: Small-time GF, transient cases Up: Rectangular coordinates. Transient 1-D. Previous: Semi-infinite body, transient 1-D.
Kevin D. Cole
2002-12-31