Plate, steady 1-D.

X11 Plate, G = 0 (Dirichlet) at x = 0 and x = L.

$$\left\vert\vphantom{ \,x^{\prime }}\right. \prime \left.\vphantom{ \,x^{\prime }}\right. \left\{\vphantom{ \begin{array}{cc} x(1-x^{\prime })/L & \text{f... ...\prime } \\ x^{\prime }(1-x)/L & \text{for }x>x^{\prime } \end{array} }\right. \begin{array}{cc} x(1-x^{\prime })/L & \text{for }x<x^{\prime } \\ x^{\prime }(1-x)/L & \text{for }x>x^{\prime } \end{array} \left.\vphantom{ \begin{array}{cc} x(1-x^{\prime })/L & \text{fo... ...\prime } \\ x^{\prime }(1-x)/L & \text{for }x>x^{\prime } \end{array} }\right.$$

X12 Plate, G = 0 (Dirichlet) at x = 0 and $\partial$G/$\partial$x = 0 (Neumann) at x = L.

$$\left\vert\vphantom{ \,x^{\prime }}\right. \prime \left.\vphantom{ \,x^{\prime }}\right. \left\{\vphantom{ \begin{array}{cc} x & \text{for }x<x^{\prime } \\ x^{\prime } & \text{for }x>x^{\prime } \end{array} }\right. \begin{array}{cc} x & \text{for }x<x^{\prime } \\ x^{\prime } & \text{for }x>x^{\prime } \end{array} \left.\vphantom{ \begin{array}{cc} x & \text{for }x<x^{\prime } \\ x^{\prime } & \text{for }x>x^{\prime } \end{array} }\right.$$

X13 Plate, G = 0 (Dirichlet) at x = 0 and k$\partial$G/$\partial$x + h₂G = 0 (Neumann) at x = L Note B₂ = h₂L/k

$$\left\vert\vphantom{ \,x^{\prime }}\right. \prime \left.\vphantom{ \,x^{\prime }}\right. \left\{\vphantom{ \begin{array}{cc} x[1-B_{2}(x^{\prime }/L)/(1+... ...prime }[1-B_{2}(x/L)/(1+B_{2})] & \text{for }x>x^{\prime } \end{array} }\right. \begin{array}{cc} x[1-B_{2}(x^{\prime }/L)/(1+B_{2})] & \text{for... ... \\ x^{\prime }[1-B_{2}(x/L)/(1+B_{2})] & \text{for }x>x^{\prime } \end{array} \left.\vphantom{ \begin{array}{cc} x[1-B_{2}(x^{\prime }/L)/(1+B... ...prime }[1-B_{2}(x/L)/(1+B_{2})] & \text{for }x>x^{\prime } \end{array} }\right.$$

X21 Plate, $\partial$G/$\partial$x = 0 (Neumann) at x = 0 and G = 0 (Dirichlet) at x = L.

$$\left\vert\vphantom{ \,x^{\prime }}\right. \prime \left.\vphantom{ \,x^{\prime }}\right. \left\{\vphantom{ \begin{array}{cc} L-x^{\prime } & \text{for }x<x^{\prime } \\ L-x & \text{for }x>x^{\prime } \end{array} }\right. \begin{array}{cc} L-x^{\prime } & \text{for }x<x^{\prime } \\ L-x & \text{for }x>x^{\prime } \end{array} \left.\vphantom{ \begin{array}{cc} L-x^{\prime } & \text{for }x<x^{\prime } \\ L-x & \text{for }x>x^{\prime } \end{array} }\right.$$

X22 Plate, $\partial$G/$\partial$x = 0 (Neumann) at both sides. Note that this geometry requires a pseudo GF, denoted H. The temperature solution found from a pseudo GF requires that the total volumetric heating sums to zero and the spatial average temperature in the body must be supplied as a known condition.

$$\left\vert\vphantom{ \,x^{\prime }}\right. \prime \left.\vphantom{ \,x^{\prime }}\right. \left\{\vphantom{ \begin{array}{cc} \lbrack x^{2}+(x^{\prime })^... ...x^{2}+(x^{\prime })^{2}]/(2L)-x & \text{for }x>x^{\prime } \end{array} }\right. \begin{array}{cc} \lbrack x^{2}+(x^{\prime })^{2}]/(2L)-x^{\prime... ... \lbrack x^{2}+(x^{\prime })^{2}]/(2L)-x & \text{for }x>x^{\prime } \end{array} \left.\vphantom{ \begin{array}{cc} \lbrack x^{2}+(x^{\prime })^{... ...x^{2}+(x^{\prime })^{2}]/(2L)-x & \text{for }x>x^{\prime } \end{array} }\right.$$

X23 Plate, $\partial$G/$\partial$x = 0 (Neumann) at x = 0 and k$\partial$G/$\partial$x + h₂G = 0 (convection) at x = L. Note B₂ = h₂L/k

$$\left\vert\vphantom{ \,x^{\prime }}\right. \prime \left.\vphantom{ \,x^{\prime }}\right. \left\{\vphantom{ \begin{array}{cc} L(1+1/B_{2}-x^{\prime }/L) &... ...^{\prime } \\ L(1+1/B_{2}-x/L) & \text{for }x>x^{\prime } \end{array} }\right. \begin{array}{cc} L(1+1/B_{2}-x^{\prime }/L) & \text{for }x<x^{\prime } \\ L(1+1/B_{2}-x/L) & \text{for }x>x^{\prime } \end{array} \left.\vphantom{ \begin{array}{cc} L(1+1/B_{2}-x^{\prime }/L) & ... ...^{\prime } \\ L(1+1/B_{2}-x/L) & \text{for }x>x^{\prime } \end{array} }\right.$$

X31 Plate, - k$\partial$G/$\partial$x + h₁G = 0 (convection) at x = 0 and G = 0 (Dirchlet) at x = L. Note B₁ = h₁L/k

$$\left\vert\vphantom{ \,x^{\prime }}\right. \prime \left.\vphantom{ \,x^{\prime }}\right. \left\{\vphantom{ \begin{array}{cc} (B_{1}x-B_{1}xx^{\prime}/L+L-... ...e}-B_{1}xx^{\prime}/L+L-x)/(1+B) & \text{for }x>x^{\prime} \end{array} }\right. \begin{array}{cc} (B_{1}x-B_{1}xx^{\prime}/L+L-x^{\prime})/(1+B) ... ...}x^{\prime}-B_{1}xx^{\prime}/L+L-x)/(1+B) & \text{for }x>x^{\prime} \end{array} \left.\vphantom{ \begin{array}{cc} (B_{1}x-B_{1}xx^{\prime}/L+L-x... ...e}-B_{1}xx^{\prime}/L+L-x)/(1+B) & \text{for }x>x^{\prime} \end{array} }\right.$$

X32 Plate, - k$\partial$G/$\partial$x + h₁G = 0 (convection) at x = 0 and $\partial$G/$\partial$x = 0 (Neumann) at x = L. Note B₁ = h₁L/k

$$\left\vert\vphantom{ \,x^{\prime }}\right. \prime \left.\vphantom{ \,x^{\prime }}\right. \left\{\vphantom{ \begin{array}{cc} L(1/B_{1}+x/L) & \text{for }... ... } \\ L(1/B_{1}+x^{\prime }/L) & \text{for }x>x^{\prime } \end{array} }\right. \begin{array}{cc} L(1/B_{1}+x/L) & \text{for }x<x^{\prime } \\ L(1/B_{1}+x^{\prime }/L) & \text{for }x>x^{\prime } \end{array} \left.\vphantom{ \begin{array}{cc} L(1/B_{1}+x/L) & \text{for }x... ... } \\ L(1/B_{1}+x^{\prime }/L) & \text{for }x>x^{\prime } \end{array} }\right.$$

X33 Plate, - k$\partial$G/$\partial$x + h₁G = 0 (convection) at x = 0 and k$\partial$G/$\partial$x + h₂G = 0 (convection) at x = L. Note B₁ = h₁L/k and B₂ = h₂L/k

$$\left\vert\vphantom{ \,x^{\prime }}\right. \prime \left.\vphantom{ \,x^{\prime }}\right. {\frac{% (B_{1}B_{2}x+B_{1}x-B_{1}B_{2}xx^{\prime }/L-B_{2}x^{\prime }+B_{2}L+L)}{ (B_{1}B_{2}+B_{1}+B_{2})}} \prime$$ =

$${\frac{(B_{1}B_{2}x^{\prime }+B_{1}x^{\prime }-B_{1}B_{2}xx^{\prime }/L-B_{2}x+B_{2}L+L)}{(B_{1}B_{2}+B_{1}+B_{2})}} \prime$$ =