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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 - ) small:
 GX11(x, t | x,) = [4(t - )]-1/2 x exp - - exp -

b. Best convergence for (t - ) large:

 GX11(x, t | x,) = exp - sin(m)sin(m)

X12 Plate, with G = 0 (Dirichlet) at x = 0 and G/x = 0 (Neumann) at x = L.
a. Best convergence for (t - ) small:
 GX12(x, tx, ) = [4(t - )]-1/2 x (- 1)nexp - - exp -

b. Best convergence for (t - ) large:
 GX12(x, tx, ) = exp - sin()sin() where = (2m - 1)(/2), for m = 1, 2,....

X13 Plate with G = 0 (Dirichlet) at x = 0 and kG/x + hG = 0 (convection) at x = L.
a. For (t - )/L2 0.022 use the following approximation:
 GX13(x, tx, ) [4(t - )]-1/2exp - -exp - + exp - - exp + h2 x erfc + h

b. For all values, but best for (t - )/L2 large:
 GX13(x, tx, ) = exp - x sin()sin() with cot = - B and where B = hL/k.

X14 Plate with G = 0 (Dirichlet) at x = 0 and kG/x + (cb)2G/t = 0 (thin film) at x = L.
 GX14(x, tx, ) = exp - x sin()sin() with tan = ,  > 0 for m = 0, 1, 2... and where C2 = .

X15 Plate with G = 0 (Dirichlet) at x = 0 and kG/x + h2G + (cb)2G/t = 0 (thin film with convection) at x = L.

 GX15(x, tx, ) & = & exp - sin()sin()

 with Nm = , eigencondition (B2 - C2)tan = - ,     > 0 for m = 0, 1, 2... and where C2 = and B2 = .

X21 Plate, with G/x = 0 (Neumann) at x = 0 and G = 0 (Dirichlet) at x = L.
a. Best convergence for (t - ) small:
 GX21(x, tx, ) = [4(t - )]-1/2 x (- 1)nexp - + exp -

b. Best convergence for (t - ) large:
 GX21(x, tx, ) = exp - cos()cos() where = (2m - 1)(), for m = 1, 2,....

X22 Plate, with G/x = 0 (Neumann) at x = 0 and at x = L.
a. Best convergence for (t - ) small:
 GX22(x, tx, ) = [4(t - )]-1/2 x exp - + exp -

b. Best convergence for (t - ) large:

 GX22(x, tx, ) = 1 + 2exp - cos(m)cos(m)

X23 Plate, with G/x = 0 (Neumann) at x = 0 and kG/x + h2G = 0 (convection) at x = L.
a. For small values of (t - )/L2 0.022 use the following approximation:
 GX23(x, tx, ) [4(t - )]-1/2exp - +exp - + exp - - expB2 + B22 x erfc + B2

b. For any value of (t - )/L2 but best for (t - )/L2 > 0.022:
 GX23(x, tx, ) = exp - x cos()cos() with tan = B2 and where B2 = h2L/k.

X24 Plate with G/x = 0 (Neumann) at x = 0 and kG/x + (cb)2G/t = 0 (thin film) at x = L.
 GX24(x, tx, ) = + exp - Nm = , with tan = - C2 and where C2 = .

X25 Plate with G/x = 0 (Neumann) at x = 0 and kG/x + h2G + (cb)2G/t = 0 (thin film with convection) at x = L.

 GX25(x, tx, ) = exp -

 with Nm = , eigencondition tan = B2 - C2 and where B2 = , C2 = and D = (B2 - C2).

X31 Plate, with - kG/x + hG = 0 (convection) at x = 0 and G = 0 (Dirichlet) at x = L.
a. For small values of (t - )/L2 0.022 use the following approximation:
 GX31(x, tx, ) [4(t - )]-1/2exp - +exp - - exp - - exp + x erfc + (t - )

b. For any value of (t - )/L2 but best for (t - )/L2 > 0.022:
 GX31(x, tx, ) = exp - x sin(1 - )sin(1 - ) where cot = - B and B = hL/k.

X32 Plate, with - kG/x + hG = 0 (convection) at x = 0 and G/x = 0 (Neumann) at x = L.
a. For small values of (t - )/L2 0.022 use the following approximation:
 GX32(x, tx, ) [4(t - )]-1/2exp - +exp - + exp - - exp + x erfc + (t - )

b. For any value of (t - )/L2 but best for (t - )/L2 > 0.022:
 GX32(x, tx, ) = exp - x cos(1 - )cos(1 - ) with tan = B and where B = hL/k.

X33 Plate, with - kG/x + h1G = 0 (convection) at x = 0 and kG/x + h2G = 0 (convection) at x = L.
a. For small values of (t - )/L2 0.022 use the following approximation:
 GX33(x, tx, ) [4(t - )]-1/2exp - +exp - + exp - - exp + x erfc + (t - ) - exp + x erfc + (t - )

b. For any value of (t - )/L2 but best for (t - )/L2 > 0.022:
 GX33(x, tx, ) = exp - x cos(x/L) + B1sin(x/L) x with tan = and where B1 = , B2 = .

X34 Plate, with - kG/x + h2G = 0 (convection) at x = 0 and kG/x + (cb)2 G/t = 0 (thin film) at x = L.
 GX34(x, tx, ) = exp - where   Xm(x) = B1sin(x/L) + cos(x/L) Nm = + C2 + x + 2C2B1 + tan C2B12 - + B1 with tan = and where   B1 = , C2 = .

X35 Plate, with - kG/x + h1G = 0 (convection) at x = 0 and kG/x + h2G + (cb)2 G/t = 0 (thin film with convection) at x = L.
 GX35(x, tx, ) = exp - where   Xm(x) = B1sin(x/L) + cos(x/L) Nm = + C2 + x + 2C2B1 + tan C2B12 - + B1 with tan = and   B1 = , B2 = , C2 = .

X41 Plate, with - kG/x + (cb)1G/t = 0 (thin film) at x = 0 and G = 0 (Dirichlet) at x = L.
 GX41(x, tx, ) = exp - x sin(1 - )sin(1 - ) Nm = C1 + C1 + 1/2 where tan = 1/C1, C1 = .

X42 Plate, with - kG/x + (cb)1G/t = 0 (thin film) at x = 0 and G/x = 0 (Neumann) at x = L.
 GX42(x, tx, ) = + exp - where   Xm(x) = cos(x/L) - C1sin(x/L) N0 = 1 + C1,  Nm = C1 + C1 + 1/2 where cot = -1/C1, C1 = .

X51 Plate, with - kG/x + hG + (cb)1G/t = 0 (thin film) at x = 0 and G = 0 (Dirichlet) at x = L.
 GX51(x, tx, ) = exp - where   Xm(x) = Dmsin(x/L) + cos(x/L) Nm = Dm2 + Dm/ + 2C1 + 1 , Dm =   - C1 where tan = , C1 = , B =

X52 Plate, with - kG/x + hG + (cb)1G/t = 0 (thin film) at x = 0 and G/x = 0 (Neumann) at x = L.
 GX52(x, tx, ) = exp - where   Xm(x) = Dmsin(x/L) + cos(x/L) Nm = Dm2 + + 2C1 + 1, Dm =   - C1 where tan = Dm, C1 = , B =

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