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Small-time GF, transient cases XIJ


This section was written by J. V. Beck and R. McMasters as part of a code verification project.


Summary of tables; full tables given below.
Table Contents
1A Small-Time GF
1B Small-Time GF, compact form
2A Boundary GF evaluated at n' = 0
2B Derivatives of boundary GF evaluated at n' = 0
3A Boundary GF evaluated at n' = L
3B Derivative of boundary GF evaluated at n' = L
4A Integrals of GF w.r.t. x
4B Derivative w.r.t. x of integrals of GF




Notation used in tables.
Fundamental heat conduction solution:

K(w) = $\displaystyle {\frac{1}{(4 \pi \alpha u)^{1/2}}}$e-w2/(4$\scriptstyle \alpha$u) (1)

The following function is needed for convective boundary conditions:

Hi(w) = exp$\displaystyle \left[\vphantom{\frac{h_i w}{k}+\frac{h_i^2}{k^2} \alpha u }\right.$$\displaystyle {\frac{h_i w}{k}}$ + $\displaystyle {\frac{h_i^2}{k^2}}$$\displaystyle \alpha$u$\displaystyle \left.\vphantom{\frac{h_i w}{k}+\frac{h_i^2}{k^2} \alpha u }\right]$erfc$\displaystyle \left[\vphantom{\frac{w}{\sqrt{4 \alpha u}}+\frac{h_i}{k} \sqrt{\alpha u } }\right.$$\displaystyle {\frac{w}{\sqrt{4 \alpha u}}}$ + $\displaystyle {\frac{h_i}{k}}$$\displaystyle \sqrt{\alpha u }$ $\displaystyle \left.\vphantom{\frac{w}{\sqrt{4 \alpha u}}+\frac{h_i}{k} \sqrt{\alpha u } }\right]$;    i = 0,  L (2)

Complentary error function (short notation):

E(w) = erfc$\displaystyle \left[\vphantom{\frac{w}{\sqrt{4 \alpha u}} }\right.$$\displaystyle {\frac{w}{\sqrt{4 \alpha u}}}$ $\displaystyle \left.\vphantom{\frac{w}{\sqrt{4 \alpha u}} }\right]$ (3)

Note that in the above definitions the dependence on time variable u is suppressed. For $ \alpha$u/L2 < 0.05 the values given in the following tables are accurate to 9 significant digits.




Table 1A. Small time Green's functions. The GX00, GX10, GX20 and GX30 cases are exact. The other cases include up to two reflections. The expressions for single reflections are obtained by dropping the terms which are function of either 2L - x + x' or 2L + x - x. Expressions for the y-direction are found by replacing x by y, x' by y' and L by W. Similar results are obtained for the z-direction by using z, z' and H.

Number Equation

1A.X00

 GX10(x, x', u) = K(x - x')

1A.X10

 GX10(x, x', u) = K(x - x') - K(x + x')

1A.X11

 GX11(x, x', u) $ \approx$ K(x - x') - K(x + x') - K(2L - x - x')
  + K(2L - x + x') + K(2L + x - x')

1A.X12

 GX12(x, x', u) $ \approx$ K(x - x') - K(x + x') + K(2L - x - x')
  - K(2L - x + x') - K(2L + x - x')

1A.X13

 GX13(x, x', u) $ \approx$ K(x - x') - K(x + x') + K(2L - x - x')
  - K(2L - x + x') - K(2L + x - x') - (hL/k)HL(2L - x - x')
  + (hL/k)HL(2L - x + x') + (hL/k)HL(2L + x - x')

1A.X20

 GX20(x, x', u) = K(x - x') + K(x + x')

1A.X21

 GX21(x, x', u) $ \approx$ K(x - x') + K(x + x') - K(2L - x - x')
  - K(2L - x + x') - K(2L + x - x')

1A.X22

 GX22(x, x', u) $ \approx$ K(x - x') + K(x + x') + K(2L - x - x')
  + K(2L - x + x') + K(2L + x - x')

1A.X23

 GX23(x, x', u) $ \approx$ K(x - x') + K(x + x') + K(2L - x - x')
  - (hL/k)HL(2L - x - x') + K(2L - x + x') + K(2L + x - x')
  - (hL/k)HL(2L - x + x') - (hL/k)HL(2L + x - x')

1A.X30

 GX30(x, x', u) = K(x - x') + K(x + x') - (h0/k)H0(x + x')

1A.X31

 GX31(x, x', u) $ \approx$ K(x - x') + K(x + x') - (h0/k)H0(x + x')
  - K(2L - x - x') - K(2L - x + x') - K(2L + x - x')
  + (h0/k)H0(2L - x + x') + (h0/k)H0(2L + x - x')

1A.X32

 GX32(x, x', u) $ \approx$ K(x - x') + K(x + x') - (h0/k)H0(x + x')
  + K(2L - x - x') + K(2L - x + x') + K(2L + x - x')
  - (h0/k)H0(2L - x + x') - (h0/k)H0(2L + x - x')

1A.X33a

 GX33(x, x', u) $ \approx$ K(x - x') + K(x + x') - (h0/k)H0(x + x') +
  K(2L - x - x') - (hL/k)HL(2L - x - x') + K(2L - x + x')
  + K(2L + x - x') - (h0/k)H0(2L - x + x')
  - (h0/k)H0(2L + x - x') - (hL/k)HL(2L - x + x')
  - (hL/k)HL(2L + x - x') + JH(L, x, x')
  where for h0 $ \neq$ hL,

1A.X33b

 JH(L, x, x') = (2h0hL/(k(hL - h0)))[H0(2L - x + x')
  + H0(2L + x - x') - HL(2L - x + x') - HL(2L + x - x')],
  and for h0 = hL

1A.X33c

 JH(L, x, x') = (2h02/k2){4$ \alpha$u[K(2L - x + x') + K(2L + x - x')]
  - (2L - x + x' + 2(h0/k)$ \alpha$u)H0(2L - x + x')
  - (2L + x - x' + 2(h0/k)$ \alpha$u)H0(2L + x - x')}


Table 1B. Small time Green's functions in compact form.
Number Equation

1B.X10

 GX10(x, x', u) = K(x - x') - K(x + x')
1B.X11 GX11(x, x', u) $ \approx$ GX10(x, x', u) - $ \Delta$GX12,
  $ \Delta$GX12 $ \equiv$ K(2L - x - x') - K(2L - x + x') - K(2L + x - x')

1B.X12

 GX12(x, x', u) $ \approx$ GX10(x, x', u) + $ \Delta$GX12

1B.X13

 GX13(x, x', u) $ \approx$ GX12(x, x', u) - $ \Delta$GX13,
  $ \Delta$GX13 $ \equiv$ (hL/k)[HL(2L - x - x') - HL(2L - x + x')
  - HL(2L + x - x')]

1B.X20

 GX20(x, x', u) = K(x - x') + K(x + x')

1B.X21

 GX21(x, x', u) $ \approx$ GX20(x, x', u) - $ \Delta$GX22,
  $ \Delta$GX22 $ \equiv$ K(2L - x - x') + K(2L - x + x') + K(2L + x - x')

1B.X22

 GX22(x, x', u) $ \approx$ GX20(x, x', u) + $ \Delta$GX22    (see 1B.X21)

1B.X23

 GX23(x, x', u) $ \approx$ GX22(x, x', u) - $ \Delta$GX23,
  $ \Delta$GX23 $ \equiv$ (hL/k)[HL(2L - x - x') + HL(2L - x + x')
  + HL(2L + x - x')]

1B.X31

 GX31(x, x', u) $ \approx$ GX21(x, x', u) - $ \Delta$GX31,
  $ \Delta$GX31 $ \equiv$ (h0/k)[H0(x + x') - H0(2L - x + x') - H0(2L + x - x')]

1B.X32

 GX32(x, x', u) $ \approx$ GX22(x, x', u) - $ \Delta$GX32,
  $ \Delta$GX32 $ \equiv$ (h0/k)[H0(x + x') + H0(2L - x + x') + H0(2L + x - x')]

1B.X33

 GX33(x, x', u) $ \approx$ GX32(x, x', u) - $ \Delta$GX23 + JH(L, x, x')
  (see Table 1A, entry X33, for JH)
Some Limiting Relations for Table 1B.
-As hi goes to zero, Hi(z) goes to E(z). As hi goes to infinity, Hi(z) goes to 0.
-As hi goes to zero, (hi/k)Hi(z) goes to 0. As hi goes to infinity, (hi/k)Hi(z) goes to 2K(z).
-As hi goes to infinity, (hi/k)[(hi/k)Hi(z) - 2K(z)] goes to -2zK(z)/(2$ \alpha$u).
-As h0 goes to zero, JH(L, x, x') goes to 0. As h0 goes to infinity, JH(L, x, x') goes to 2(hL/k)[HL(2L - x + x') + HL(2L + x - x')].
-As hL goes to zero, JH(L, x, x') goes to 0. As hL goes to infinity, JH(L, x, x') goes to 2(h0/k)[H0(2L - x + x') + H0(2L + x - x')].
-As u goes to 0 for x $ \neq$ 0, E(z) goes to 0 and Hi(z) goes to 0.



Table 2A. Boundary Green's functions for boundary conditions of the 2nd and 3rd kinds and negative of the derivative of the Green's function (with respect to the outward pointing normal) for the 1st kind of boundary condition. Evaluated at x' = 0. (Notice for this boundary n' = x'.) Simpler and still accurate equations are given if the functions of 2L + x are dropped.

Number Equation
2A.X10 - $ \partial$GX10(x, 0, u)/$ \partial$n' = ($ \alpha$u)-1xK(x)

2A.X11

 - $ \partial$GX11(x, 0, u)/$ \partial$n' $ \approx$ (2$ \alpha$u)-1[2xK(x)
  -2(2L - x)K(2L - x) + (2L + x)K(2L + x)],    x $ \neq$ 0

2A.X12

 - $ \partial$GX12(x, 0, u)/$ \partial$n' $ \approx$ (2$ \alpha$u)-1[2xK(x) +
  2(2L - x)K(2L - x) - (2L + x)K(2L + x)],    x $ \neq$ 0

2A.X13

 - $ \partial$GX13(x, 0, u)/$ \partial$n' $ \approx$ (2$ \alpha$u)-1[2xK(x) + 2(2L - x)K(2L - x)
  - (2L + x)K(2L + x)] + 2(hL/k)2HL(2L - x) - (4hL/k)K(2L - x)
  - (hL/k)2HL(2L + x) + (2hL/k)K(2L + x),    x $ \neq$ 0

2A.X20

 GX20(x, 0, u) = 2K(x)

2A.X21

 GX21(x, 0, u) $ \approx$ 2K(x) - 2K(2L - x) - K(2L + x)

2A.X22

 GX22(x, 0, u) $ \approx$ 2K(x) + 2K(2L - x) + K(2L + x)

2A.X23

 GX23(x, 0, u) $ \approx$ 2K(x) + 2K(2L - x) + K(2L + x)
  -2(hL/k)HL(2L - x) - (hL/k)HL(2L + x)

2A.X30

 GX30(x, 0, u) = 2K(x) - (h0/k)H0(x)

2A.X31

 GX31(x, 0, u) $ \approx$ 2K(x) - 2K(2L - x) - K(2L + x)
  - (h0/k)H0(x) + (h0/k)H0(2L - x) + (h0/k)H0(2L + x)

2A.X32

 GX32(x, 0, u) $ \approx$ 2K(x) + 2K(2L - x) + K(2L + x)
  - (h0/k)H0(x) - (h0/k)H0(2L - x) - (h0/k)H0(2L + x)

2A.X33a

 GX33(x, 0, u) $ \approx$ 2K(x) + 2K(2L - x) + K(2L + x)
  - (h0/k)H0(x) - (h0/k)H0(2L - x) - 2(hL/k)HL(2L - x)
  - (h0/k)H0(2L + x) - (hL/k)HL(2L + x) + JH(L, x, 0)

where for h0 $ \neq$ hL,

2A.X33b

 JH(L, x, 0) = (2h0hL/(k(hL - h0)))[H0(2L - x) + H0(2L + x)
  - HL(2L - x) - HL(2L + x)]

and for h0 = hL,

2A.X33b

 JH(L, x, 0) = (2h02/k2){2($ \alpha$u/$ \pi$)1/2(exp[- (2L - x)2/(4$ \alpha$u)
  + exp[- (2L + x)2/(4$ \alpha$u)]) - [2L - x + 2(h0/k)$ \alpha$u]H0(2L - x)
  - [2L + x + 2(h0/k)$ \alpha$u]H0(2L + x)}
Notes for Table 2A. If the surface temperature is given (x = 0 surface is nonhomogeneous), that is for X11, X12 and X13, we need not calculate the surface temperature since it is known. Notice

I = $\displaystyle \alpha$ $\displaystyle \int^{t}_{u=0}$$\displaystyle {\frac{x}{\alpha u}}$K(x, u)  du = $\displaystyle \alpha$ erfc$\displaystyle \left[\vphantom{ \frac{x}{(4\alpha t)^{1/2}} }\right.$$\displaystyle {\frac{x}{(4\alpha t)^{1/2}}}$ $\displaystyle \left.\vphantom{ \frac{x}{(4\alpha t)^{1/2}} }\right]$

as x goes to 0, I goes to $ \alpha$, which clearly is not zero. Also note that as x goes to 0, contributions to the integral only occurs for very small values of u. This means that

B = $\displaystyle \lim_{x\rightarrow 0}^{}$$\displaystyle \int^{t}_{u=0}$$\displaystyle {\frac{x}{\alpha u}}$K(x, u)  C(u)  du = C(0)

Some limiting conditions:
-As hi goes to zero, Hi(z) goes to E(z). As hi goes to infinity, Hi(z) goes to 0.
-As hi goes to zero, (hi/k)Hi(z) goes to 0. As hi goes to infinity, (hi/k)Hi(z) goes to 2K(z).
-As hi goes to infinity, (hi/k)[(hi/k)Hi(z) - 2K(z)] goes to -2zK(z)/(2$ \alpha$u).


Table 2B. Derivative wrt x of Green's functions for boundary conditions of the 2nd and 3rd kinds and negative of mixed second partial derivative wrt x and n' of the Green's function for the 1st kind of boundary condition. Small time forms. All evaluated at n' = 0. Simpler and still accurate equations are given if the functions of 2L + x and 3L - x are dropped.
Number Equation
2B.X10 - $ \partial^{2}_{}$GX10(x, 0, u)/$ \partial$x$ \partial$n' = ($ \alpha$u)-1K(x)[1 - x2(2$ \alpha$u)-1]

2B.X11

 - $ \partial^{2}_{}$GX11(x, 0, u)/$ \partial$x$ \partial$n' $ \approx$ ($ \alpha$u)-1[K(x)(1 - x2(2$ \alpha$u)-1)
  + K(2L - x)(1 - (2L - x)2(2$ \alpha$u)-1) + (1/2)K(2L + x)(1 - (2L + x)2(2$ \alpha$u)-1)]

2B.X12

 - $ \partial^{2}_{}$GX12(x, 0, u)/$ \partial$x$ \partial$n' $ \approx$ ($ \alpha$u)-1[K(x)(1 - x2(2$ \alpha$u)-1)
  - K(2L - x)(1 - (2L - x)2(2$ \alpha$u)-1) - (1/2)K(2L + x)(1 - (2L + x)2(2$ \alpha$u)-1)]

2B.X13

 - $ \partial^{2}_{}$GX13(x, 0, u)/$ \partial$x$ \partial$n' $ \approx$ ($ \alpha$u)-1[K(x)(1 - x2(2$ \alpha$u)-1)
  - K(2L - x)(1 - (2L - x)2(2$ \alpha$u)-1)] + 4(hL/k)K(2L - x)[(hL/k)
  - (2L - x)(2$ \alpha$u)-1] + 2(hL/k)K(2L + x)[(hL/k) - (2L + x)(2$ \alpha$u)-1]
  -2(hL/k)3HL(2L - x) - (hL/k)3HL(2L + x)

2B.X20

 $ \partial$GX20(x, 0, u)/$ \partial$x = - ($ \alpha$u)-1xK(x); see integral at bottom of Table 2A.

2B.X21

 $ \partial$GX21(x, 0, u)/$ \partial$x $ \approx$ - ($ \alpha$u)-1[xK(x) + (2L - x)K(2L - x)
  - (1/2)(2L + x)K(2L + x)]

2B.X22

 $ \partial$GX22(x, 0, u)/$ \partial$x $ \approx$ - ($ \alpha$u)-1[xK(x) - (2L - x)K(2L - x)
  + (1/2)(2L + x)K(2L + x)]

2B.X23

 $ \partial$GX23(x, 0, u)/$ \partial$x $ \approx$ - ($ \alpha$u)-1[xK(x) - (2L - x)K(2L - x)
  + (1/2)(2L + x)K(2L + x)] - 4(hL/k)K(2L - x) + 2(hL/k)2HL(2L - x)
  +2(hL/k)K(2L + x) - (hL/k)2HL(2L + x)

2B.X30

 $ \partial$GX30(x, 0, u)/$ \partial$x = - ($ \alpha$u)-1xK(x) + 2(h0/k)K(x) - (h0/k)2H0(x)

2B.X31

 $ \partial$GX31(x, 0, u)/$ \partial$x $ \approx$ - ($ \alpha$u)-1[xK(x) + (2L - x)K(2L - x)
  - (1/2)(2L + x)K(2L + x)] + 2(h0/k)[K(x) + K(2L - x)
  - K(2L + x)] - (h0/k)2[H0(x) + H0(2L - x) - H0(2L + x)]

2B.X32

 $ \partial$GX32(x, 0, u)/$ \partial$x $ \approx$ - ($ \alpha$u)-1[xK(x) - (2L - x)K(2L - x)
  + (1/2)(2L + x)K(2L + x)] + 2(h0/k)[K(x) - K(2L - x) + K(2L + x)]
  - (h0/k)2[H0(x) - H0(2L - x) + H0(2L + x)]

2B.X33a

 $ \partial$GX33(x, 0, u)/$ \partial$x $ \approx$ - ($ \alpha$u)-1[xK(x)
  - (2L - x)K(2L - x) + (1/2)(2L + x)K(2L + x)] + 2(h0/k)[K(x)
  - K(2L - x) + K(2L + x)] - (h0/k)2[H0(x) - H0(2L - x)
  + H0(2L + x)] - 4(hL/k)K(2L - x) + 2(hL/k)K(2L + x)
  + (hL/k)2[2HL(2L - x) - HL(2L + x)] + DJH(L, x, 0)
  where for h0 $ \neq$ hL,
2B.X33b DJH(L, x, 0) = (2h0hL/(k(hL - h0)))[- (h0/k)(H0(2L - x)
  - H0(2L + x)) + (hL/k)(HL(2L - x) - HL(2L + x)]

and for h0 = hL,

2B.X33c

 JH(L, x, 0) = (2h02/k2){4(h0/k)$ \alpha$u[- K(2L - x) + K(2L + x)]
  + [1 + (h0/k)(2L - x) + 2(h0/k)2$ \alpha$u]H0(2L - x)
  - [1 + (h0/k)(2L + x) + 2(h0/k)2$ \alpha$u]H0(2L + x)}


Table 3A. Boundary Green's functions for boundary conditions of the 2nd and 3rd kinds and negative of the derivative of the Green's function with respect to n' for the 1st kind of boundary condition. Small time forms and evaluated at n' = x' = L.
Number Equation
3A.X11 - $ \partial$GX11(x, L, u)/$ \partial$n' $ \approx$ (2$ \alpha$u)-1[2(L - x)K(L - x)
  -2(L + x)K(L + x) + (3L - x)K(3L - x)],    x $ \neq$ L

3A.X12

 GX12(x, L, u) $ \approx$ 2K(L - x) - 2K(L + x) - K(3L - x)

3A.X13

 GX13(x, L, u) $ \approx$ 2K(L - x) - 2K(L + x) - K(3L - x)
  - (hL/k)HL(L - x) + (hL/k)HL(L + x) + (hL/k)HL(3L - x)

3A.X21

 - $ \partial$GX21(x, L, u)/$ \partial$n' $ \approx$ (2$ \alpha$u)-1[2(L - x)K(L - x)
  +2(L + x)K(L + x) - (3L - x)K(3L - x)],    x $ \neq$ L
3A.X22 GX22(x, L, u) $ \approx$ 2K(L - x) + 2K(L + x) + K(3L - x)

3A.X23

 GX23(x, L, u) $ \approx$ 2K(L - x) + 2K(L + x) + K(3L - x)
  - (hL/k)[HL(L - x) + HL(L + x) + HL(3L - x)]

3A.X31

 - $ \partial$GX31(x, L, u)/$ \partial$n' $ \approx$ (2$ \alpha$u)-1[2(L - x)K(L - x)
  +2(L + x)K(L + x) - (3L - x)K(3L - x)] - 4(h0/k)K(L + x)
  +2(h0/k)2H0(L + x) + 2(h0/k)K(3L - x)
  - (h0/k)2H0(3L - x),    x $ \neq$ L

3A.X32

 GX32(x, L, u) $ \approx$ 2K(L - x) + 2K(L + x) + K(3L - x)
  -2(h0/k)H0(L + x) - (h0/k)H0(3L - x)

3A.X33a

 GX33(x, L, u) $ \approx$ 2K(L - x) + 2K(L + x) + K(3L - x)
  -2(h0/k)H0(L + x) - (h0/k)H0(3L - x) - (hL/k)HL(L - x)
  - (hL/k)HL(L + x) - (hL/k)HL(3L - x) + JH(L, x, L)
  where for h0 $ \neq$ hL,

3A.X33b

 JH(L, x, L) $ \approx$ (2h0hL/(k(hL - h0)))[H0(3L - x) + H0(L + x)
  - HL(3L - x) - HL(L + x)]
  and for h0 = hL,

3A.X33c

 JH(L, x, x') $ \approx$ (2h02/k2)[4$ \alpha$u[K(3L - x) + K(L + x)]
  - (3L - x + 2(h0/k)$ \alpha$u)H0(3L - x)
  - (L + x + 2(h0/k)$ \alpha$u)H0(L + x)]


Table 3B. Derivative wrt x of Green's functions for Boundary Conditions of the 2nd and 3rd Kinds and Negative of Mixed Second Partial Derivative wrt x and n' of the Green's function for the 1st Kind of Boundary Condition. Small Time Forms. All evaluated at n' = L. Simpler and still accurate equations are given if the functions of 2L + x and 3L - x are dropped.
Number Equation
3B.X11 - $ \partial^{2}_{}$GX11(x, L, u)/$ \partial$x$ \partial$n' $ \approx$ - ($ \alpha$u)-1[K(L - x)(1 - (L - x)2(2$ \alpha$u)-1)
  + K(L + x)(1 - (L + x)2(2$ \alpha$u)-1)
  + (1/2)K(3L - x)(1 - (3L - x)2(2$ \alpha$u)-1)]

3B.X12

 $ \partial$GX12(x, L, u)/$ \partial$x $ \approx$ ($ \alpha$u)-1[(L - x)K(L - x)
  + (L + x)K(L + x) - (1/2)(3L - x)K(3L - x)]

3B.X13

 $ \partial$GX13(x, L, u)/$ \partial$x $ \approx$ ($ \alpha$u)-1[(L - x)K(L - x) + (L + x)K(L + x)
  - (1/2)(3L - x)K(3L - x)] - 2(h0/k)[K(L - x) + K(L + x)
  - K(3L - x)] + (h0/k)2[H0(L - x) + H0(L + x) - H0(3L - x)]

3B.X21

 - $ \partial^{2}_{}$GX21(x, L, u)/$ \partial$x$ \partial$n' $ \approx$ - ($ \alpha$u)-1[K(L - x)(1 - (L - x)2(2$ \alpha$u)-1)
  - K(L + x)(1 - (L + x)2(2$ \alpha$u)-1)
  - (1/2)K(3L - x)(1 - (3L - x)2(2$ \alpha$u)-1)]

3B.X22

 $ \partial$GX22(x, L, u)/$ \partial$x $ \approx$ ($ \alpha$u)-1[(L - x)K(L - x)
  - (L + x)K(L + x) + (1/2)(3L - x)K(3L - x)]

3B.X23

 $ \partial$GX23(x, L, u)/$ \partial$x $ \approx$ ($ \alpha$u)-1[(L - x)K(L - x) - (L + x)K(L + x)
  + (1/2)(3L - x)K(3L - x)] - 2(hL/k)[K(L - x) - K(L + x)
  + K(3L - x)] + (hL/k)2[HL(L - x) - HL(L + x) + HL(3L - x)]

3B.X31

 - $ \partial^{2}_{}$GX31(x, L, u)/$ \partial$x$ \partial$n' $ \approx$ - ($ \alpha$u)-1[K(L - x)(1 - (L - x)2(2$ \alpha$u)-1)
  - K(L + x)(1 - (L + x)2(2$ \alpha$u)-1)
  - (1/2)K(3L - x)(1 - (3L - x)2(2$ \alpha$u)-1)]
  -4(h0/k)K(L + x)[(hL/k) - (L + x)(2$ \alpha$u)-1]
  -2(hL/k)K(3L - x)[(hL/k) - (3L - x)(2$ \alpha$u)-1]
  +2(hL/k)3HL(L + x) + (hL/k)3HL(3L - x)

3B.X32

 $ \partial$GX32(x, L, u)/$ \partial$x $ \approx$ ($ \alpha$u)-1[(L - x)K(L - x) - (L + x)K(L + x)
  + (1/2)(3L - x)K(3L - x)] + 2(h0/k)[2K(L + x) - K(3L - x)]
  - (h0/k)2[2H0(L + x) - H0(3L - x)]

3B.X33a

 $ \partial$GX33(x, L, u)/$ \partial$x $ \approx$ ($ \alpha$u)-1[(L - x)K(L - x) - (L + x)K(L + x)
  + (1/2)(3L - x)K(3L - x)] + 2(h0/k)[2K(L + x) - K(3L - x)]
  - (h0/k)2[2H0(L + x) - H0(3L - x)] - 2(hL/k)[K(L - x)
  - K(L + x) + K(3L - x)] + (hL/k)2[HL(L - x) - HL(L + x)
  + HL(3L - x)] + DJH(L, x, L)
  where for h0 $ \neq$ hL,

3B.X33b

 DJH(L, x, L) = (2h0hL/(k(hL - h0)))[(h0/k)(H0(L + x)
  - H0(3L - x)) - (hL/k)(HL(L + x) - HL(3L - x)]

and for h0 = hL,

3B.X33c

 DJH(L, x, L) = (2h02/k2){4(h0/k)$ \alpha$u[K(L + x) - K(3L - x)]
  - [1 + (h0/k)(L + x) + 2(h0/k)2$ \alpha$u]H0(L + x)
  + [1 + (h0/k)(3L - x) + 2(h0/k)2$ \alpha$u]H0(3L - x)}


Table 4A. Integrals of small time form of Green's functions, from x' = 0 to L for the finite cases and from x' = 0 to $ \infty$ for the XI0 cases. The X10, X20, X22 and X30 results are exact. Note that erfc (2/(4 . 0.05)1/2) $ \approx$ 2.54E - 10 and erfc (3/(4 . 0.05)1/2) $ \approx$ 2.38E - 21 so for dimensionless times less than 0.05 the E(2L + x) and E(3L - x) terms can be dropped. Note that we define

E(z) = erfc$\displaystyle \left[\vphantom{ \frac{z}{\sqrt{4 \alpha u}} }\right.$$\displaystyle {\frac{z}{\sqrt{4 \alpha u}}}$ $\displaystyle \left.\vphantom{ \frac{z}{\sqrt{4 \alpha u}} }\right]$

Also we observe that Hi(z) $ \leq$ E(z) for z > 0. As u $ \rightarrow$ 0 and x is nonzero, E(x) $ \rightarrow$ 0 and H(x) $ \rightarrow$ 0. For every boundary condition except the first, the integrated Green's function goes to unity as u $ \rightarrow$ 0 for all x.
Number Equation
4A.X10 $ \int^{\infty}_{}$GX10(x, x', u)dx' = 1 - E(x) = erf[x/(4$ \alpha$u)1/2]

4A.X11

 $ \int^{L}_{}$GX11(x, x', u)dx' $ \approx$ 1 - E(x) - E(L - x) + E(L + x) + E(2L - x) -
  (1/2)[E(2L + x) + E(3L - x)]

4A.X12

 $ \int^{L}_{}$GX12(x, x', u)dx' $ \approx$ 1 - E(x) - E(2L - x) + (1/2)[E(2L + x)
  + E(3L - x)]

4A.X13

 $ \int^{L}_{}$GX13(x, x', u)dx' $ \approx$ 1 - E(x) - E(L - x) + E(L + x)
  + E(2L - x) + HL(L - x) - HL(L + x) - 2HL(2L - x)
  - (1/2)[E(2L + x) + E(3L - x)] + HL(2L + x) + HL(3L - x)

4A.X20

 $ \int^{\infty}_{}$GX20(x, x', u)dx' = 1

4A.X21

 $ \int^{L}_{}$GX21(x, x', u)dx' $ \approx$ 1 - E(L - x) - E(L + x) + (1/2)[E(2L + x)
  + E(3L - x)]

4A.X22

 $ \int^{L}_{}$GX22(x, x', u)dx' $ \approx$ 1

4A.X23

 $ \int^{L}_{}$GX23(x, x', u)dx' $ \approx$ 1 - E(L - x) - E(L + x)
  + HL(L - x) + HL(L + x) + (1/2)[E(2L + x) + E(3L - x)]
  - [HL(2L + x) + HL(3L - x)]

4A.X30

 $ \int^{\infty}_{}$GX30(x, x', u)dx' = 1 - E(x) + H0(x)

4A.X31

 $ \int^{L}_{}$GX31(x, x', u)dx' $ \approx$ 1 - E(x) - E(L - x) + E(L + x) + E(2L - x)
  + H0(x) - 2H0(L + x) - H0(2L - x) - (1/2)[E(2L + x) + E(3L - x)] +
  H0(2L + x) + H0(3L - x)

4A.X32

 $ \int^{L}_{}$GX32(x, x', u)dx' $ \approx$ 1 - E(x) - E(2L - x) + H0(x) + H0(2L - x)
  + (1/2)[E(2L + x) + E(3L - x)] - [H0(2L + x) + H0(3L - x)]

4A.X33a

 $ \int^{L}_{}$GX33(x, x', u)dx' $ \approx$ 1 - E(x) - E(L - x) - E(L + x) - E(2L - x)
  + H0(x) + H0(2L - x) + HL(L + x) + HL(L - x) + IJH(L, x)
  where for h0 $ \neq$ hL,

4A.X33b

 IJH(L, x) = 2E(L + x) + 2E(2L - x) - 2E(2L + x) - 2E(3L - x)
  - (2hL/(hL - h0))[H0(2L - x) + H0(L + x) - H0(2L + x)
  - H0(3L - x)] + (2h0/(hL - h0))[HL(2L - x) + HL(L + x)
  - HL(2L + x) - HL(3L - x)]

and where for h0 = hL = h,

4A.X33c

 IJH(L, x) = 2E(L + x) + 2E(2L - x) - 2E(2L + x) - 2E(3L - x)
  +2H0(2L - x)[- 1 + 2$ \alpha$u(h/k)2 + h(2L - x)/k]
  +2H0(L + x))[- 1 + 2$ \alpha$u(h/k)2 + h(L + x)/k]
  -2H0(2L + x)[- 1 + 2$ \alpha$u(h/k)2 + h(2L + x)/k]
  -2H0(3L - x)[- 1 + 2$ \alpha$u(h/k)2 + h(3L - x)/k]
  - (8h$ \alpha$u/k)[K(2L - x) + K(L + x) - K(2L + x) - K(3L - x)]


Table 4B. Derivative wrt x of integrals of small time form of Green's functions, from x' = 0 to L.
Number Equation

4B.X11

 $ \partial$[$ \int^{L}_{}$GX11(x, x', u)dx']/dx $ \approx$ 2[K(x) - K(L - x) - K(L + x)
  + K(2L - x)] + K(2L + x) - K(3L - x)

4B.X12

 $ \partial$[$ \int^{L}_{}$GX12(x, x', u)dx']/dx $ \approx$ 2[K(x) - K(2L - x)]
  - K(2L + x) + K(3L - x)

4B.X13

 $ \partial$[$ \int^{L}_{}$GX13(x, x', u)dx']/dx $ \approx$ 2[K(x) - K(2L - x)] - K(2L + x)
  + K(3L - x) - hL/k[HL(L - x) + HL(L + x) - 2HL(2L - x)
  - HL(2L + x) + HL(3L - x)]

4B.X21

 $ \partial$[$ \int^{L}_{}$GX21(x, x', u)dx']/dx $ \approx$ - 2K(L - x) + 2K(L + x)
  - K(2L + x) + K(3L - x)

4B.X22

 $ \partial$[$ \int^{L}_{}$GX22(x, x', u)dx']/dx = 0

4B.X23

 $ \partial$[$ \int^{L}_{}$GX23(x, x', u)dx']/dx $ \approx$ K(2L + x) - K(3L - x)
  - hL/k[HL(L - x) - HL(L + x) + HL(2L + x) - HL(3L - x)]

4B.X31

 $ \partial$[$ \int^{L}_{}$GX31(x, x', u)dx']/dx $ \approx$ - 2K(L - x) + 2K(L + x)
  - K(2L + x) + K(3L - x) + h0/k[H0(x) - 2H0(L + x)
  + H0(2L - x) + H0(2L + x) - H0(3L - x)]

4B.X32

 $ \partial$[$ \int^{L}_{}$GX32(x, x', u)dx']/dx $ \approx$ K(2L + x) - K(3L - x)
  + h0/k[H0(x) - H0(2L - x) - H0(2L + x) + H0(3L - x)]

4B.X33a

 $ \partial$[$ \int^{L}_{}$GX33(x, x', u)dx']/dx $ \approx$ K(2L + x) - K(3L - x)
  + h0/k[H0(x) - H0(2L - x) - H0(2L + x) + H0(3L - x)]
  - hL/k[HL(L - x) - HL(L + x) + HL(2L + x) - HL(3L - x)] + DIJH

where for h0 $ \neq$ hL,

4B.X33b

 DIJH = 2h0hL/k(hL - h0)[- H0(L + x) + H0(2L - x)
  + H0(2L + x) - H0(3L - x) + HL(L + x) - HL(2L - x)
  - HL(2L + x) + HL(3L - x)]

and for h0 = hL = h,

4B.X33c

 DIJH = 2h2/k2{4$ \alpha$u[- K(L + x) + K(2L - x) + K(2L + x)
  - K(3L - x)](L + x + 2h$ \alpha$u/k)H(L + x)
  - (2L - x + 2h$ \alpha$u/k)H(2L - x) - (2L + x + 2h$ \alpha$u/k)H(2L + x)
  + (3L - x + 2h$ \alpha$u/k)H(3L - x)]}

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Next: Cylindrical Coordinates. Transient 1-D. Up: Rectangular coordinates. Transient 1-D. Previous: Plate, transient 1-D.
Kevin D. Cole
2002-12-31