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FIGURE 3.6 (A) Numerical solution for the heat flux as given by Eq. (15). The exact
function (solid lines) is compared with the output of Eq. (16) with N = 215 (dashed lines).
The boundary points are x = 0 and x =1/3. (B) Plot of the error |q ( t k ) −q k |.
3.4 CONCLUSION
The present technique yields accurate numerical results whenever the
Fourier transform can be applied to solve a particular problem. For in-
stance, the standard Cauchy problem for the heat equation given in Eq.
(2) can be solved by using this technique for a large number of values of
t and α with numerical stability, and the inhomogeneous heat equation
can be attempted to be solved in the same way with an extra dimension
for the time integral corresponding to the inhomogeneous term. However,
this is not the case for the inverse problem (1) since, as noted earlier, the
exponential term in Eq. (9) becomes greater as the backward time t , N , or
n are increased. A similar situation occurs in the last example of the previ-
ous section since
(
)
()
becomes greater as or N is
increased. Therefore, in the cases where these parameters are large, the
method is unable to give a result.
1/
G
ωω
=
exp
2
ω
KEYWORDS
Boundary element method
Fast Fourier transform
Gauss transform
Heat equation
Inverse problems
XFT
 
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