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3.1 INTRODUCTION
In technological and scientific applications related to heat transfer, it is of-
ten used to solve the inverse of the heat conduction problem: the so-called
backward heat conduction problem (BHCP). This is also called retrospec-
tive heat conduction problem, and it is one of the cases in the general clas-
sification of inverse heat conduction problems [1]. It is the inverse of the
initial boundary value problem for the heat equation; and for this reason,
it is also called a final boundary value problem. This problem consists in
finding the initial temperature distribution given the final distribution. It is
an ill-posed problem since the solution does not have a continuous depen-
dence on the data [2], and it may have no solution at all [3]. Besides this, it
is a singular boundary value problem if the domain is unbounded as in the
present case. Many methods of solving this problem in bounded domains
can be found in the literature; regularization, mollification, and functional
methods are popular techniques [4]. Some of these methods may be used
together with additional special techniques to treat the case of unbounded
domains [5-10].
In this chapter, we present a fast and easy-to-implement method for
solving a BHCP in unbounded domains. This is a boundary element meth-
od that uses a new algorithm to compute the Fourier transform of quadrati-
cally integrable functions defi ned in
R
n
with no need of artifi cial boundary
conditions imposed at the computational domain or regularized solutions
that are frequently used; this algorithm is called extended Fourier trans-
form (XFT) [11] and has complexity
O
(
N
n
logN
) in
n
dimensions when
the same number of nodes
N
is taken for all dimensions. An accurate and
weakly stable numerical solution of the BCHP in unbounded domains can
be obtained by applying this algorithm, and the convolution theorem to the
integral transform that gives the solution of the direct problem.
In Section 3.2, we fi rst describe the basic tools by solving the BHCP in
R
n
. In Section 3.3, we restrict
n
= 2, 3 and give some examples of BCHPs.
At the end of the section, we show how this technique can also be applied
to other kind of inverse heat conduction problems.
3.2
THE PROBLEM AND SOLUTION
Consider the function
u
(
x, t
) where
and
t
is a real number. The in-
verse problem to solve is the following: given
f
(
x
), find
u
(
x,
0) such that
x
n
∈
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