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the heating process and the estimation of heat flux using an inverse technique for
slab surface [2].
Different discretization methods [3–6] are available in the literature for solving the
direct problems. On the other hand, handling of inverse problems is very strenuous,
because these are ill-conditioned from mathematical point of view. Ill-conditioned
means small deviation of input data can cause large deviation on output value.
Generally, two categories of methods are available in literature one deterministic
and other stochastic for the solution of inverse problems [7]. Deterministic methods
are gradient-based methods, which are efficient and accurate. A stochastic method is
generally used for finding global solution, which usually consumes a large iteration
time. Least-squares method [8], genetic algorithm [9], Tikhonov regularization technique [10], particle swarm optimization method [11], Levenberg–Marquardt method
[12], and Kalman filter method [13] have been used in literature.
In this work, the MATLAB R2018a code has been developed based on the CGM
algorithm for inverse problems. The objective of the work is to show that the algorithm
is robust, efficient, and accurate, which is justified by comparing it with the actual
experiment. The sections of this article are arranged as follows. The one-dimensional
transient heat conduction problem is outlined in Sect. 2. The inverse problem is
outlined in Sect. 3. The conjugate gradient method (CGM) for optimization is outlined
in Sect. 4. The computation procedure for CGM is outlined in Sect. 5. The experiment
setup and results are outlined in Sect. 6. Finally, the conclusive remarks are given.
2 One-Dimensional Transient Heat Conduction Problem
The governing equations for one-dimensional transient heat conduction (direct)
problem are given as
ρc
∂ T (x, t)
∂t
=
∂
∂ x
k
∂ T (x, t)
∂ x
in the domain 0 < x < l, for t > 0
(1a)
T (x, 0) = T (x) in the domain 0 < x < l, at t = 0
(1b)
k
∂ T (0, t)
∂ x
= q(t) for t > 0, at x = 0
(1c)
T (l, t) = T (t) for t > 0, at x = l
(1d)
Here, k is the thermal property of the heated body; ρ is the density; l is the length
of the heated body; c is the mass-specific; x is the coordinate along the x-axis; T is the
temperature; t is the time; q is the heat flow. It can be shown that thermal properties
are not temperature-dependent. The solution of this problem is done using the finite
volume approach programmed in MATLAB. FVM is considered as an accurate and
P. Sathavara et al.
the heating process and the estimation of heat flux using an inverse technique for
slab surface [2].
Different discretization methods [3–6] are available in the literature for solving the
direct problems. On the other hand, handling of inverse problems is very strenuous,
because these are ill-conditioned from mathematical point of view. Ill-conditioned
means small deviation of input data can cause large deviation on output value.
Generally, two categories of methods are available in literature one deterministic
and other stochastic for the solution of inverse problems [7]. Deterministic methods
are gradient-based methods, which are efficient and accurate. A stochastic method is
generally used for finding global solution, which usually consumes a large iteration
time. Least-squares method [8], genetic algorithm [9], Tikhonov regularization technique [10], particle swarm optimization method [11], Levenberg–Marquardt method
[12], and Kalman filter method [13] have been used in literature.
In this work, the MATLAB R2018a code has been developed based on the CGM
algorithm for inverse problems. The objective of the work is to show that the algorithm
is robust, efficient, and accurate, which is justified by comparing it with the actual
experiment. The sections of this article are arranged as follows. The one-dimensional
transient heat conduction problem is outlined in Sect. 2. The inverse problem is
outlined in Sect. 3. The conjugate gradient method (CGM) for optimization is outlined
in Sect. 4. The computation procedure for CGM is outlined in Sect. 5. The experiment
setup and results are outlined in Sect. 6. Finally, the conclusive remarks are given.
2 One-Dimensional Transient Heat Conduction Problem
The governing equations for one-dimensional transient heat conduction (direct)
problem are given as
ρc
∂ T (x, t)
∂t
=
∂
∂ x
k
∂ T (x, t)
∂ x
in the domain 0 < x < l, for t > 0
(1a)
T (x, 0) = T (x) in the domain 0 < x < l, at t = 0
(1b)
k
∂ T (0, t)
∂ x
= q(t) for t > 0, at x = 0
(1c)
T (l, t) = T (t) for t > 0, at x = l
(1d)
Here, k is the thermal property of the heated body; ρ is the density; l is the length
of the heated body; c is the mass-specific; x is the coordinate along the x-axis; T is the
temperature; t is the time; q is the heat flow. It can be shown that thermal properties
are not temperature-dependent. The solution of this problem is done using the finite
volume approach programmed in MATLAB. FVM is considered as an accurate and