Heat Transfer Asian Research
A Novel Effective Approach for Solving Nonlinear Heat Transfer
Equations
H. Aminikhah1 and M. Hemmatnezhad2
1
Department of Mathematics, Faculty of Sciences, University of Guilan, Rasht, Iran
2
Faculty of Mechanical Engineering, Takestan Branch, Islamic Azad University, Takestan, Iran
In this paper, a novel analytic technique, namely the Laplace transform new
homotopy perturbation method (LTNHPM), is applied for solving the nonlinear
differential equations arising in the field of heat transfer. This approach is a new
modification to the homotopy perturbation method based on the Laplace transform.
Unlike the previous approach implemented by the present authors for these problems,
the present method does not consider the initial approximation as a power series. The
nonlinear convective radiative cooling equation and nonlinear equation of conduction
heat transfer with the variable physical properties are chosen as illustrative examples.
The exact solution has been found for the first case and for the others; results with
remarkable accuracy have been achieved which verify the efficiency as well as
accuracy of the presented approach. © 2012 Wiley Periodicals, Inc. Heat Trans Asian
Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj).
DOI 10.1002/htj.20411
Key words: homotopy perturbation method, Laplace transform, nonlinear
differential equations, heat transfer
1. Introduction
In the last few years, analytical asymptotic techniques for solving nonlinear problems have
attracted extensive research activities by engineers and mathematicians. Recently, many new analyti-
cal techniques have been widely applied to a wide class of nonlinear differential equations arising in
various fields of science. One of these methods which attracted great attention due to its versatility
and straightforwardness is the homotopy perturbation method (HPM). A large number of publications
have been hitherto conducted on the application of this technique since it was first introduced by He
in 1992 [1 14].
In this method, which is based upon the Taylor series expansion, the linear and nonlinear
problem is transferred to an infinite number of sub-problems and then the solution is approximated
by the sum of the solutions of the first several sub-problems. To date, several modifications have been
given for this scheme in order to improve its efficiency as well as accuracy [15 20]. Recently,
Aminikhah and his co-associates proposed a new form of the homotopy perturbation method (NHPM)
to solve the ordinary differential equations [21]. In their approach, the solution is considered as an
© 2012 Wiley Periodicals, Inc.
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infinite series which converges rapidly to the exact solutions. Afterwards, they solved the system of
ordinary differential equations [22], nonlinear Blasius equation [23], quadratic Riccati differential
equation [24], stiff systems of ordinary differential equations [25], and the nonlinear equations arising
in heat transfer problems [26 28] using the aforementioned technique.
In the present work, a new modification of HPM based on the Laplace transform is proposed,
which we call the Laplace transform new homotopy perturbation method (LTNHPM) and then is
applied to the nonlinear heat transfer differential equations. The obtained results in comparison with
the exact solutions are in excellent agreement. As would be observed, the method is effective and
simple to apply and it is a promising method for solving several other differential equations in various
fields of study.
Nomenclature
A: general differential operator
f(r): known analytic function
H(U, p): homotopy expression
L: linear operator
L: Laplace transform operator
N: nonlinear operator
p: homotopy embedding parameter
u: approximate solution
u0: initial approximation
Greek Symbols
¸: dimensionless operator
Ä: dimensionless time
µ: small parameter
2. Basic Ideas of the LTNHPM
To illustrate the basic ideas of this method, let us consider the following nonlinear differential
equation
(1)
with the following initial conditions
(2)
where A is a general differential operator and f(r) is a known analytical function. The operator A can
be divided into two parts, L and N, where L is a linear and N is a nonlinear operator. Therefore, Eq.
(1) can be rewritten as
(3)
Based on NHPM [22], we construct a homotopy U(r, p) : &! × [0, 1] R, which satisfies
(4)
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or equivalently,
(5)
where p " [0, 1] is an embedding parameter and u0 is an initial approximation for the solution of Eq.
(1). Clearly, Eqs. (4) and (5) give
(6)
(7)
Applying the Laplace transform to both sides of Eq. (5), we arrive at
(8)
Using the differential property of the Laplace transform we have
(9)
or
(10)
Finally, applying the inverse Laplace transform to both sides of Eq. (10), one can successfully
reach the following
(11)
According to the HPM, we can first use the embedding parameter p as a small parameter, and
assume that the solutions of Eq. (11) can be represented as a power series in p as
(12)
Now let us rewrite Eq. (11) using Eq. (12) as
(13)
Therefore, equating the coefficients of p with the same power leads to
(14)
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Suppose that the initial approximation has the form U(0) = u0 = Ä…0, U2 (0) = Ä…1, . . . ,
U(n-1)(0) = Ä…n-1, therefore the exact solution may be obtained as follows
(15)
3. Applications
3.1 Cooling of a lumped system by combined convection and radiation
Consider the problem of combined convective radiative cooling of a lumped system as
considered by Aminikhah and Hemmatnezhad [27]. The corresponding governing equation of this
cooling problem after some simplification is as follows
(16)
The exact solution of the above equation was found to be of the form
Expanding ¸(Ä) using the Taylor expansion about Ä = 0 gives
(17)
3.1.1 Application of LTNHPM
To solve Eq. (16) by means of LTNHPM, we construct the following homotopy
(18)
or
(19)
Applying the inverse Laplace transform to both sides of the above equation yields
(20)
From the initial condition, we consider ¸0 = Åš(0) = 1. Therefore, Eq. (20) can be rewritten as
the following equation
(21)
Suppose the solution of Eq. (16) to have the following form
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(22)
in which Åši(Ä) are unknown functions to be determined using the inverse Laplace transform.
Collecting the same powers of p and equating their coefficients, results in
(23)
Solving the above equations for Åši(Ä), i = 0, 1, . . . leads to the following results
Therefore we arrive at the solution of Eq. (16) as
and this, in the limit of infinitely many terms, yields the exact solution of Eq. (16) as given by Eq.
(17).
3.2 Cooling of a lumped system with variable specific heat
Consider the cooling of a lumped system exposed to a convective environment as considered
by Aminikhah and Hemmatnezhad [27]. The corresponding governing equation of this cooling
problem after some simplification is as follows
(24)
The Taylor expansion of the exact solution of Eq. (24) about Ä = 0 can be readily obtained using the
software Maple as
(25)
3.2.1 Application of LTNHPM
To solve Eq. (24), by means of LTNHPM, we construct the following homotopy
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(26)
or
(27)
Applying the Laplace transform to both sides of above equation gives
(28)
Taking ¸0 = Åš(0) = 1, Eq. (28) is simplified as
(29)
Suppose the solution of Eq. (24) to have the following form
(30)
In a similar manner as presented above, by putting Eq. (30) into Eq. (29) and equating the
coefficients of the same power of p, we arrive at
(31)
Solving the above equations using the inverse Laplace transform gives
and the solution of Eq. (24) is then achieved by putting p = 1 into Eq. (30) as
(32)
Figure 1 illustrates the variation of 2nd, 4th, 6th, and 10th-order approximations over Ä for µ
= 0.5 which was obtained from Eq. (32). The results are also compared to the exact solution given by
Eq. (25). As can be seen from this figure the 10th-order approximation in comparison with the exact
solution shows good conformation. A comparison of the present results with those obtained via the
new homotopy perturbation method (NHPM) of Ref. 28 and the exact solutions are shown in Table
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Fig. 1. Variation of ¸(Ä) over Ä for the second example (µ = 0.5). [Color figure can be viewed in
the online issue, which is available at wileyonlinelibrary.com/journal/htj.]
1. As can be seen the results are in good agreement with those obtained via other methods especially
for small values of µ. However, for larger values (µ e" 0.5), the difference between the present result
and the exact solution becomes greater.
4. Conclusion
A new form of the homotopy perturbation method based on the Laplace transform is adopted
for solving two nonlinear differential equations arising in heat transfer problems. Unlike the previous
approach implemented by the present authors, named NHPM, the present technique does not need
the initial approximation to be defined as a power series. Also, unlike the well known HPM there is
no need to solve several recurrence differential equations here. The solution approximations can be
readily obtained using the inverse Laplace transform. Comparison of the results obtained with the
exact solutions clarify that LTNHPM is a promising method for solving several other differential
Table 1. Comparison of the Present Results with Those Obtained via NHPM
and the Exact Solutions
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equations in various fields of study. The computations corresponding to the examples have been
performed using Maple 12.
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