Approximate polynomial solutions for nonlinear heat transfer problems using the squared remainder minimization method

International Communications in Heat and Mass Transfer 39 (2012) 1336–1341

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Approximate polynomial solutions for nonlinear heat transfer problems using the squared remainder minimization method☆ Bogdan Căruntu ⁎, Constantin Bota “Politehnica” University of Timişoara, Dept. of Mathematics, P-ta Victoriei, 2, Timişoara, 300006, Romania

a r t i c l e

i n f o

Available online 30 August 2012 Keywords: Heat transfer problem Nonlinear differential equation Approximate analytical polynomial solution

a b s t r a c t In the present paper we introduce a new approximation method, called the squared remainder minimization method (SRMM), which allows us to compute analytical approximate polynomial solutions for several nonlinear heat transfer problems. The applications illustrate the validity of the method and a comparison with previous results emphasizes the accuracy of the method. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction Most natural phenomena, including heat transfer phenomena, are inherently nonlinear and, as a consequence, are best modeled using nonlinear equations. With the exception of a number of simple particular cases, these equations can not be solved analytically using traditional methods. If the exact solution of a nonlinear problem can not be found and a numerical solution of the problem is not sufficient, an approximate analytical solution must be computed. To this end, various approximation techniques were employed, the best known ones including perturbation-type methods such as the homotopy perturbation method (HPM) introduced by He ([1–3]) and the homotopy analysis method (HAM) introduced by Liao ([4–6]). These methods eliminate the necessity of a small parameter which rendered the traditional perturbation method largely inefficient for problems containing strong nonlinearities. HPM, HAM and many other methods (i.e. [7–9,19]) were successfully employed to find approximate analytical solutions for some well-known nonlinear heat transfer problems modeled by using both ordinary differential equations ([8–16]) and partial differential equations ([17–19]). In the present paper we will study several problems from the first category, namely: -1 The cooling of a lumped system involving combined modes of convection and radiation heat transfer ([8,10–15]) -2 The heat transfer with conduction in a slab of a material with temperature dependent thermal conductivity ([14,15,9]) -3 The temperature distribution equation in a thick rectangular fin radiating to free space ([12,13,8]) ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail addresses: [email protected] (B. Căruntu), [email protected] (C. Bota). 0735-1933/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2012.08.002

In the next section we will introduce a new approximation method which allows us to determine analytical approximate polynomial solutions for all of the above problems. In the third section we will compare our approximate solutions with approximate solutions presented in ([8–10,12–14,11,15]). The computations show that our method allows us to obtain approximations with an error relative to the exact or numerical solution smaller than the errors obtained using other methods. 2. The squared reminder minimization method In this section we introduce the squared reminder minimization method (SRMM), which allows us to find analytical approximate polynomial solutions for a problem consisting of a nonlinear differential equation of order n and some boundary conditions:
 ðnÞ ðn−1Þ ðn−2Þ ð1Þ u ðt Þ ¼ F u ðt Þ; u ðt Þ; …; u ðt Þ; uðt Þ; t : α n−1 uðn−1Þ ðaÞ þ α n−2 uðn−2Þ ðaÞ þ … þ α 1 uð1Þ ðaÞ þ α 0 uðaÞ ¼ 0 βn−1 uðn−1Þ ðbÞ þ βn−2 uðn−2Þ ðbÞ þ … þ β1 uð1Þ ðbÞ þ β0 uðbÞ ¼ 0

ð1Þ

ð2Þ

Here F is a continuous function, t ∈ [a,b] and αi,βi are real constants. We consider the operator: ðnÞ

DðuÞ ¼ u


 ðn−1Þ ðn−2Þ ð1Þ ðt Þ−F u ðt Þ; u ðt Þ; …; u ðt Þ; uðt Þ; t :

ð3Þ

If uapp is an approximate solution of the Eq. (1), the error obtained by replacing the exact solution u with the approximation uapp is given by the remainder:

 R t; uapp ¼ D uapp ðt Þ ; t∈½a; b

ð4Þ

B. Căruntu, C. Bota / International Communications in Heat and Mass Transfer 39 (2012) 1336–1341

We will find approximate polynomial solutions uapp of Eqs. (1) and (2) on the [a,b] interval, solutions which satisfy the following conditions: 
   R t; uapp b ε

ð5Þ

1337

0 • Using the constants c00,c10,…,cm thus determined, we consider the polynomial:

T n ðt Þ ¼

m X

0 k

ð11Þ

ck t

k¼0 n−1Þ n−2Þ 1Þ α n−1 uðapp ðaÞ þ α n−2 uðapp ðaÞ þ … þ α 1 uðapp ðaÞ þ α 0 uapp ðaÞ ¼ 0 n−1Þ n−2Þ 1Þ βn−1 uðapp ðbÞ þ βn−2 uðapp ðbÞ þ … þ β1 uðapp ðbÞ þ β0 uapp ðbÞ ¼ 0

ð6Þ

Definition 1. We call an ε-approximate polynomial solution of the problem (1) and (2) an approximate polynomial solution uapp satisfying the relations (5) and (6). Definition 2. We call a weak δ-approximate polynomial solution of the problem (1) and (2) an approximate polynomial solution uapp satisfying the relation: b 
   ∫ R t; uapp dt ≤ δ a

The following convergence theorem holds: Theorem 1. If the sequence of polynomials Pm(t) converges to the solution of the problem (1) and (2), then the sequence of polynomials Tm(t) from Eq. (10) satisfies the property: b

2

lim ∫ R ðt; T m Þdt ¼ 0

m→∞

a

Moreover, ∀> 0; ∃m0 ∈N such that ∀m∈N; m > m0 it follows that Tm(t) is a weak ε-approximate polynomial solution of the problem (1) and (2). Proof. Based on the way the coefficients of polynomial Tm(t) are computed and taking into account the relations (8)–(11), the following inequality holds:

together with the initial condition (6). Definition 3. We consider the sequence of polynomials Pm(t) = a0 + a1t + … + amt m,ai ∈ R, i = 0,1,…,m satisfying the conditions:

b

b

2

a

α n−1 P ðmn−1Þ ðaÞ þ α n−2 P ðmn−2Þ ðaÞ þ … þ α 1 P ðm1Þ ðaÞ þ α 0 P m ðaÞ ¼ 0; ðn−1Þ ðn−2Þ ð1Þ βn−1 P m ðbÞ þ βn−2 P m ðbÞ þ … þ β1 P m ðbÞ þ β0 P m ðbÞ ¼ 0:

a

It follows that: b

We call the sequence of polynomials Pm(t) convergent to the solution of the problem (1) and (2) if limm→∞ DðP m ðt ÞÞ ¼ 0. We will find a weak ε-polynomial solution of the type: u ˜ ðt Þ ¼

m X

k

ck t ; m > n;

ð7Þ

k¼0

where the constants c0,c1,…,cm are calculated using the steps outlined in the following. • By substituting the approximate solution (7) in the Eq. (1) we obtain the following expression: Rðt; c0 ; c1 ; …; cm Þ ¼ Rðt˜; u Þ ¼ ð8Þ

0 0 such that ℜ(t,c00,c10,…,cm )= If we could find the constants c00,c10,…,cm 0 for any t ∈ [a,b] and the equivalents of Eq. (2):

α n−1 u˜

ðn−1Þ

ðaÞ þ α n−2 u˜

¼ 0; βn−1 u˜ ¼0

ðn−1Þ

ðn−2Þ

ð1Þ

ðaÞ þ … þ α 1 u˜ ðaÞ þ α 0 u˜ ðaÞ ðn−2Þ

ðbÞ þ βn−2 u˜

ð1Þ

ðbÞ þ … þ β1 u˜ ðbÞ þ β0 u˜ ðbÞ

ð9Þ

0 are also satisfied, then by substituting c00,c10,…,cm in Eq. (7) we obtain the exact solution of Eqs. (1) and (2). In general this situation is rarely encountered in polynomial approximation methods. • Next we attach to the problem (1) and (2) the following real functional:

b

2

J ðc2 ; c3 ; …; cm Þ ¼ ∫ R ðt; c0 ; c1 ; …; cm Þdt

ð10Þ

a

where c0,c1 are computed as functions of c2,c3,…,cm by using the initial conditions (9). 0 • We compute the values of c20,c30,…,cm as the values which give the minimum of the functional (10) and the values of c00,c10 again as functions 0 of c20,c20,…,cm by using the initial conditions.

b

2

2

0≤ lim ∫ R ðt; T m ðt ÞÞdt≤ lim ∫ R ðt; P m ðt ÞÞdt ¼ 0: m→∞

a

m→∞

a

We obtain: b

2

lim ∫ R ðt; T m ðt ÞÞdt ¼ 0:

m→∞


 ¼ ˜u ðnÞ ðt Þ−F ˜u ðn−1Þ ðt Þ; ˜u ðn−2Þ ðt Þ; …; ˜u ð1Þ ðt Þ; ˜u ðt Þ; t

2

0≤∫ R ðt; T m ðt ÞÞdt≤∫ R ðt; P m ðt ÞÞdt; ∀m∈N:

a

From this limit we obtain that ∀ε > 0; ∃m0 ∈N such that ∀m∈N; m > m0 it follows that Tm(t) is a weak ε‐approximate polynomial solution of the problem (1) and (2) q.e.d. Remark 1. Any ε-approximate polynomial solution of the problem (1) and (2) is also a weak ε 2·(b − a)-approximate polynomial solution, but the opposite is not always true. It follows that the set of weak approximate solutions of the problem (1) and (2) also contains the approximate solutions of the problem. Taking into account the above remark, in order to find ε-approximate polynomial solutions of the problem (1) and (2) by SRMM we will first determine weak approximate polynomial solutions, ũapp. If |R(t,ũapp)|b  then ũapp is also an ε-approximate polynomial solution of the problem. 3. Applications 3.1. Application 1: the cooling of a lumped system involving combined modes of convection and radiation heat transfer We consider a lumped system of combined convective–radiative heat transfers. The specific heat coefficient is a linear function of temperature ([10,11]): c ¼ ca ð1 þ βðT−T a ÞÞ where β is a constant and ca is the specific heat at Ta. The problem which describes the cooling process of the system is: ρVc


 dT 4 4 þ hAðT−T a Þ þ Eσ A T −T s ¼ 0 dτ T ð0Þ ¼ T i

1338

B. Căruntu, C. Bota / International Communications in Heat and Mass Transfer 39 (2012) 1336–1341

In order to obtain the nondimensional form of the problem, we perform the following changes of variables: u¼

T T τ ðhAÞ Eσ T 3i T ; us ¼ s : ;u ¼ a ;t ¼ ; ε ¼ βT i ; ε2 ¼ Ti a Ti ρVca 1 h Ti

Assuming for simplicity ua = us = 0, we obtain the following problem:

ð1 þ ε 1 uÞ

du 4 þ u þ ε2 u ¼ 0 dt uð0Þ ¼ 1

ð12Þ

3.1.2. Application 1, case 2: ε1 ≠ 0, ε2 = 0 If ε2 = 0, the problem (12) becomes: ð1 þ ε1 uÞ

3.1.1. Application 1, case 1: ε1 ≠ 0, ε2 ≠ 0 In ([10]), by using the HPM, Ganji and Rajabi computed the following approximate solution of Eq. (12): uHPM ¼

It is easy to see that the approximate solution given by SRMM is much closer to the numerical solution that the previous ones from ([10,11]), while, at the same time, it has a much simpler form. The precision of our method is clearly illustrated in Fig. 1, which presents the relative error functions on the [0.1] interval. The graphical representation of the relative error corresponding to HPM is represented by a dotted line, the one corresponding to HAM by a dashed line and the one corresponding to SRMM by a solid line.


3 −3t 2 −2t ε  7 −5t 2 −7t 2 −2t ε 1 − 2 ε1 þ e ε 1 −e ε 1 −2e e ε2 ε1 þ e ε2 þ 3
12  2 9
 ε 1 −4t 4 −4t ε þ e−t ε 1 − 2 þ e ε 2 þ e ε1 − 2 ε2 þ 3 3 3 3 !
3ε 2 ε  17ε 2 ε 1 2ε 22 4
ε  þ e−t − 1 þ 2 ε 1 − 2 ε 1 þ − − ε 1 − 2 ε 2 þ e−t 2 3 12 9 3 3

In ([11]), by using the HAM, Abbasbandy computed the following approximate solution of Eq. (12):

uð0Þ ¼ 1

3 −3t 2 2 2 −7t 2 2 17 −5t 2 e ε1 ℏ þ e ε2 ℏ − e ε1 ε2 ℏ þ 2

9
12 ε    þ e−2t ε1 ℏ2 þ 2ℏ −2ε1 ε1 − 2 ℏ2 þ 3 
 4 ε  1
2 2 ε1 − 2 ε2 ℏ þ ε2 −ℏ −2ℏ þ þ e−4t 3 3 ! !2 ε21 7ε2 ε1 2ε22 2
ε  −t − þ þe ℏ − ε1 − 2 ðℏ þ 2Þℏ þ 1 2 12 9 3

ð13Þ

The problem (13) admits an exact solution, ue, which is the solution of the algebraic equation εu + ln(u) + t = 0. In ([12]), by using the HPM, Ganji computed an approximate solution of problem (13): uHPM ¼

−3t

3e 2

−2t

−2e

þ

−t

e 2

!


 2 −t −2t −t ε1 þ e ε 1 þ e −e

In ([13]), by using the HAM, Abbasbandy computed an approximate solution of problem (13): uHAM

uHAM ¼

du þu¼0 dt

¼

 
 3 −3t 2 2 −2t 2 2 −t 1 2 2 ε 1 ℏðℏ þ 1Þ−2ε 1 ℏ þ e e ε1 ℏ þ e ε1 ℏ −ε1 ℏðℏ þ 1Þ þ 2 2 þe

−2t

−t

−t

ε1 ℏ−e ε1 ℏ þ e

In ([8]), by using the optimal homotopy asymptotic method (OHAM), Marinca and Herişanu computed an approximate solution of problem (13): −4t

uOHPM ¼ −0:666667e

þ 3e

−3t

−2t

−4e

þ 2:66666e

−t

Using the SRMM, we computed the following second order polynomial approximate solution of Eq. (13): Using the SRMM, we computed the following third order polynomial approximate solution of Eq. (12): 3

2

uSRM ¼ −0:218521t þ 0:620281t −0:954465t þ 1: In order to compare our solution uSRMM with the previous solutions uHPM and uHAM, since Eq. (12) does not have a known exact solution, we computed for each approximate solution the relative error as the difference (in absolute value) between the approximate solution and the numerical solution given by the Wolfram Mathematica software (as an interpolation function). Table 1 presents the values of the relative errors computed for a set of equidistant values of t on the [0,1] interval and for the values of the parameters ε1 = 1, ε2 = 1, ℏ = − 0.1:

2

uSRMM ¼ 0:0690567t −0:501923t þ 1: We compare again our solution uSRMM with the previous solutions given by uHPM, uHAM and uOHAM: This time we computed for each approximate solution the relative error as the difference (in absolute value) between the approximate solution and the exact solution ue. Table 2 presents the values of the relative errors for the value of the parameters ε1 = 1, ℏ = − 0.6. The approximate solution given by

HPM

0.07 0.06

Table 1 Comparison of HPM, HAM and SRMM for ε1 = 1, ε2 = 1. t

HPM

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.2092 7.24274 4.27601 2.47275 1.38919 7.49580 3.81870 1.78647 7.30380

10−1 10−1 10−1 10−1 10−2 10−2 10−2 10−3

HAM

0.05

HAM

SRMM 3rd deg.

1.866 10−2 5.82112 10−3 6.18631 10−3 1.70979 10−2 2.68093 10−2 3.52867 10−2 4.25329 10−2 4.85751 10−2 5.34593 10−2

1.70402 10−3 4.66941 10−4 1.19348 10−3 2.12971 10−3 1.99599 10−3 9.44512 10−4 5.48908 10−4 1.78989 10−3 1.93210 10−3

SRM 3 rd deg.

0.04 0.03 0.02 0.01 0.4

0.6

0.8

1.0

Fig. 1. Comparison of HPM, HAM and SRMM for ε1 = 1, ε2 = 1.

B. Căruntu, C. Bota / International Communications in Heat and Mass Transfer 39 (2012) 1336–1341 Table 2 Comparison of HPM, HAM, OHAM and SRMM for ε1 = 1. t

HPM

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

3.35016 4.34569 4.02973 3.07182 1.88650 7.17017 3.06275 1.12534 1.72676

HAM 10−2 10−2 10−2 10−2 10−2 10−3 10−3 10−2 10−2

3.67248 1.95406 4.09182 5.41548 5.54168 4.43298 2.24327 7.82736 4.37277

10−5 10−3 10−3 10−3 10−3 10−3 10−3 10−4 10−3

HPM

OHAM

SRMM 2nd deg.

3.70848 10−2 5.36624 10−2 5.65858 10−2 5.11379 10−2 4.11884 10−2 2.94297 10−2 1.76287 10−2 6.85581 10−3 2.32003 10−3

1.56178 8.02446 1.13819 1.66928 1.73074 1.37979 6.73670 3.30402 1.57520

10−3 10−5 10−3 10−3 10−3 10−3 10−4 10−4 10−3

SRMM is much closer to the exact solution that the previous ones from ([12,13,8]), while, at the same time, it has a much simpler form. Fig. 2 presents the relative error functions on the [0,1] interval. The graphical representation of the relative error corresponding to HPM is represented by a dotted line, the one corresponding to HAM by a dotted-dashed line, the one corresponding to OHAM by a dashed line and the one corresponding to SRMM by a solid line. 3.1.3. Application 1, case 3: ε1 = 0, ε2 ≠ 0 If ε1 = 0, the problem (12) becomes: du 4 þ u þ ε2 u ¼ 0 dt uð0Þ ¼ 1

ð14Þ

In ([14]), by using the HPM, Rajabi, Ganji and Taherian computed an approximate solution of problem (14): uHPM ¼

  2
−7t 1
−4t −4t −t 2 −t −t e −2e e −e ε2 þ ε2 þ e þe 9 3

¼

4 −9t 2 2 1 −4t 2 2 2 −6t 1 −t 2 −5t 2 2 e ε2 ℏ − e ε2 ℏ − e ε2 ℏ − e ε2 ℏ − e ε2 ℏ þ 45 5 15 5 5 þe

−t

Using the SRMM, we computed the following 5th order polynomial approximate solution of Eq. (14): 4

HAM

0.008 OHAM

0.006

SRM 2nd deg.

0.004 0.002

0.2

0.4

0.6

0.8

1.0

Fig. 2. Comparison of HPM, HAM, OHAM and SRMM for ε1 = 1.

Again the approximate solution given by SRMM is much closer to the numerical solution that the previous ones from ([14,15]), while, at the same time, it has a much simpler form. Fig. 3 presents the relative error functions on the [0,1] interval. The graphical representation of the relative error corresponding to HPM is represented by a dotted line, the one corresponding to HAM by a dashed line and the one corresponding to SRMM by a solid line. We remark the fact that, for each of the three cases of the Application 1, computations were also performed for other values of the parameters ε1 and ε2 and, in general, the larger the values of the parameters, the more SRMM outperforms the other methods.

We consider the process of one-dimensional conduction in a slab of thickness L, with the two faces maintained at uniform temperatures T1 and T2, T1 > T2. The thermal conductivity k is a linear function of temperature ([9,14,15]): k ¼ k2 ð1 þ ηðT−T 2 ÞÞ

2

1 2 2  ℏ 2ε ℏ þ ε2 ℏ þ 2 þ 2 9 5 3

5

0.010

3.2. Application 2: heat transfer with conduction in a slab of a material with temperature dependent thermal conductivity

In ([15]), by using the HAM, Domairry and Nadim computed an approximate solution of problem (14): uHAM

1339

3

2

uSRMM ¼ −1:1979t þ 3:86534t −4:98357t þ 3:53843t −1:92784t þ 1:

The comparison between our solution uSRM and the previous solutions given by uHPM and uHAM is presented in Table 3. Since no exact solution for the Eq. (14) is known, we computed the relative error as the difference (in absolute value) between the approximate solution and the numerical solution given by the Mathematica Wolfram software. Table 3 presents the values of the relative errors for the value of the parameters ε2 = 1, ℏ = − 0.8:

where η is a constant and k2 is the thermal conductivity at T2. The problem which describes the process is:   d dT k ¼ 0; x∈½0; L dx dx T ð0Þ ¼ T 1 ; T ðLÞ ¼ T 2 :

HPM

0.030

HAM

0.025

Table 3 Comparison of HPM, HAM and SRMM for ε2 = 1.

SRM 5 th deg.

0.020 t

HPM

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2.24219 9.47661 1.79917 2.51387 3.00464 3.27857 3.37783 3.35023 3.23752

HAM 10−3 10−3 10−2 10−2 10−2 10−2 10−2 10−2 10−2

1.18868 8.71208 1.25548 1.54052 2.09832 3.04594 4.35783 5.95087 7.72966

SRMM 5th deg. 10−4 10−4 10−3 10−3 10−3 10−3 10−3 10−3 10−3

6.08673 1.25683 3.12286 9.67612 1.12410 1.78443 8.14607 7.43428 3.59748

10−5 10−3 10−4 10−4 10−3 10−4 10−4 10−4 10−4

0.015 0.010 0.005 0.2

0.4

0.6

0.8

Fig. 3. Comparison of HPM, HAM and SRMM for ε2 = 1.

1.0

1340

B. Căruntu, C. Bota / International Communications in Heat and Mass Transfer 39 (2012) 1336–1341

In order to obtain the nondimensional form of the problem, taking into account the fact that k1 is the thermal conductivity at T1, we introduce the dimensionless variables: θ¼

T−T 2 x k −k2 ; y ¼ ; ε ¼ ηðT 1 −T 2 Þ ¼ 1 : L T 1 −T 2 k2

We obtain the following problem:
2 dθ d2 θ ε dt ; y∈½0; 1 − 2¼ 1 þ εθ dy θð0Þ ¼ 1; θð1Þ ¼ 0:

3.3. Application 3: temperature distribution in a thick rectangular fin radiating in free space

Equivalently:  2 2 d θ dθ þε ¼ 0; y∈½0; 1 2 dt dy θð0Þ ¼ 1; θð1Þ ¼ 0: ð1 þ εθÞ

ð15Þ

The problem (15) admits an exact solution, θe:

2

In ([14]), by using the HPM, Rajabi, Ganji and Taherian computed an approximate solution of problem (15): 2

y ε þ 2

2

y −

3

!

We consider the nondimensional form of the temperature distribution problem for a uniformly thick rectangular fin radiating in free space with nonlinearity of high order ([12,13,8]): d θ 4 −εθ ¼ 0 dy2 dθ θð1Þ ¼ 1; ð0Þ ¼ 0: dy

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −1 þ 1−ðy−1Þεðε þ 2Þ : θe ¼ ε

θHPM ¼ −

For the case of the Application 2, the strong nonlinearity of the equation contributed to the fact that the degree of our approximate polynomial solution must be chosen much higher than in the case of the Application 1. Still, the conclusion is the same: we can compute by SRMM an approximation better and simpler that the previous ones in ([14,15,9]). Fig. 4 presents the relative error functions on the [0,1] interval. The graphical representation of the relative error corresponding to HAM by a dotted line, the one corresponding to GAM (obtained by interpolating the results in [9]) by a dashed line and the one corresponding to SRMM by a solid line. The relative error corresponding to HPM is too large in comparison to the other ones, so its graphical representation was omitted from the picture.

2

y yε yε 2 ε − þ −y þ 1: 2 2 2

In ([12]), by using the HPM, Ganji computed the following approximate solution of Eq. (16):

θHPM ¼ In ([15]), by using the HAM, Domairry and Nadim computed an approximate solution of problem (15):

θSRMM ¼ −0:194382y8 þ 0:576857y7 −0:72868y6 þ 0:440894y5 − −0:173141y4 −0:0296625y3 −0:141884y2 −0:750001y þ 1: We compare again our solution θSRMM with the previous solutions θHPM, θHAM and θGAM. We computed for each approximate solution the relative error as the difference (in absolute value) between the approximate solution and the exact solution θe. Table 4 presents the values of the relative errors for the value of the parameters ε=1, ℏ=−0.5: Table 4 Comparison of HPM, HAM, GAM and SRMM for ε1 = 1. HAM

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.90384 10−2 2.79089 10−2 2.91817 10−2 2.53201 10−2 1.86388 10−2 1.12397 10−2 4.90488 10−3 9.11064 10−4 3.24575 10−4

8.65938 9.11085 3.06686 1.32005 3.01383 5.23970 7.52988 8.91106 7.55043

GAM 10−5 10−5 10−4 10−3 10−3 10−3 10−3 10−3 10−3

5.40617 1.08915 1.56862 2.00531 2.18301 2.16974 1.88752 1.50641 8.42510

  1
2 1
2 y −1 εℏ− y −1 εðℏ þ 1Þℏþ 2 2  1
4 2 2 2 þ y −6y þ 5 ε ℏ þ 1: 6

θHAM ¼ −

In ([9]), by using the generalized approximation method (GAM), Khan computed an approximate solution of problem (15). The expression of the approximate solution is not given explicitly, just its values for a set of values of y on the [0,1] interval, for several values of the parameter ε. Using the SRMM, we computed the following eight order polynomial approximate solution of Eq. (15):

HPM

! 2  y 1 1
4 2 2 y −6y þ 5 ε þ 1 − εþ 6 2 2

In ([13]), by using the HAM, Abbasbandy computed an approximate solution of Eq. (16):

1 3 2 2 1 2 2 1 2 2 2 2 2 2 θHAM ¼ − y ε ℏ þ y ε ℏ þ y εℏ þ y εℏ− yε ℏ − 2 2 2 1 2 − yεℏ −yεℏ−y þ 1: 2

t

ð16Þ

In ([8]), by using the optimal homotopy asymptotic method (OHAM), Marinca and Herişanu computed a 6th order approximate polynomial solution of the problem (16): 6

4

2

θOHAM ¼ 0:000023293y þ 0:0202783y þ 0:204645y þ 0:775053

0.0001

HAM

0.00008

GAM

0.00006

SRMM 8 th deg.

SRMM 8th deg. 10−6 10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−6

5.69231 2.59835 1.24162 4.79476 3.75054 5.50051 4.65280 6.42001 2.69815

10−7 10−6 10−6 10−6 10−7 10−6 10−7 10−6 10−6

0.00004 0.00002

0.2

0.4

0.6

0.8

Fig. 4. Comparison of HPM, HAM, GAM and SRMM for ε = 1.

1.0

B. Căruntu, C. Bota / International Communications in Heat and Mass Transfer 39 (2012) 1336–1341 Table 5 Comparison of HPM, HAM, OHAM and SRMM for ε = 1. t

HPM

HAM

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

5.47359 10−1 5.27037 10−1 4.93714 10−1 4.48203 10−1 3.91624 10−1 3.25387 10−1 2.51159 10−1 1.70822 10−1 8.64095 10−2

3.39452 3.32133 3.19805 3.02193 2.78745 2.48429 2.09446 1.58783 9.15554

OHAM 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−3

3.88930 3.29144 2.33350 1.08117 3.56592 1.80809 3.00939 3.56040 2.85895

SRMM 4th deg. 10−3 10−3 10−3 10−3 10−4 10−3 10−3 10−3 10−3

8.98806 2.48399 3.24245 2.63685 8.81475 1.25384 2.73888 2.72332 1.19728

10−5 10−4 10−4 10−4 10−5 10−4 10−4 10−4 10−4

1341

The applications presented clearly illustrate the accuracy of the method, since for all the problems we were able to compute better approximations than the ones computed in previous papers. We remark again that while, due to the number of problems solved, we only included the cases were the ε parameters were equal to one, computations performed for other values of the parameters show that SRMM still has the advantage over the other methods and, moreover, the larger the values of ε, the more SRMM outperforms the other methods. Finally, we mention that while in this paper we solved only ordinary differential equations, SRMM can be easily adapted for other types of nonlinear differential and integral equations, and thus it can be considered a powerful tool for the computation of approximate polynomial solutions for nonlinear problems.

HAM

0.010 References OHAM

0.008

SRMM 4 th deg.

0.006 0.004 0.002

0.2

0.4

0.6

0.8

1.0

Fig. 5. Comparison of HAM, OHAM and SRMM for ε = 1.

Using the SRMM, we computed the following 4th order polynomial approximate solution of Eq. (16): 4

3

2

θSRMM ¼ 0:0696406y −0:0473588y þ 0:198583y þ 0:779135: We compare again our solution θSRM with the previous solutions θHPM, θHAM and θOHAM: Since no exact solution for the Eq. (16) is known, we computed the relative error as the difference (in absolute value) between the approximate solution and the numerical solution given by the Mathematica Wolfram software. Table 5 presents the values of the relative errors for the value of the parameters ε = 1, ℏ = −0.4. Again, the approximate solution given by SRMM is closer to the numerical solution that the previous ones from ([12,13,8]), while, at the same time, it has a simpler form. Fig. 5 presents the relative error functions on the [0,1] interval. The graphical representation of the relative error corresponding to HAM is represented by a dotted line, the one corresponding to OHAM by a dashed line and the one corresponding to SRMM by a solid line. As in the case of the previous application, the relative error corresponding to HPM is too large in comparison to the other ones, so its graphical representation was omitted from the picture. 4. Conclusions The squared remainder minimization method (SRMM) was introduced as a straightforward and efficient method to compute approximate polynomial solutions for nonlinear heat transfer problems.

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