Pandey et al. Journal of Theoretical and Applied Physics 2012, 6:45 http://www.jtaphys.com/content/6/1/45
RESEARCH Open Access Semi-analytic algorithms for the electrohydrodynamic flow equation Ram K Pandey1,2, Vipul K Baranwal1, Chandra S Singh1 and Om P Singh1*
Abstract In this paper, we consider the nonlinear boundary value problem for the electrohydrodynamic (EHD) flow of a fluid in an ion-drag configuration in a circular cylindrical conduit. This phenomenon is governed by a nonlinear second-order differential equation. The degree of nonlinearity is determined by a nondimensional parameter α.Wepresenttwo semi-analytic algorithms to solve the EHD flow equation for various values of relevant parameters based on optimal homotopy asymptotic method (OHAM) and optimal homotopy analysis method. In 1999, Paullet has shown that for large α, the solutions are qualitatively different from those calculated by Mckee in 1997. Both of our solutions obtained by OHAM and optimal homotopy analysis method are qualitatively similar with Paullet’s solutions. Keywords: Optimal homotopy asymptotic method (OHAM), Optimal homotopy analysis method, Electrohydrodynamic flow, Square residual error, Gauss quadrature MSC: 34B15, 34B16, 76E30
Background solutions given by Equations 4 and 6 depending on the The electrohydrodynamic flow of a fluid in an ion-drag value of the nonlinearity control parameter α. configuration in a circular cylindrical conduit is gov- For α << 1 and assuming a solution of the form erned by a nonlinear second-order ordinary differential X1 αn ; α : equation. Perturbation solutions of fluid velocities for urðÞ¼ unðÞr ð3Þ different orders of nonlinearities were given by McKee n¼0 et al. [1]. In their study, a description of the problem Mckee et al. [1] obtained the O(α3) perturbation solu- was presented in which the governing equations were tion as I0ðÞHr reduced to the following nonlinear boundary value prob- urðÞ¼; α 1 À þ α½þðÞu1ðÞþHr C1 I0ðÞþHr v1ðÞHr K0ðÞHr I0ðÞH lem (BVP): 2 α ½ðÞu2ðÞþHr C2 I0ðÞþHr v2ðÞHr K0ðÞHr : d2u 1 du u ð4Þ þ þ H2 1 À ¼ 0; 0 < r < 1; ð1Þ dr2 r dr 1 À αu Similarly, for α >> 1, the authors [1] proposed that the solution to the BVP could be expanded in the series of subject to boundary conditions the form X1 0 Àn u ðÞ¼0 0; uðÞ¼1 0; ð2Þ urðÞ¼ α unðÞr; α ð5Þ n¼0 where u(r)isthefluidvelocity,r is the radial distance from with an O(1) leading-order term and obtained the per- the centre of the cylindrical conduit, H is the Hartman turbation solution as electric number and the parameter α is a measure of the 0 1 Zr strength of the nonlinearity. In [1], the authors used a H2 1 ÀÁ1 logðÞ 1 À s2 π2 urðÞ¼; α 1 þ 1 À r2 þ @2 ds þ A: regular perturbation technique to obtain two perturbation 4 α α2 s 6 0 * Correspondence: [email protected] ð6Þ 1Department of Applied Mathematics, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, India Full list of author information is available at the end of the article
© 2012 Pandey et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Pandey et al. Journal of Theoretical and Applied Physics 2012, 6:45 Page 2 of 10 http://www.jtaphys.com/content/6/1/45
Paullet [2] proved the existence and uniqueness of the so- lution to the BVP (1) and (2) in the following theorem: 1400 Theorem 1 α 2 ≠ . For any > 0 and any H 0, there exists 1200 a solution to the BVP (1) and (2). Furthermore, this solu- 1000 tion is monotonically decreasing and satisfies 0 < u(r) < E3 1/(α + 1) for all r ∈ (0,1). 800 Remark 1. By a solution of Equations 1 and 2, we mean 600 2 afunctionu(r) ∈ C[0,1] \ C (0,1) that satisfies Equation 1 400 for 0 < r < 1 along with Equation 2. In order for such a 200 function to be a solution, we must necessarily have u(r)< 2 0 1/α on (0,1); if u(r) ever equals 1/α,itisnolongerC , 0.40 0.35 0.30 0.25 0.20 0.15 0.10 owing to the term u(r)/(1− αu(r)) in Equation 1 [2]. C1 Paullet [2] claimed an error in the perturbation and Figure 1 Square residual error E for the third-order OHAM numerical solutions given in [1] for large values of α. 3 (C2 = −0.1809055, C3 = 0.275378). This stems from the fact that for large α,thesolutions are O(1/α), not O(1) as proposed in the perturbation ex- pansionusedin[1].Forα << 1, our solutions obtained method) as compared to the deviations in ℏ (in the case by the two semi-analytic algorithms (proposed in the of HAM). A comparison is made between OHAM, opti- ‘Application of OHAM to EHD flow problem’ and mal homotopy analysis method and HAM via exact ‘Application of optimal homotopy analysis method to square residual errors. It is shown that for higher values EHD flow problem’ subsections) are in complete agree- of α and H2, the respective third- and fourth-order ment with those of [1] and [2], but for α >> 1, the pro- OHAM and optimal homotopy analysis method solu- posed solution profiles are similar to those of [2]. Thus, tions are more accurate than the 20th-order HAM solu- based on our work in this paper, we support Paullet’sso- tions. Further, the central processing unit (CPU) time is lution profiles for α >> 1. also calculated and compared for these methods, estab- Recently, Mastroberardino [3] proposed an analytical lishing the superiority of OHAM and optimal homotopy method based on the homotopy analysis method (HAM) analysis method over the HAM solution. Also, the solu- to find the solutions of Equations 1 and 2 for α ∈ (0,1] tion profiles shown for α = 4, 10, H2 = 1 and α = 4, 10, and H2 up to 4. The author [3] has shown that the H2 = 10 by Figures 3 and 4 respectively match Paullet’s homotopy perturbation method (HPM) yields a diver- solution profiles shown in Figures one and two of [2] for gent solution for all of the cases considered. The HAM the corresponding values of the parameters. solutions are quite accurate for lower values of the para- meters α and H2, but the accuracy decreases rather fast Analysis of the method for higher values of these parameters even though fairly Optimal homotopy asymptotic method higher order (20 to be precise) solutions were consid- Since the last two decades, homotopy perturbation ered, as shown in Table one of [3]. Further, from Figure method [4] and homotopy analysis method [5] based on two of [3], we observe that even a slight deviation from the topological concept of homotopy have become very the optimal value of ℏ causes a huge square residual popular in solving nonlinear ordinary/partial differential error for α = 0.5, 1 and H2 = 4. This, along with the qualitative difference between the solution profiles of 1000 Mckee et al. [1] and Paullet [2] for α >> 1, motivated us to look for algorithms giving accurate solutions for 800 higher values of the parameters as well. 600 The aim of the present work is to propose two algo- E4 rithms for the solutions of the above BVP (1) and (2) for 400 all values of relevant parameters using optimal homo- 200 topy asymptotic method (OHAM) and optimal homo- topy analysis method. We show that even the third- and 0 fourth-order solutions obtained from OHAM and opti- 200 mal homotopy analysis method, respectively, are highly 0.10 0.08 0.06 0.04 0.02 0.00 accurate for α >> 1. From Figures 1 and 2, we see that c1 the square residual errors E3/E4 are stable even for larger Figure 2 Square residual error E4 of the fourth-order optimal deviations from the optimal value of C1 (in the case of HAM (c2 = 0.0242099, c3 = −6.19495). OHAM) or c1 (in the case of optimal homotopy analysis Pandey et al. Journal of Theoretical and Applied Physics 2012, 6:45 Page 3 of 10 http://www.jtaphys.com/content/6/1/45
where, A = L + N, L is a linear operator, N is a nonlinear operator, r denotes the independent variable, u(r)isan unknown function, f(r) is a known function and B is a boundary operator. A homotopy h(ϕ(r,q),q): R × [0,1] → R is constructed satisfying
ðÞ1 À q ½¼LðÞþ’ðÞr; q frðÞ HqðÞ½LðÞþ’ðÞr; q frðÞ þ NðÞ’ðÞr; q ; BðÞ¼’ðÞr; q 0; ð8Þ
Figure 3 OHAM solution u(r) for H2 = 1 and α = 4 (red), α =10 ∈ (blue). where, q [0,1] is an embedding parameter, H(q) is a non- zero auxiliary function for q ≠ 0andH(0) = 0. As the em- bedding parameter q increases from 0 to 1, the ϕ(r,q)varies equations [6,7]. Later, in 2008, Marinca et al. [8-11] from the initial approximation u0(r)tothesolutionu(r). introduced a new analytical method known as OHAM The auxiliary function H(q) is chosen as to solve a variety of nonlinear problems. This method is 2 3 straightforward and reliable, and it does not need to look HqðÞ¼qC1 þ q C2 þ q C3 þ ⋯; ð9Þ for ℏ curves like HAM. This method provides us a con- venient way to control the convergence of the series so- ⋯ lution and allows the adjustment of the convergence where C1, C2, C3, are constants to be determined. It is region wherever it is needed via unspecified number of very important to choose these constants properly since convergence control parameters. These parameters are the convergence of the solution depends on them. ϕ determined in such a way that the optimal values are Expanding (r,q) in a power series with respect to the yielded unlike the ℏ curve method used in HAM. The parameter q, we get OHAM solution generally agrees with the exact solution at larger domains as compared to HPM and HAM solu- 2 ϕðÞ¼r; q; C1; C2; ⋯ u0ðÞþr u1ðÞr; C1 q þ u2ðÞr; C1; C2 q þ ⋯: tions. OHAM is based on a generalized zeroth-order 10 deformation equation (8) and does not consider the mth- ð Þ order deformation equation like HAM. We apply OHAM to the following nonlinear differen- Substituting Equation 10 into Equation 8 and equating tial equation: the coefficients of like powers of q, we obtain the follow- ing equations: AurðÞþðÞ frðÞ¼0; BuðÞ¼0; ⇔LurðÞþðÞ frðÞ þ NurðÞ¼ðÞ 0; BuðÞ¼0; LuðÞþ0ðÞr frðÞ¼0; BuðÞ¼0 0; ð11Þ ð7Þ
LuðÞ¼1ðÞr LuðÞþ0ðÞr frðÞþC1½LuðÞ0ðÞr ð12Þ þ N0 ðÞþu0ðÞr frðÞ; BuðÞ¼1 0;
LuðÞ¼2ðÞr LuðÞþ1ðÞr C1½LuðÞþ1ðÞr N1ðÞu0ðÞr ; u1ðÞr
þ C2½LuðÞþ0ðÞr N0ðÞþu0ðÞr frðÞ; BuðÞ2 ¼ 0; ð13Þ
LuðÞ¼3ðÞr LuðÞ2ðÞr
þ C1½LuðÞþ2ðÞr N2ðÞu0ðÞr ; u1ðÞr ; u2ðÞr
þ C2½LuðÞþ1ðÞr N1ðÞu0ðÞr ; u1ðÞr 2 Figure 4 OHAM solution u(r) for H = 10 and α = 4 (red), þ C3½LuðÞþ0ðÞr N0ðÞþu0ðÞr frðÞ; BuðÞ3 α = 10 (blue). ¼ 0; ð14Þ Pandey et al. Journal of Theoretical and Applied Physics 2012, 6:45 Page 4 of 10 http://www.jtaphys.com/content/6/1/45
⋮ ∂ ∂ ∂ Em Em ⋯ Em : LuðÞmðÞr ¼ LuðÞþmÀ1ðÞr C1½LuðÞmÀ1 ¼ ¼ ¼ ¼ 0 ð20Þ ∂C1 ∂C2 ∂Cm þNmÀ1ðÞu0; u1; ⋯; umÀ1
þC2½LuðÞþmÀ2 NmÀ2ðÞu0; u1; ⋯; umÀ2 Substituting the optimal values of C ’s obtained from þ⋯ þ CmÀ1½LuðÞþ1ðÞr N1ðÞu0ðÞr ; u1ðÞr i Equation 20 into Equation 17, the mth-order approxi- þCm½LuðÞþ0ðÞr N0ðÞþu0ðÞr frðÞ; mate solution ũm is obtained. BuðÞ¼m 0; ð15Þ As discussed in [12], computing En(C1, C2, C3,⋯, Cn) directly with a symbolic computational software is im- practical. Thus, we approximate Equation 19 using a m where Nm(u0, u1,⋯, um) is the coefficient of q in the ex- Gaussian quadrature with eight nodes followed by min- pansion of N(φ(r,q)) about the embedding parameter q. imizing Equation 19 using the Mathematica function The above equations are called the zeroth-, first-, Minimize; the optimal values of these convergence con- second- and mth-order problems, respectively. trol parameters are obtained. As q → 1, in Equation 10, OHAM faces the practical problem of computing higher order iterates since as many number of parameters Cm are X1 to be computed as the order of iterates. The method is urðÞ¼; Ci u0ðÞþr urðÞ; Ci : ð16Þ well suited for the electrohydrodynamic (EHD) problem k¼1 as shown by the various solution profiles and tables.
Application of OHAM to EHD flow problem Truncating Equation 16 at level k = m, the mth-order Choosing L ¼ d2 and f = H2 and using Equation 11, the solution is given by dr2 zeroth-order problem for Equation 1 with boundary con- ditions (2)
uemðÞ¼r; C1; C2; ⋯; Cm u0ðÞr Xm ; ; ; ⋯; : þ uiðÞr C1 C2 Ci d2u ðÞr 0 þ H2 ¼ 0; u 0ðÞ¼1 0; u ðÞ0 i¼1 dr2 0 0 ð17Þ ¼ 0givesu0ðÞr 1 ÀÁ ¼ H2 À H2r2 : ð21Þ 2 Substituting Equation 17 into Equation 7, one gets the following residual: As the fourth-order approximate solution gives a very RmðÞ¼r; C1; C2; ⋯; Cm LðÞuemðÞr; C1; C2; ⋯; Cm accurate solution even for higher values of the nonli- þ frðÞ nearity parameter α and the Hartmann electric number þ NðÞuemðÞr; C1; C2; ⋯; Cm : H. These iterates are obtained from Equations 12 to 14, ð18Þ and the first three iterates are listed below:
First-order problem: If Rm = 0, then ũm will be the exact solution, which does not happen in practice, especially in nonlinear pro- blems. In order to find the optimal values of Ci, i =1,2, ⋯ 3, , we first construct the functional (called the square d2u ðÞr d2u ðÞr d2u ðÞr 1 du ðÞr 1 ¼ 0 þ H2 þ C 0 þ 0 residual error) dr2 dr2 1 dr2 r dr 2 α α d u0ðÞr du0ðÞr Zb À u0ðÞr À u0ðÞr dr2 r dr ; ; ⋯; 2 ; ; ; ⋯; ; EmðÞ¼C1 C2 Cm R ðÞr C1 C2 Cm dr 2 þH ðÞ1 À ðÞ1 þ α u0ðÞr ; a 0 ð19Þ u1 ðÞ¼1 0; u1ðÞ¼0 0: ð22Þ
([a,b] being the domain of the problem), and then minimizing it, we get Second-order problem: Pandey et al. Journal of Theoretical and Applied Physics 2012, 6:45 Page 5 of 10 http://www.jtaphys.com/content/6/1/45
2 2 2 1 ÀÁ d u2ðÞr d u1ðÞr d u1ðÞr 1 du1ðÞr u ðÞ¼Àr H2 À1 þ r2 ½25; 200ðÞC þ C ¼ þ C1 þ 3 ; 2 3 dr2 dr2 dr2 r dr ÀÁ604ÀÁ800 2 2 α ; 2 d2u ðÞr d2u ðÞr  12 þ H À5 þ r ðÞÀ1 þ þ 1 680C1 f720 Àαu ðÞr 1 À αu ðÞr 0 ÀÁ 0 2 1 2 2 α 2 α 4 dr dr þ10H 47 À 77ÀÁþ r ðÞÀ7 þ 13 ÀÁþ H ð61 À 260Á α α du1ðÞr du0ðÞr þ199α2 À 14r2 1 À 5α þ 4α2 þ r4 1 À 10α þ 9α2 g À u0ðÞr À u1ðÞr r dr r dr 3 ; ; ; α 2 þC1 f1 209 600À þ 5 600Hð193 À 424 þ r ðÀ23 2 α α 4 ; ; α ; α2 ÀH ðÞ1 þ u0ðÞr þ62ÀÁÞÞ þ 56H 4 853 À 26 310 þ 25 597 Âà þr2 À922 þ 5; 940α À 6; 278α2 þ r4ð53 À 710α 2 ; 0 ; : Á À þC2 Au0 þ H u2 ðÞ¼1 0 u2ðÞ¼0 0 α2 6 ; ; α ; α2 þ897 ÞþHÀÁ20 775 À 203 959 þ 456 665 ; α3 4 ; α ; α2 ; α3 ð23Þ À273 ÀÁ481 r 405 À 9 989 þ 31 835 À 22 251 6 α α2 α3 2 ; þ15r À1 þ 49 À 299 þ 191 Áþ r ð4 845 ; α ; α2 ; α3 þ56 861 À 139 315 þ 87 299À Þg þ1; 680C f180ðÞþ 1 þ 4C H4C 61 À 260α þ 199α2 Third-order problem: ÀÁ1 2 ÀÁÁ2 2 α α2 4 α α2 À14r 1ÀÁÀ 5 þ 4 þ r 1 À 10 þ 9 2 2 α α þ5H ð3 À5 þ rÁ ðÞþÀ1 þ 2C2ð47 À 77 2 þr ðÞÀ7 þ 13α g: ð27Þ d2u ðÞr d2u ðÞr d2u ðÞr 1 du ðÞr 3 ¼ 2 þ C 2 þ 2 dr2 dr2 1 dr2 r dr 2 2 2 d u2ðÞr d u1ðÞr d u0ðÞr Substituting the above iterations in Equation 17, the Àαu0ðÞr À αu1ðÞr À αu2ðÞr dr2 dr2 dr2 mth-order approximate solution is obtained as α du ðÞr α du ðÞr α du ðÞr À u ðÞr 2 À u ðÞr 1 À u ðÞr 0 0 1 2 e ; ; ; ; ⋯; ; r dr r dr r dr umðÞ¼r C1 C2 C3 Cm u0ðÞþr u1ðÞr C1 à 2 d u1ðÞr 1 du1ðÞr ; ; ÀH2ðÞ1 þ α u ðÞr þ C þ þu2ðÞr C1 C2 2 2 dr2 r dr þ⋯umðÞr; C1; C2; C3; ⋯; Cm : 2 2 d u1ðÞr d u0ðÞr α du1ðÞr Àαu0ðÞr À αu1ðÞr À u0ðÞr ð28Þ dr2 dr2 r dr α du0ðÞr 2 À u1ðÞr À H ðÞ1 þ α u0ðÞr From Equation 18, the mth-order residual is r ÂÃdr 2 0 2 þ C3 Au0 þ H ; u3 ðÞ¼1 0; u3ðÞ¼0 0: ð24Þ d uem 1 duem RmðÞ¼r; C1; C2; C3; ⋯; Cm þ dr2 r dr ue þ H2 1 À m 1 À αuem Solving Equations 22 to 24, we get the first three iter- m ¼ 1; 2; 3⋯: ð29Þ ates as follows: Substituting Equation 29 in Equation 19 and computing the square residual error Em numerically using the Gauss ÀÁÂÃÀÁ 1 2 2 2 2 quadrature formulae with eight node points followed by u1ðÞ¼r H C1 À1 þ r 12 þ H À5 þ r ðÞÀ1 þ α ; 24 minimizing Em, the optimal values of the convergence ⋯ ð25Þ control parameters C1, C2, C3, , Cm are obtained.
Optimal homotopy analysis method ÀÁ The optimal homotopy analysis method was first proposed 1 2 2 u2ðÞ¼r H À1 þ r 30ðÞC1 þ C2 by Liao [12] containing exactly three convergence control 720 ÀÁÀÁ parameters at any level of approximation in contrast to 2 2 α Â 12nþ H À5 þ r ðÞÀ1 þ OHAM. The optimal homotopy analysis method is based 2 2 2 þC1 720 þ 10H ð47 À 77α þ r ðÞÀ7 þ 13α on a generalized zeroth-order deformation equation (31). ÀÁ 4 α2 2 α α2 Liao [12] used special deformation functions which are þH 61 À 260 þ 199 À 14r 1 À 5 þ 4 ÀÁo determined completely by only one characteristic param- þr4 1 À 10α þ 9α2 ; ð26Þ eter |c2|<1and|c3| < 1, respectively. To illustrate the procedure, consider the nonlinear equation Pandey et al. Journal of Theoretical and Applied Physics 2012, 6:45 Page 6 of 10 http://www.jtaphys.com/content/6/1/45
Table 1 Comparison of exact square residual errors at α = 0.5 and different values of H2 2 2 2 2 Auxiliary parameters Ci α = 0.5, H = 0.5 α = 0.5, H = 1 α = 0.5, H = 2 α = 0.5, H = 4
OHAM C1 −0.108084 −0.128187 −0.064627 −0.0236008
C2 −0.478377 −0.295243 −0.564313 −0.78968
C3 0.527761 0.270073 0.724121 1.19611 −12 −8 −6 −4 E3 7.527668 × 10 1.51604 × 10 5.41811 × 10 7.36637 × 10
Optimal homotopy analysis method c1 −0.566753 −0.374439 −0.0316471 −0.276117
c2 0.0199658 −0.321575 −0.357213 0.389223
c3 −0.0812378 −0.391183 −8.72346 −0.37244 −12 −8 −6 −4 E4 2.40146 × 10 8.90742 × 10 4.68025 × 10 2.05172 × 10 HAM ℏ −0.375 −0.276 −0.275 −0.205 −12 −9 −8 −5 E20 7.772 × 10 1.230 × 10 5.319 × 10 4.568 × 10
: X1 Nur½¼ðÞ 0 ð30Þ ; μ m; A1ðÞ¼q c2 mðÞc2 q m¼1 X1 ; σ m; Marinca and Herisanu [9] constructed a general form B1ðÞ¼q c3 mðÞc3 q ð34Þ of the zeroth-order deformation equation: m¼1
ðÞ1 À BqðÞL½¼ϕðÞÀr; q u0ðÞr c1AqðÞN½ϕðÞr; q ; r∈Ω; q∈½0; 1 ;
ð31Þ where |c2|<1and|c3| < 1 are constants called the conver- gence control parameters, and where N is a nonlinear operator, L is a linear operator and μ ; μ A(q)andB(q) are called deformation functions satisfying 1ðÞ¼c2 1 À c2 mðÞc2 mÀ1 ¼ ðÞ1 À c2 c2 ; m > 1; ð35Þ AðÞ¼0 BðÞ¼0 0; AðÞ¼1 BðÞ¼1 1; ð32Þ
σ1ðÞ¼c3 1 À c3; σmðÞc3 the Taylor series of which: mÀ1 ¼ ðÞ1 À c3 c3 ; m > 1: ð36Þ X1 X1 μ m; σ m; AqðÞ¼ mq BqðÞ¼ mq ð33Þ Using these deformation functions, the zeroth-order m¼1 m¼1 deformation equation takes the form exist and are convergent |q| ≤ 1. ðÞ1 À B1ðÞq; c3 L½’ðÞÀr; q u0ðÞr There are infinite number of deformation functions sat- ¼ c1AqðÞ; c2 N½’ðÞr; q ; r∈Ω; q∈½0; 1 ; ð37Þ isfying the properties (32) and (33). For the sake of com- putational efficiency, we use the following one-parameter where c1 ≠ 0 is an auxiliary parameter called the conver- deformation functions: gence control parameter. Thus, we have at most three
Table 2 Comparison of exact square residual errors at α = 1 and different values of H2 2 2 2 2 Auxiliary parameters Ci α = 1, H = 0.5 α = 1, H = 1 α = 1, H = 2 α = 1, H = 4
OHAM C1 −0.377725 −0.35981 −0.30792 −0.168637
C2 0.0420993 0.0623337 0.06334952 −0.1809055
C3 −0.01487 −0.0164235 −0.00274481 0.275378 −11 −9 −7 −4 E3 2.591989 × 10 1.55047 × 10 1.05203 × 10 1.17518 × 10
Optimal homotopy analysis method c1 −0.835839 −0.950907 −0.022845 −0.0455211
c2 0.587823 −0.411825 −0.3021 0.0242099
c3 0.546788 −0.037081 −12.3406 −6.19495 −11 −9 −4 −4 E4 1.74228 × 10 −1.0616 × 10 1.5504 × 10 4.27912 × 10 HAM ℏ −0.303 −0.292 −0.254 −0.198 −11 −9 −6 −4 E20 4.634 × 10 4.996 × 10 2.363 × 10 3.461 × 10 Pandey et al. Journal of Theoretical and Applied Physics 2012, 6:45 Page 7 of 10 http://www.jtaphys.com/content/6/1/45
Table 3 Comparison of exact square residual error at α = 1.5 and different values of H2 2 2 2 2 Auxiliary parameters Ci α = 1.5, H = 0.5 α = 1.5, H = 1 α = 1.5, H = 2 α=1.5, H = 4
OHAM c1 −0.429884 −0.443299 −0.6898 −0.499435
c2 0.0742146 0.139540 0.0298953 0.07564
c3 −0.0128337 −0.02021099 0.5407644 1.06037 −9 −7 −6 −4 E3 2.0763995 × 10 3.8790 × 10 2.6342 × 10 1.5866 × 10
Optimal homotopy analysis method c1 −0.675245 −0.800339 −0.341231 −0.0427887
c2 0.2038278 0.741063 −0.301007 0.111194
c3 −0.030015 −1.36725 −0.623547 −7.00114 −8 −5 −5 −3 E4 1.3619 × 10 2.563 × 10 6.744 × 10 1.6 × 10 HAM ℏ −0.393372 −0.0164281 −0.6237173 −0.32296 −8 −4 E9 8.195638 × 10 8.1234 × 10 0.177825 0.551513
"# convergence control parameters: c1, c2 and c3. As the mXÀ1 ϕ χ σ embedding parameter q increases from 0 to 1, (r;q) LumðÞÀr m mÀk ðÞc3 uk ðÞr deforms continuously from the initial guess u0(r) to the k¼1 exact solution u(r) since ϕ(r;0) = u (r) and ϕ(r;1) = u(r). mXÀ1 0 μ ; Note that ϕ(r;q) is determined by the auxiliary oper- ¼ c1 mÀk ðÞc2 Nu½k ðÞr ð39Þ k¼0 ator L, the initial guess u0(r) and the convergence con- trol parameters c , c and c , and we have great freedom 1 2 3 subject to the given boundary conditions. to choose them. Assuming that all of them are so prop- erly chosen that ϕ(r;q) has the Taylor series representa- X1 m tion as ’ðÞ¼r; q u0ðÞþr umðÞr q ; which converges Application of optimal homotopy analysis method to EHD m¼1 flow problem at q = 1, thus the solution by optimal homotopy analysis Now, we apply the optimal homotopy analysis method as method will be given as developed above to the EHD flow (Equations 1 and 2). Assuming the initial guess u0(r) = 0 and the linear oper- X1 d2 ator L ¼ 2, Equation 39 becomes urðÞ¼u0ðÞþr umðÞr ; ð38Þ dr m¼1 "# m 2 mXÀ1 1 ∂ ’ðÞr;q d where umðÞ¼r m! ∂qm q¼0 is the mth-order homotopy u ðÞÀr χ σ ðÞc u ðÞr dr2 m m mÀk 3 k derivative [9]. k¼1 mXÀ1 Taking the mth-order homotopy derivative on both ¼ c μ ðÞc Nu½ðÞr ; ð40Þ sides of Equation 37, we get the mth-order deformation 1 mÀk 2 k k¼0 equation:
where 0.005
0.000 00 1 0 Nu½¼k ðÞr uk ðÞþr uk ðÞr R4()r ÀÁr 2 χ α 0.005 þ H 1 À k À ðÞ1 þ ukÀ1ðÞr Xk 00 À α fguiðÞr ukÀi ðÞr 0.010 i¼1 α Xk 0 0.0 0.2 0.4 0.6 0.8 1.0 À fguiðÞr ukÀi ðÞr ; ð41Þ r i 0 r ¼ 2 Figure 5 Residual R4(r) for H =4,α = 0.5 (red - OHAM, blue - optimal HAM). and σm(c3)andμm(c2) are given in Equations 35 and 36. Pandey et al. Journal of Theoretical and Applied Physics 2012, 6:45 Page 8 of 10 http://www.jtaphys.com/content/6/1/45
0.03
0.02
0.01
0.00 R4()r 0.01
0.02
0.03
0.04
0.0 0.2 0.4 0.6 0.8 1.0 r Figure 6 Residual R (r) for H2 =4,α = 1 (red - OHAM, 4 Figure 8 Optimal HAM solution u(r) for H2 = 10 and α = 0.1 blue - optimal HAM). (red), 0.5 (blue), 1 (green).
ÀÁ Using Equations 40 and 41, we compute the first four 1 2 2 u4ðÞ¼Àr H c1ðÞÀÀ1 þ c2 1 þ r iterates as we get a very satisfactory solution from these  604ÀÁ; 800  302; 400 1 þ c þ c 3 À c 2ðÞÀÀ1 þ c c À c c four iterates only. These are 2 À 2 2 3 3 2 3 þ25; 200c ðÞÀ1 þ c 3 þ 3c 2 À 4c 2ðÞÀ1 þ c 1ÁÀÁ2 ÀÁ2 2 3 1 ÀÁ À4c þ c 2 À24 þ H2 À5 þ r2 ðÞ1 þ α u ðÞ¼Àr H2c ðÞÀÀ1 þ c 1 þ r2 ; 3 3 1 1 2 2 2 þ2; 520ðÞÀ1 þ c2 ðÞ1 þ c2 À c3 f1; 440 ÀÁ ÀÁ 2 2 4 2 2 1 2 2 þH 61 À 14r þ r ðÞ1 þ α À 20H ðÀ31 À 46α u2ðÞ¼Àr H c1ðÞÀÀ1 þ c2 1 þ r Á 24 2 3 3 þr ðÞ5 þ 8α gþc1 ðÞÀ1 þ c2 fÀ2; 419; 200  ð12 þ ðÞþ1 þ c2 À c3 c1ðÞÀ1 þ c2 ÀÁ ÀÁÀÁ 6 2 4 6 3 2 2 α ; þ15H À1; 385 þ 323r À 27r þ r ðÞ1 þ α ÂÀ24 þ H À5 þ r ðÞ1 þ ÀÁ þ11; 200H2 À143 À 281α þ r2ðÞ19 þ 43α ÀÁ 4 2 1 2 2 À112H ð2; 884 þ 8; 633α þ 5; 749α u3ðÞ¼Àr H c1ðÞÀÀ1 þ c2 1 þ r ÀÁ Â720 ÀÁ 4 α α2 2 þ2r 17 þ 79 þ 62  360 1 þ c2 þ c2 À c3 À c2c3 ÀÁÁ 2 ; α ; α2 : Âþ60c ðÞÀ1 þ c ðÞ1 þ c À c Àr 566 þ 1 867 þ 1 301 g ÀÁ1 ÀÁ2 2 3 ÂÀ24 þ H2 À5 þ r2 ðÞ1 þ α ð42Þ È ÀÁ þc 2ðÞÀ1 þ c 2 1; 440 þ H2 61 À 14r2 þ r4 2 2 ÀÁÉ The fourth-order optimal homotopy analysis method α 2 2 α 2 α ; ÂðÞ1 þ À 20H À31 À 46 þ r ðÞ5 þ 8 solution û4(r) is obtained by substituting Equation 42 in Equation 38 and is given by
0.20 X4 u^4ðÞ¼r; c1; c2; c3 uiðÞr; c1; c2; c3 : ð43Þ 0.15 i¼0
0.10 The optimal values of parameters c1, c2 and c3 are computed by minimizing the square residual error Em defined by Equation 19. 0.05
Convergence of the solutions 0.00 In this section, we discuss the convergence of the third- 1.0 0.5 0.0 0.5 1.0 order OHAM solution and the fourth-order optimal homo- u r H2 α Figure 7 Optimal HAM solution ( )for =1and =0.1 topy analysis method solution given in Equations 28 and (red-dotted-top), 0.5 (blue-solid-middle), 1 (green-dashed-bottom). 43, respectively. The convergence of these two solutions Pandey et al. Journal of Theoretical and Applied Physics 2012, 6:45 Page 9 of 10 http://www.jtaphys.com/content/6/1/45
Table 4 CPU times incurred in calculating the iterations Case 1 by OHAM, optimal homotopy analysis method and HAM For α << 1, the solution profiles obtained by optimal Iterations CPU times (s) homotopy analysis method are depicted in Figures 7 and 4 iterates by optimal homotopy analysis method 2.108 8. The similar profiles are obtained by using OHAM as 3/4 iterates by OHAM 1.217/3.226 well. From the figures, it is clear that the profiles obtained by our approach are the same as that obtained 19 iterates by HAM 1,765 in [1].
Case 2 depend on their respective convergence control parameters For α >> 1, in Figures 3 and 4, we present OHAM solu- C1, C2, C3 and c1, c2, c3. The values of these parameters are tions of the BVP (1) and (2) for values of α = 4, 10 and obtained by using the least square method given in the H2 = 1 and α = 4, 10 and H2 = 10, respectively. In the ‘Analysis of the method’ section. From Figures 1 and 2, we case of α = 4 (respectively, α = 10), the solutions are see that the square residual errors E3/E4 arestableevenfor bounded above by 1 / (α +1)=0.2(respectively,1/ larger deviations from the optimal values of C1 (OHAM) (α + 1) = 0.09) and are in agreement with those of 2 or c1 (optimal homotopy analysis method) as compared to Paullet [2]. As noted in [1], for large H ,thesolutions the deviations in ℏ (HAM) from its optimal value. From should tend to u(r)=1/(α +1),andsuchbehaviouris Figure two of [3], we see that when ℏ moves towards the evident in Figures eight and nine. This is in contrast to left of its optimal value −0.198 and approaches −0.30, the Figures twelve and thirteen of [1] where for H2 =10and 2 square residual error E20 shoots up from its minimum H = 100, the solutions do not exhibit the proper − value 3.461 × 10 4 to a value greater than 107.So,fora limiting behaviour and also cross the singularity at 1 / α − relatively smaller variation of the order 10 1 in ℏ,thereis and thus are not C2. So, our calculations support Paullet’s a huge variation of the order 1011 in the square residual numerical results for α >> 1. error E20, whereas in our proposed algorithm based on OHAM, a similar variation in the value of C1 about its Comparison with HAM solutions − optimal value 0.168637 causes no appreciable change in Tables 1, 2 and 3 display the square residual errors E3 E3. From Figure 2, we conclude that the sensitivity of E4 (OHAM), E4 (optimal homotopy analysis method) and E20 (optimal homotopy analysis method) with respect to the (HAM) at various values of parameters α and H2.Itis − variation in c1 about its optimal value 0.0455211 is much observed that the square residual errors E3/E4 obtained by lesser compared to the sensitivity of E20 (HAM) but is OHAM/optimal homotopy analysis method are compa- larger than the sensitivity of E3 (OHAM). rable with E20 (obtained by HAM) for lower values of α 2 The optimal values of the convergence control para- and H and are significantly smaller than E20 for higher meters for all the cases considered are obtained by min- values of α and H2. Table 4 shows the ratio of CPU times imizing Equation 19 using the Mathematica function incurred to find the 3, 4 and 19 iterates by OHAM, optimal Minimize and are given in Tables 1, 2 and 3. In addition, homotopy analysis method and HAM, respectively. Further, we plot the residual functions R4(r) for both the pro- the CPU time for HAM is very large compared to those for posed algorithms in Figures 5 and 6 for the parameters OHAM and optimal homotopy analysis method. α = 0.5, H2 = 4 and α =1,H2 = 4, respectively. These plots demonstrate the accuracy of OHAM and optimal Conclusions homotopy analysis method solutions over the HAM so- In this paper, we propose two semi-analytical algorithms lution [3]. As was done by Mastroberardino in [3], the based on OHAM and optimal homotopy analysis residuals have been plotted as a function of r for the op- method to obtain semi-analytical solutions for a non- timal values of the convergence control parameters C1, linear boundary value problem governing electrohydro- C2, C3 and c1, c2, c3 (given in Tables 1, 2 and 3) and not dynamic flow, though the nonlinearity confronted in this as a function of these convergence control parameters problem is in the form of a rational function posing a for a fixed value of r as this is a better illustration of significant challenge in regard to obtaining analytical/ convergence. semi-analytical solutions. Earlier in 1997, Mckee et al. [1] gave numerical solutions to the EHD flow in a circu- lar cylindrical conduit described by Equations 1 and 2 Discussion of solution profiles for various values of H2 and α with the perturbation In this section, we give the various fourth-order solution expansions of the solutions for small and large values of profiles for small and large values of α. For α << 1, the parameter α. Later in 1999, Paullet [2] provided a rigor- solution profiles obtained by the two proposed methods ous result concerning the existence, uniqueness and are similar to that obtained by Mckee et al. [1]. qualitative properties of the solutions of Equations 1 and Pandey et al. Journal of Theoretical and Applied Physics 2012, 6:45 Page 10 of 10 http://www.jtaphys.com/content/6/1/45
2 2 for any α >0andH ≠ 0. Paullet’s solution [2] for 11. Herisanu, N, Marinca, V: Accurate analytical solution to oscillators with α << 1 matches with that of Mckee et al. [1], but there was discontinuities and fractional power restoring force by means of the optimal homotopy asymptotic method. Compt. Math. Appl. 60, 1607–1615 a difference between the solutions of Paullet and Mckee for (2010) α >> 1. Our solutions obtained from the two proposed 12. Liao, S: An optimal homotopy-analysis approach for strongly nonlinear algorithms support Paullet’s solutions for α >> 1. For differential equations. Comm. Nonlinear Sci. Numer. Simulat. 15, 2003–2016 (2010) α << 1, all the solutions obtained by Mckee et al. [1], by Paullet [2] and from our proposed algorithms are in good doi:10.1186/2251-7235-6-45 agreement. For lower values of α and H2, the solutions Cite this article as: Pandey et al.: Semi-analytic algorithms for the electrohydrodynamic flow equation. Journal of Theoretical and Applied obtained by OHAM, optimal homotopy analysis method Physics 2012 6:45. and HAM are compatible, whereas for higher values of α and H2, the algorithm based on OHAM gives better results than the one based on optimal homotopy analysis method, and the optimal homotopy analysis method solutions are better than the HAM solutions.
Competing interests The authors declare that they have no competing interests.
Authors' contributions RKP proposed the algorithm and searched the EHD problem. RKP, VKB and CSS applied the algorithm to solve the problem. OPS supervised the work and along with RKP analysed the convergence of the solution for the larger values of the nonlinear parameters α and H2. All authors have read and approved the final manuscript.
Acknowledgements The first, second and third authors acknowledge the financial support from CSIR-SRF, UGC-SRF and BHU-JRF, respectively.
Author details 1Department of Applied Mathematics, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, India. 2Department of Mathematics, Government Gundadhur Degree College, Kondagaon, Chhattisgarh, India.
Received: 20 April 2012 Accepted: 1 December 2012 Published: 22 December 2012
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