Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method for Blasus Equaton M. Sajd 1, Z. Abbas 2 and N. Al and T. Javed 3 1 Theoretcal Plasma Physcs Dvson PINSTECH P.O. Nlore, Islamabad 44000, Pakstan sajdqau2002@yahoo.com 2 Department of Mathematcs The Islama Unversty of Bahawalpur Bahawalpur 63100, Pakstan za_qau@yahoo.com 3 Department of Mathematcs and Statstcs Internatonal Islamc Unversty Islamabad 44000, Pakstan nasral_qau@yahoo.com; tarq_17pk@yahoo.com Receved: Aprl 11, 2014; Accepted: January 26, 2015 Abstract The objectve of ths paper s to present the hybrd varatonal teraton method. The proposed algorthm s based on the combnaton of varatonal teraton and shootng methods. In the proposed algorthm the entre doman s dvded nto subntervals to establsh the accuracy and convergence of the approxmate soluton. It s found that n each subnterval a three term approxmate soluton usng varatonal teraton method s suffcent. The proposed hybrd varatonal teraton method offers not only numercal values, but also closed form analytc solutons n each subnterval. The method s mplemented usng an example of the Blasus equaton. The results show that a hybrd varatonal teraton method s a powerful technque for solvng nonlnear problems. Keywords: Varatonal teraton method; shootng method; Blasus equaton; subntervals MSC 2010: 34G20, 76D10, 76M30 223
224 M. Sajd et al. 1. Introducton In 1997, He (1997) proposed the varatonal teraton method (VIM) for solvng nonlnear dfferental equatons. Snce then the VIM has been extensvely used for solvng ths type of dfferental equatons. In ths paper, a hybrd varatonal teraton method s proposed to solve the well known Blasus equaton (1979) whch descrbes the flow over a flat plate. Blasus solved the equaton usng a seres expanson method. The numercal soluton whch uses the Runge-Kutta method was provded by Toepfer n 1912. A more accurate numercal soluton of the problem s gven by Howarth (1938) and Ozsk (1977). Yu and Chen (1998) provded the soluton of the Blasus equaton usng the dfferental transform method. Lao and Campo (2002) found the analytcal solutons of the temperature dstrbuton n Blasus vscous flow problems usng the homotopy analyss method. The soluton of Blasus equatons usng varatonal teraton method was frst gven by He (1999). In a later study Wazwaz (2007) provded another soluton usng varatonal teraton method and more recently Ayesm and Ny (2011) provded ther own. In the present paper we have revsted the Blasus equaton for the soluton usng the proposed hybrd varatonal teraton method where ts accuracy and convergence are establshed by subdvdng the doman nto subntervals. The detals of the proposed algorthm are gven n the next secton. 2. Hybrd varatonal teraton method The dmensonless form of the Blasus equaton s gven by (1979) f + 1 2 ff = 0, (1) subject to boundary condtons f(0) = 0, f (0) = 0, lm f () = 1, (2) n whch f s a dmensonless velocty and s a dmensonless ndependent varable. For the soluton we use the shootng method n combnaton wth a varatonal teraton method. The boundary value problem gven n Equatons (1) and (2) s converted nto an ntal value problem usng the shootng method (1979) by assumng f (0) = s. (3) Here s s a mssng condton whch wll be computed through the soluton process. Dfferentatng Equatons (1)-(3) wth respect to s one gets g + 1 2 (gf + fg ) = 0, (4) g(0) = 0, g (0) = 0, g (0) = 1. (5) The ntal value problems gven n Equatons (1)-(5) can be transformed as follows
AAM: Intern. J., Vol. 10, Issue 1 (June 2015) 225 f = F, F = G, G = 1 fg, (6) 2 g = Y, Y = Z, Z = 1 (gg + fz), (7) 2 f(0) = 0, F(0) = 0, G(0) = s, (8) g(0) = 0, Y(0) = 0, Z(0) = 1. (9) For the numercal computatons we replace by a number and dvde the doman 0 nto subntervals each represented by H such that H =, = 1,2,3,. (10) For fxed length subntervals each subnterval s represented by[( 1)H, H], The ntal value problem n each subnterval takes the form = 1,2,3,. df dg = F, = Y, df dy = G, = Z, dg dz = 1 = 1 2 f G (11) 2 (g G + f Z ), (12) and ntal condtons n the frst subnterval are f 1 (0) = 0, F 1 (0) = 0, G 1 (0) = s, (13) g 1 (0) = 0, Y 1 (0) = 0, Z 1 (0) = 1. (14) The numercal values computed at the end pont n the th subnterval are the ntal values n the ( + 1)st subnterval. Accordng to varatonal teraton method (1997) the correcton functonal of system (11) and (12) can be constructed as follows () = f n () + λ 1 (ξ) { df n (ξ) F n (ξ)} dξ, (15) f n+1 F n+1 ( 1)H () = F n () + λ 2 (ξ) { df n (ξ) G n (ξ)} dξ, (16) G n+1 ( 1)H () = G n () + λ 3 (ξ) { dg n (ξ) + 1 2 f n (ξ)g n (ξ)} dξ, (17) g n+1 ( 1)H () = g n () + λ 4 (ξ) { dg n (ξ) Y n (ξ)} dξ, (18) Y n+1 ( 1)H () = Y n () + λ 5 (ξ) { dy n (ξ) Z n (ξ)} dξ, (19) Z n+1 ( 1)H () = Z n () + λ 6 (ξ) { dz n (ξ) + 1 2 [f n (ξ)z n (ξ) + g n (ξ)g n (ξ)]} dξ, (20) ( 1)H
226 M. Sajd et al. where λ j : j = 1,2,3,, 6 are general Lagrange multplers and terms subscrbed denote restrcted varatons,.e., δf = δf = δg = δg = δy = δz = 0. (21) Makng the correcton functonal (15)-(20) statonary, one can obtan the followng statonary condtons dλ j (ξ) = 0, 1 + λ j (ξ) ξ= = 0, j = 1,2,3,, 6. (22) The Lagrange multplers take the values λ j = 1, j = 1,2,3,, 6. (23) Usng the above values n the correcton functonal gven n Equatons (15)-(20), one yelds () = f n (( 1)H) + F n (ξ) dξ, (24) f n+1 F n+1 ( 1)H ( 1)H () = F n (( 1)H) + G n (ξ) dξ, (25) () = G n (( 1)H) 1 f 2 n (ξ)g n (ξ) dξ, (26) G n+1 g n+1 ( 1)H ( 1)H () = g n (( 1)H) + Y n (ξ) dξ, (27) () = Y n (( 1)H) + Z n (ξ) dξ, (28) Y n+1 Z n+1 ( 1)H ( 1)H () = Z n (( 1)H) 1 {f 2 n (ξ)z n (ξ) + g n (ξ)g n (ξ)} dξ. (29) Wth the startng ntal approxmatons f 0 1 (0) = F 0 1 (0) = g 0 1 (0) = Y 0 1 (0) = 0, G 0 1 (0) = s, Z 0 1 (0) = 1. (30) Now the soluton proceeds as follows. Frst an approxmate value of s s chosen and the teraton formulas (24)-(29) are evaluated for = 1 wth three terms.e. n = 1,2,3. Then the values of the functons are evaluated at the fnal pont of the frst subnterval.e. = H. These values are the ntal condtons for the second subnterval. The process s repeated and an analytc soluton s evaluated at each subnterval. A zero fndng algorthm s chosen to evaluate the correct value of s, whch leads to F 0 N ( ) = 1. 3. Numercal results and dscusson The procedure proposed n the prevous secton s mplemented n MATHEMATICA for fndng the numercal values of the velocty feld and the mssng condton. In each subnterval a three term soluton usng varatonal teraton method s computed. The hybrd varatonal teraton method not only provdes the numercal values but also the analytc soluton of the Blasus
AAM: Intern. J., Vol. 10, Issue 1 (June 2015) 227 equaton n each subnterval. The obtaned numercal values for the functons f, f and f are gven n Tables 1-3, respectvely. The results obtaned usng hybrd varatonal teraton method agrees well wth the Howarth (1938) soluton. However, the results presented wth the standard varatonal ternaton method by He (1999) have very hgh percentage error. Hence the presented hybrd varatonal teraton method s more accurate than the standard varatonal teraton method. In a recent study Yun (2010) reported the mssng value of f (0) = 0.33205733621519630. However, n the present study by mplementng proposed hybrd varatonal teraton method we have found f (0) = 0.332057337331755. The presented results prove that the hybrd varatonal teraton method s an effectve method for solvng nonlnear boundary value problems. It s hoped that the applcaton of ths proposed algorthm wll lead to many nterestng results for future studes. Table 1. Comparson of the present numercal values of f wth Howarth (1938) exact soluton and He (1999) standard VIM soluton Present Howarth (1938) He (1999) 0 0 0 0 1 0.16557 0.16557 0.19319 2 0.65003 0.65003 0.67940 3 1.39682 1.39682 1.39106 4 2.30576 2.30576 2.24573 4.8 3.08534 3.08534 2.98719 5 3.28329 3.28329 3.17448 6 4.27964 4.27964 4.14688 7 5.27926 5.27926 5.13359 8 6.27923 6.27923 6.12796 8.8 7.07923 7.07923 6.92593 Table 2. Comparson of the present numercal values of f wth Howarth (1938) exact soluton and He (1999) standard VIM soluton Present Howarth (1938) He (1999) 0 0 0 0 1 0.32979 0.32979 0.35064 2 0.62977 0.62977 0.61218 3 0.84605 0.84605 0.79640 4 0.95552 0.95552 0.90185 4.8 0.98779 0.98779 0.94744 5 0.99115 0.99115 0.95523 6 0.99868 0.99868 0.988032 7 0.99992 0.99992 0.99158 8 1 1 0.99618 8.8 1 1 0.99813
228 M. Sajd et al. Table 3. Comparson of the present numercal values of f wth Howarth (1938) exact soluton and He (1999) standard VIM soluton Present Howarth (1938) He (1999) 0 0.33206 0.33206 0.54360 1 0.32301 0.32301 0.27141 2 0.26675 0.26675 0.22784 3 0.16136 0.16136 0.14117 4 0.06424 0.06424 0.07469 4.8 0.02187 0.02187 0.04184 5 0.01591 0.01591 0.03600 6 0.00240 0.00240 0.01645 7 0.00022 0.00022 0.00724 8 0.00001 0.00001 0.00310 8.8 0 0 0.00155 4. Concluson The hybrd varatonal teraton method s proposed n ths paper. For valdaton purpose the Blasus flow problem s consdered. The developed algorthm combnes the features of shootng and varatonal teraton methods. Its comparson wth exstng methods s very favorable. The present approach also provdes more accurate results when compared to the standard varatonal teraton method. Acknowledgment We are thankful to the revewers for ther useful comments. The frst author acknowledges the fnancal support provded by AS-ICTP durng hs vst for ths work as a Junor Assocate. REFERENCES Ayesm Y. M., Ny O. O. (2011). Computatonal analyss of the nonlnear boundary layer flow over a flat plate usng varatonal teratve method (VIM), Amer. J. Comput, Appl. Math., Vol., 1 pp. 94-97. He J. H. (1997). A new approach to nonlnear partal dfferental equatons, Commun. NonLnear Sc. Numer. Smul., Vol., 2 pp. 230-235. He J. H. (1999). Approxmate analytcal soluton of Blasus equaton, Commun. Non-Lnear Sc. Numer. Smul., Vol., 4 pp. 75-78. Howarth L. (1938). On the soluton of the lamnar boundary layer equatons, Proc. Roy. Soc. London, A164. Lao S. J., Campo A. (2002). Analytcal solutons of the temperature dstrbuton n Blasus vscous flow problems., Vol., 453 pp. 411-425.
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