یک روش تحلیلی بهینه برای حل مسائل مقدار مرزی غیرخطی بر پایه روش تغییر پارامتر
محورهای موضوعی : آمار
1 - گروه ریاضی، دانشگاه آزاد اسلامی، واحد نکا، نکا، ایران
2 - گروه ریاضی، دانشگاه آزاد اسلامی، واحد قائمشهر، قائمشهر، ایران
کلید واژه: Optimal variation of parameter, Boundary value problems, Sturm- Liouville, Airy and Quantum mechanical ha,
چکیده مقاله :
در این مقاله، نویسندگان یک الگوریتم تحلیلی همگرای اصلاح شده را برای جواب مسائل مقدار مرزی و مقدار اولیه غیرخطی به واسطه روش تغییر پارامتر ارائه میکنند و بطور خلاصه روش تغییر پارامتر بهینه مینامند. این روش براساس تعبیه یک پارامتر و یک عملگر کمکی، یک مزیت محاسباتی برای همگرایی جوابهای تقریبی معادلات دیفرانسیل غیرخطی مهیا میکند. همگرایی توسعه یافته مذ کور نشان داده شده و جزییات آن نیز مورد بحث قرار میگیرد. علاوه بر این، یک روش مناسب برای انتخاب مقدار بهینه پارامتر کمکی در نظر گرفته میشود که تحت مینیممسازی خطا روی دامنه مساله میباشد. موثر بودن روش و دقت الگوریتم پیشنهادی، با اجرا روی مسائل فیزیکی همچون مساله استورم- لیوویل، مساله ایری و مساله نوسانگر هارمونیک کوانتومی نشان داده میشود. نتایج عددی و شکلهای بدست آمده بوضوح دقت الگوریتم و همگرایی آن را منعکس میکند.
In this paper, the authors present a modified convergent analytic algorithm for the solution of nonlinear boundary value problems by means of a variable parameter method and briefly, the method is called optimal variable parameter method. This method, based on the embedding of a parameter and an auxiliary operator, provides a computational advantage for the convergence of the approximate solutions of nonlinear differential equations. The developed convergence has been shown and its details are discussed. Additionally, a convenient method is considered for selecting an optimal value of the auxiliary parameter, via minimizing the residual error over the domain of problem. The effectiveness of the method and the accuracy of the proposed algorithm are illustrated by the implementation of physical problems such as Sturm-Liouville problem, Airy equation, and Quantum mechanical harmonic oscillator problem. The numerical results and obtained demonstrate clearly reflect the accuracy of the method and its convergence.
[1] Cole JD. Perturbation methods in applied mathematics. Waltham (MA): Blaisdell; 1968.
[2] Murdock JA. Perturbation: theory and methods. New York: John Wiley & Sons; 1991.
[3] He JH. Homotopy perturbation technique. Comput Methods Appl Mech Eng 1999;178:257–62.
[4] He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003;135:73–9.
[5] Noor MA, Mohyud-Din ST. Homotopy perturbation method for solving Thomas fermi equation using pade approximants. Int J Nonlinear Sci 2009;8:27–31.
[6] Noor MA, Mohyud-Din ST. An explicit Homotopy perturbation method for solving sixth order boundary value problems. Comput Math Appl
2008;55:2953–72.
[7] Zhou JK. Differential transformation and its applications for electrical circuits,Wuhan. China: Huazhong University Press; 1986 [in Chinese].
[8] Batiha A-M, Batiha B. Differential transformation method for a reliable treatment of the nonlinear biochemical reaction model. Adv Stud Biol 2011;3:355–60.
[9] Khan Y, Svoboda Z, Smarda Z. Solving certain classes of Lane-Emden type equations using the differential transformation method. Adv Difference Eqs 2012;2012. article 174.
[10] Adomian G. Solving frontier problems of physics: the decomposition method, vol. 60. Boston, Mass (USA): Kluwer Academic; 1994.
[11] Adomian G. A review of the decomposition method and some recent results for nonlinear equations. Math Comp Mod 1990;13:17–43.
[12] Cherruault Y, Adomian G. Decomposition method. A new proof of convergence. Math Comp Mod 1993;18:103.
[13] Kaya D. Explicit and numerical solutions of some fifth-order KdV equations by decomposition method. Appl Math Comput 2003;144:353–63.
[14] Siddiqi SS, Twizell EH. Spline solutions of linear sixth-order boundary value problems. Int J Comput Math 1996;60:295–304.
[15] He JH. Variational iteration method-Some recent results and new interpretations. J Comp Appl Math 2007;207:3–17.
[16] He JH. Variational iteration method. A kind of non-linear analytical technique, some examples. Internat J Nonlinear Mech 1999;34(4):699–708.
[17] He JH. Variational iteration method for autonomous ordinary differential systems. Appl Math Comput 2000;114(2–3):115–23.
[18] He JH, Wu XH. Construction of solitary solution and compaction-like solution by variational iteration method. Chas SoltonFract 2006;29(1):108–13.
[19] He JH. Some asymptotic methods for strongly nonlinear equation. Int J
Mod Phys 2006;20(10):1144–99. 10.
[20] He JH. The variational iteration method for eighth-order initial boundary value
problems. PhysScr 2007;76(6):680–2.
[21] Finlayson BA. The method of weighted residuals and variational principles. New York: Academic Press; 1972.
[22] Zienkiewicz OC, Taylor RL, Auth JZ. The finite element method: its basis and fundamentals. Butterworth-Heinemann; 2013.
[23] Ma WX, You Y. Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions. Trans Am Math Soc 2004;357:1753–78.
[24] Ma WX, You Y. Rational solutions of the Toda lattice equation in Casoratianform. Chaos, Solitons Fractals 2004;22:395–406.
[25] Ma WX, Wu HY, He JS. Partial differential equations possess in Frobeniuintegrable decompositions. PhysLett A 2007;364:29–32.
[26] Noor MA, Mohyud-Din ST, Waheed A. Variation of parameters method for solving fifth-order boundary value problems. Appl Math InfSci2008;2:135–41.
[27] Mohyud-Din ST, Noor MA, Waheed A. Variation of parameter method for solving sixth-order boundary value problems. Commun Korean Math Soc2009;24:605–15.
[28] Mohyud-Din ST, Noor MA, Waheed A. Variation of parameter method for initial and boundary value problems. World ApplSci J 2010;11:622–39.
[29] Mohyud-Din ST, Noor MA, Waheed A. Modified variation of parameters method for second-order integro-differential equations and coupled systems. World ApplSci J 2009; 6: 1139–46.
[30] Gangi DD, Tari Hafez, BakhshiJooybari M. Variational iteration method andhomotopy perturbation method for nonlinear evolution equations. ComputMath Appl 2007; 54: 1018–27.
[31] Peregrine DH. Calculations of the development of an undular bore. J Fluid Mech 1966;25:321–30.
[32] Bona JL, Pritchard WG, Scott LR. Numerical schemes for a model of nonlinear dispersive waves. J Comp Phys 1985; 60: 167–96.
[33] Ghaneai H, Hosseini MM. Variational iteration method with an auxiliary parameter for solving wave-like and heat-like equations in large domains. Comput Math Appl 2015;69 (5): 363–73.
[34] SemaryMourad S, Hassan Hany N. A new approach for a class of nonlinear boundary value problems with multiple solutions. J Assoc Arab Univ Basic ApplSci 2015;17:27–35.
[35] Zill DG, Cullen MR. Differential equations with boundary value problems. Cengage Learning, Brooks/ Cole; 2009.
[36] Boyce WE, DiPrima RC. Elementary differential equations and boundary value problems. Wiley, New York;2012.
[37] Ramos J.On the variational iteration method and other iterative
techniques for nonlinear differential equations.ApplMath Comput 2008; 199: 39–69.
[38] Hosseini MM, Mohyud-Din ST, Ghaneai H, Usman M. Auxiliary parameter in the variational iteration method and its optimal determination. Int J Nonlinear SciNumerSimul 2010;11(7):495–502.
[39] Hosseini SMM, Mohyud-Din ST, Ghaneai H. Variationaliteration method for nonlinear Age-Structured population models using auxiliary parameter. ZeitschriftfrNaturforschung 201065(12):1137–42.
[40] Liao, S. J. (1992). The proposed homotopy analysis technique for the solution of nonlinear problems (Doctoral dissertation, Ph. D. Thesis, Shanghai Jiao Tong University(
