Abstract
In this study, an effective and rapidly convergent analytical technique is introduced to obtain approximate analytical solutions for nonlinear differential equations. The technique is a combination of the optimal quasilinearization method and the Picard iteration method. The optimal quasilinearization method is used to reduce the nonlinear differential equation to a sequence of linearized differential equations and the Picard iteration method is applied to get the approximate solutions of the linearized equations arising from the optimal quasilinearization method. The convergence analysis of the technique is also discussed. To determine the efficiency and effectiveness of the technique, we consider two numerical examples from real-world applications. The proposed method can easily be extended to a wide class of nonlinear differential equations also.
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References
Adomian G (1988) A review of the decomposition method in applied mathematics. J Math Anal Appl 135(2):501–544
Agarwal RP (1986) Boundary value problems from higher order differential equations. World Scientific
Agarwal RP, Hodis S, O’Regan D (2019) 500 examples and problems of applied differential equations. Springer
Agarwal R, Thompson H, Tisdell C (2003) Difference equations in Banach spaces. Comput Math Appl 45(6–9):1437–1444
Ali J, Islam S, Islam S, Zaman G (2010) The solution of multipoint boundary value problems by the optimal homotopy asymptotic method. Comput Math Appl 59(6):2000–2006
Anderson N, Arthurs A (1985) Analytical bounding functions for diffusion problems with Michaelis–Menten kinetics. Bull Math Biol 47(1):145–153
Asaithambi N, Garner J (1989) Pointwise solution bounds for a class of singular diffusion problems in physiology. Appl Math Comput 30(3):215–222
Bellman RE, Kalaba RE (1965) Quasilinearization and nonlinear boundary-value problems. Rand Corporation
Bernfeld SR, Lakshmikantham V (1974) An introduction to nonlinear boundary value problems. Elsevier
Çağlar H, Çağlar N, Özer M (2009) B-spline solution of non-linear singular boundary value problems arising in physiology. Chaos Sol Fract 39(3):1232–1237
Cordero A, Hueso JL, Martínez E, Torregrosa JR (2011) Efficient high-order methods based on golden ratio for nonlinear systems. Appl Math Comput 217(9):4548–4556
Cuomo S, Marasco A (2008) A numerical approach to nonlinear two-point boundary value problems for ODEs. Comput Math Appl 55(11):2476–2489
Duan JS, Rach R (2011) A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations. Appl Math Comput 218(8):4090–4118
He JH (1999a) Homotopy perturbation technique. Comput Methods Appli Mech Eng 178(3):257–262
He JH (1999b) Variational iteration method-a kind of non-linear analytical technique: some examples. Int J Non-linear Mech 34(4):699–708
Kanth AR, Bhattacharya V (2006) Cubic spline for a class of non-linear singular boundary value problems arising in physiology. Appl Math Comput 174(1):768–774
Kreyszig E (1978) Introductory functional analysis with applications, vol 1. Wiley, New York
Liao SJ (1992) The proposed homotopy analysis technique for the solution of nonlinear problems. Shanghai Jiao Tong University, Shanghai
Na TY (1980) Computational methods in engineering boundary value problems, vol 145. Academic Press
Pandey R, Tomar S (2021) An effective scheme for solving a class of nonlinear doubly singular boundary value problems through quasilinearization approach. J Comput Appl Math 392:113411
Picard E (1893) Sur l’application des méthodes d’approximations successives à l’étude de certaines équations différentielles ordinaires. Journal de Mathématiques Pures et Appliquées 9:217–272
Roul P, Biswal D (2017) A new numerical approach for solving a class of singular two-point boundary value problems. Numer Algorithms 75(3):531–552
Singh M, Verma AK (2016) An effective computational technique for a class of Lane–Emden equations. J Math Chem 54(1):231–251
Tomar S (2020) A computationally efficient iterative scheme for solving fourth-order boundary value problems. Int J Appl Comput Math 6(4):1–16
Tomar S (2021) An effective approach for solving a class of nonlinear singular boundary value problems arising in different physical phenomena. Int J Comput Math: 1–18
Tomar S, Pandey RK (2019) An efficient iterative method for solving bratu-type equations. J Comput Appl Math 357:71–84
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Communicated by Jose Alberto Cuminato.
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Pandey, R.K., Tomar, S. An efficient analytical iterative technique for solving nonlinear differential equations. Comp. Appl. Math. 40, 180 (2021). https://doi.org/10.1007/s40314-021-01563-x
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DOI: https://doi.org/10.1007/s40314-021-01563-x