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Neural Computing and Applications

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01-10-2015 | Original Article

Design and application of nature inspired computing approach for nonlinear stiff oscillatory problems

Authors: Junaid Ali Khan, Muhammad Asif Zahoor Raja, Muhammed I. Syam, Shujaat Ali Khan Tanoli, Saeed Ehsan Awan

Published in: Neural Computing and Applications | Issue 7/2015

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Abstract

In this paper, meta-heuristic intelligent approaches are developed for handling nonlinear oscillatory problems with stiff and non-stiff conditions. The mathematical modeling of these oscillators is accomplished using feed-forward artificial neural networks (ANNs) in the form of an unsupervised manner. The accuracy as well as efficiency of the model is subject to the tuning of adaptive parameters for ANNs that are highly stochastic in nature. These optimal weights are carried out with swarm intelligence and pattern search methods hybridized with an efficient local search technique based on constraints minimization known as active set algorithm. The proposed schemes are validated on various stiff and non-stiff variants of the oscillator. The significance, applicability and reliability of the proposed scheme are well established based on comparison made with the results of standard numerical solver.

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Mickens RE, Oyedeji K (2010) Comments on the general dynamics of the nonlinear oscillator. Mech Syst Signal Process 24:2076–2095CrossRef

Kovacic I (2011) Forced vibrations of oscillators with a purely nonlinear power-form restoring force. J Sound Vib 330:4313–4327CrossRef

Balaram B, Narayanan MD, RajendrakumarAli PK (2012) Optimal design of multi-parametric nonlinear systems using a parametric continuation based Genetic Algorithm approach. Nonlinear Dyn 67:2759–2777CrossRef

Feng J, Zhu W-Q, Liu ZH (2011) Stochastic optimal time-delay control of quasi-integrable Hamiltonian systems. Commun Nonlinear Sci Numer Simulat 16:2978–2984MathSciNetCrossRefMATH

Erneux T, Baer SM, Mandel P (1987) Subharmonic bifurcation and bistability of periodic solutions in a periodically modulated laser. Phys Rev A 35:1165–1171CrossRef

Kaya M, Higuchi H (2010) Nonlinear elasticity and an 8-Nm working stroke of a single myosin molecules in myofilaments. Science 329(5992):686–689

Storm C, Nelson PC (2003) Theory of high-force DNA stretching and overstretching. Phys Rev E 67:051906MathSciNetCrossRef

Hamdan MN, shabaneh NH (1997) On the large amplitude free vibrations of a restained uniform beam carrying an intermediate lumped mass. J Sound Vib 199:711–736CrossRef

Mickens RE, Jackson DD (1999) oscillation in systems having velocity dependent frequencies. J Sound Vib 223:329–333MathSciNetCrossRefMATH

10.

Beatty J, Mickens RE (2005) A qualitative study of the solutions to the differential equation. J Sound Vib 283:475–477MathSciNetCrossRefMATH

11.

Mickens RE (2006) Investigation of the properties of the period for the nonlinear oscillator. J Sound Vib 292:1031–1035MathSciNetCrossRefMATH

12.

Belendz A, Hernandez A, Belendz T, Neipp C, Marques A (2006) Asymptotic representation of the period of nonlinear oscillator. J Sound Vib 299:406–408

13.

Kalmar T, Erneux T (2008) Approximating a small and large amplitude periodic orbits of the oscillator. J Sound Vib 313:806–811CrossRef

14.

Kerschen G, Worden K, Vakakis AF, Golinval JC (2006) Past, present and future of nonlinear system identification in structural dynamics. Mech Syst Signal Process 20:505–592CrossRef

15.

Raja MAZ, Sabir Z, Mahmood N, Alaidarous ES, Khan JA (2014) Stochastic solvers based on variants of genetic algorithms for solving nonlinear equations. Neural Comput Appl. doi:10.1007/s00521-014-1676-z

16.

Anastassi AA (2014) Constructing Runge–Kutta methods with the use of artificial neural networks. Neural Comput Appl 25(1):229–236CrossRef

17.

Mokeddem D, Khellaf A (2012) Optimal feeding profile for a fuzzy logic controller in a bioreactors using genetic algorithm. Nonlinear Dyn 67:2835–2845MathSciNetCrossRef

18.

Chakraverty S, Mall S (2014) Regression-based weight generation algorithm in neural network for solution of initial and boundary value problems. Neural Comput Appl 25(3–4):585–594CrossRef

19.

Raja MAZ, Qureshi IM, Khan JA (2011) Swarm intelligent optimized neural networks for solving fractional differential equations. Int J Innov Comput Inf Control 7(11):6301–6318

20.

Raja MAZ, Khan JA, Qureshi IM (2010) Heuristic computational approach using swarm intelligence in solving fractional differential equations. GECCO (Companion) 2010:2023–2026

21.

Raja MAZ, Khan JA, Qureshi IM (2011) Swarm intelligence optimized neural network for solving fractional order systems of Bagley-Tervik equation. Eng Intell Syst 1:41–51

22.

Zahoor RMA, Khan JA, Qureshi IM (2009) Evolutionary computation technique for solution of Riccati differential equation of arbitrary order. World Acad Sci Eng Technol 58:303–309

23.

Khan JA, Raja MAZ, Qureshi IM (2011) Novel approach for van der Pol oscillator on the continuous time domain. Chin Phys Lett 28(11):110205CrossRef

24.

Khan JA, Raja MAZ, Qureshi IM (2011) Hybrid evolutionary computational approach: application to van der Pol oscillator. Int J Phys Sci 6(31):7247–7261

25.

Raja MAZ, Khan JA, Ahmad SI, Qureshi IM Numeriacal treatment of Painleve equation I using neural networks and stochastic solvers. Innovation in Machine-3, Studies in Computational intelligence, book series Springer, vol 442, Sep 2012

26.

Raja MAZ, Khan JA, Ahmad SI, Qureshi IM (2012) Solution of the Painlevé equation-I using neural network optimized with swarm intelligence. Comput Intell Neurosci 2012:1–10. ID 721867

27.

Raja MAZ, Khan JA, Behloul D, Haroon T, Siddiqui AM, Samar R (2015) Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation. Appl Soft Comput 26:244–256. doi:10.1016/j.asoc.2014.10.009 CrossRef

28.

Raja MAZ, Khan JA, Shah SM, Bhahoal D, Samar R (2014) Comparison of three unsupervised neural network models for first Painleve´ Transcendent. Neural Comput Appl. Accepted 30-11-2014. doi:10.1007/s00521-014-1774-y

29.

Khan JA, Raja MAZ, Qureshi IM (2011) Numerical treatment of nonlinear Emden-Fowler equation using stochastic technique. Ann Math Artif Intell 63(2):185–207MathSciNetCrossRef

30.

Raja MAZ (2014) Unsupervised neural networks for solving Troesch’s problem. Chin Phys B 23(1):018903CrossRef

31.

Raja MAZ (2014) Stochastic numerical techniques for solving Troesch’s Problem. Inf Sci 279:860–873. doi:10.1016/j.ins.2014.04.036 MathSciNetCrossRef

32.

Raja MAZ, Samar R (2014) Numerical Treatment for nonlinear MHD Jeffery-Hamel problem using neural networks optimized with interior point algorithm. Neurocomputing 124:178–193. doi:10.1016/j.neucom.2013.07.013 CrossRef

33.

Raja MAZ, Samar R (2014) Numerical treatment of nonlinear MHD Jeffery-Hamel problems using stochastic algorithms. Comput Fluids 91:28–46MathSciNetCrossRef

34.

Raja MAZ (2014) Numerical treatment for boundary value problems of pantograph functional differential equation using computational intelligence algorithms. Appl Soft Comput 24:806–821. doi:10.1016/j.asoc.2014.08.055 CrossRef

35.

Raja MAZ (2014) Solution of the one-dimensional Bratu equation arising in the fuel ignition model using ANN optimised with PSO and SQP. Connect Sci 26(3):195–214. doi:10.1080/09540091.2014.907555 CrossRef

36.

Raja MAZ, Ahmad SI (2014) Numerical treatment for solving one-dimensional Bratu problem using neural networks. Neural Comput Appl 24(3–4):549–561. doi:10.1007/s00521-012-1261-2 CrossRef

37.

Raja MAZ, Ahmad SI, Samar R (2013) Neural network optimized with evolutionary computing technique for solving the 2-dimensional Bratu problem. Neural Comput Appl 23(7–8):2199–2210. doi:10.1007/s00521-012-1170-4 CrossRef

38.

Raja MAZ, Samar R, Rashidi MM (2014) Application of three unsupervised neural network models to singular nonlinear BVP of transformed 2D Bratu equation. Neural Comput Appl. Published online 17-06-2014. doi:10.1007/s00521-014-1641-x

39.

Raja MAZ, Ahmad SI, Samar R (2014) Solution of the 2-dimensional Bratu problem using neural network, swarm intelligence and sequential quadratic programming. Neural Comput Appl. Published online 06-07-2014. doi:10.1007/s00521-014-1664-3

40.

Raja MAZ, Khan JA, Haroon T (2014) Stochastic Numerical treatment for thin film flow of third grade fluid using unsupervised neural networks. J Chem Inst Taiwan. doi 10.1016/j.jtice.2014.10.018

41.

Raja MAZ, Manzar MA, Samar R, An Efficient computational intelligence approach for solving fractional order Riccati equations using ANN and SQP. Appl Math Model. doi:10.1016/j.apm.2014.11.024

42.

Rarisi Daniel R et al (2003) solving differential equations with unsupervised neural networks. Chem Eng Process 42:715–721CrossRef

43.

Aarts LP, Van Der Veer P (2001) Neural network method for solving the partial differential equations. Neural Process Lett 14:261–271CrossRefMATH

44.

Hornik K, Stichcombr M, White H (1990) Universal approximation of an unknown mapping and its derivatives using multilayer feed forward networks. Neural Netw 3:551–560CrossRef

45.

Meada AJ, Fernandez AA (1994) The numerical solution of linear ordinary differential equation by feedforward neural network. Math Comput Model 19:1–25CrossRef

46.

Raja MAZ, Khan JA, Qureshi IM (2011) Solution of fractional order system of Bagley-Torvik equation using evolutionary computational intelligence. J Math Probl Eng 2011:1–18

47.

Zhao HM, Chen KZ (2002) Neural network for solving systems of nonlinear equations. Acta Electronica 30:601–604

48.

Khan JA, Raja MAZ, Qureshi IM (2011) Stochastic computational approach for complex non-linear ordinary differential equations. Chin Phys Lett 28(2):020206MathSciNetCrossRef

49.

Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of the IEEE international conference on neural networks, Perth, Australia, IEEE Service Center, vol. 4. Piscataway, NJ, pp 1942–1948

50.

Lee KC, Jhang JY (2006) Application of particle swarm optimization algorithm to the optimization of unequally spaced antenna arrays. J Electromagn Wave Appl 20:2001–2012

51.

Coello CA, Pulido GT, Lechuga MS (2004) handling multiple objectives with particle swarm optimization. IEEE Trans Evolut Comput 8:256–279CrossRef

52.

Pan I, Korre A, Das S, Durucan S (2012) Chaos suppression in a fractional order financial system using intelligent regrouping PSO based fractional fuzzy control policy in the presence of fractional Gaussian noise. Nonlinear Dyn 70:2445–2461MathSciNetCrossRef

53.

Gao Z, Liao X (2012) Rational approximation for fractional-order system by particle swarm optimization. Nonlinear Dyn 67:1387–1395MathSciNetCrossRefMATH

54.

Raja MAZ, Khan JA, Qureshi IM (2010) A new stochastic approach for solution of Riccati differential equation of fractional order. Ann Math Artif Intell 60(3–4):229–250MathSciNetCrossRefMATH

55.

Sun J, Liu X (2013) A novel APSO-aided maximum likelihood identification method for Hammerstein systems. Nonlinear Dyn 73:449–462CrossRefMATH

56.

Lee KY, El-Sharkawi MA (2008) Modern heuristic optimization techniques: theory and application to power systems. Wiley, HobokenCrossRef

57.

Kennedy J, Eberhart R (2001) Swarm intelligence, 1st edn. Academic press, San Diego

58.

Hsu CH, Tsou CS, Yu FJ (2009) Multicriteria tradeoffs in inventory control using memetic particle swarm optimization. Int J Innov Comput Inf Control 5:3755–3768

59.

Hooke R, Jeeves R (1961) Direct search solution of numerical and statistical problems. J Assoc Comput Mach 8:212–229CrossRefMATH

60.

Pelillo M, Torsello A (2006) Payoff-monotonic game dynamics and the maximum clique problem. Neural Comput 18(5):1215–1258

61.

Huang Z et al (2003) Applying SVM in license plate character recognition. J Comput Eng 29(5):192–194

62.

Ghosh A, Chowdhury A, Giri R, Das S, Abraham A (2010) A hybrid evolutionary direct search technique for solving Optimal Control problems. HIS 125-130

63.

Fan LT, Hwang YH, Hwang CL (1970) Applications of modern optimal control theory to environmental control of confined spaces and life support systems. Part 3- optimal control of systems in which straight variables have equally constraints at the final process time, vol 5, no 3–4, pp 125–136

64.

Wang L (2005) Support vector machines: theory and application. Springer, Berlin

65.

Shampine LF, Reichelt MW (1997) The MATLAB ode suite. SIAM J Sci Comput 18:1–22MathSciNetCrossRefMATH

Title: Design and application of nature inspired computing approach for nonlinear stiff oscillatory problems
Authors: Junaid Ali Khan
Muhammad Asif Zahoor Raja
Muhammed I. Syam
Shujaat Ali Khan Tanoli
Saeed Ehsan Awan
Publication date: 01-10-2015
Publisher: Springer London
Published in: Neural Computing and Applications / Issue 7/2015
Print ISSN: 0941-0643
Electronic ISSN: 1433-3058
DOI: https://doi.org/10.1007/s00521-015-1841-z

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