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Solution of the 2-dimensional Bratu problem using neural network, swarm intelligence and sequential quadratic programming

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Abstract

In this paper, stochastic techniques have been developed to solve the 2-dimensional Bratu equations with the help of feed-forward artificial neural networks, optimized with particle swarm optimization (PSO) and sequential quadratic programming (SQP) algorithms. A hybrid of the above two algorithms, referred to as the PSO-SQP method is also studied. The original 2-dimensional equations are solved by first transforming them into equivalent one-dimensional boundary value problems (BVPs). These are then modeled using neural networks. The optimization problem for training the weights of the network has been addressed using particle swarm techniques for global search, integrated with an SQP method for rapid local convergence. The methodology is evaluated by applying on three different test cases of BVPs for the Bratu equations. Monte Carlo simulations and extensive analyses are carried out to validate the accuracy, convergence and effectiveness of the schemes. A comparative study of proposed results is made with available exact solution, as well as, reported numerical results.

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References

  1. Parisi DR, Mariani MC, Laborde MA (2003) Solving differential equations with unsupervised neural networks. Chem Eng Process 42(8–9):715–721

    Article  Google Scholar 

  2. Khan JA, Raja MAZ, Qureshi IM (2011) Stochastic computational approach for complex non-linear ordinary differential equations. Chin Phys Lett 28(2):020206–020209

    Article  MathSciNet  Google Scholar 

  3. Yazdi HS, Pourreza R (2010) Unsupervised adaptive neural-fuzzy inference system for solving differential equations. Appl Soft Comput 10(1):267–275

    Article  Google Scholar 

  4. Shirvany Y, Hayati M, Moradian R (2009) Multilayer perceptron neural networks with novel unsupervised training method for numerical solution of the partial differential equations. Appl Soft Comput 9(1):20–29

    Article  Google Scholar 

  5. Beidokhti RS, Malek A (2009) Solving initial-boundary value problems for systems of partial differential equations using neural networks and optimization techniques. J Franklin Inst 346(9):898–913

    Article  MathSciNet  Google Scholar 

  6. 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. doi:10.5897/IJPS11.922

    Google Scholar 

  7. Khan JA, Raja MAZ, Qureshi IM Novel (2011) Approach for van der Pol oscillator on the continuous time domain. Chin Phys Lett 28:110205. doi:10.1088/0256-307X/28/11/110205

  8. 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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Monterola Christopher, Saloma Caesar (2001) Solving the nonlinear Schrodinger equation with an unsupervised neural network. Opt Express 9(2):16

    Article  Google Scholar 

  11. 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

    Article  Google Scholar 

  12. Raja MAZ (2014) Solution of one-dimension Bratu equation arising in fuel ignition model using ANN optimized with PSO and SQP. Connect Sci. doi:10.1080/09540091.2014.907555

  13. Raja MAZ (2014) Stochastic numerical techniques for solving Troesch’s problem. Inf Sci. doi:10.1016/j.ins.2014.04.036

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Kumar Manoj, Yadav Neha (2011) Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: a survey. Comput Math Appl 62(10):3796–3811

    Article  MATH  MathSciNet  Google Scholar 

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

  18. Raja MAZ, Khan JA, Qureshi IM (2010) Evolutionary computational intelligence in solving the fractional differential equations. Lecture notes in computer science, vol 5990, part 1. Springer, ACIIDS Hue City, Vietnam, pp 231–240

  19. Raja MAZ, Khan JA, Qureshi IM (2011) Solution of fractional order system of Bagley–Torvik equation using evolutionary computational intelligence. Math Probl Eng 2011:01–18, Article ID. 765075

  20. 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):229–250

    Article  MATH  MathSciNet  Google Scholar 

  21. 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

    Article  Google Scholar 

  22. 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

  23. Sivanandam SN, Visalakshi P (2007) Multiprocessor scheduling using hybrid particle swarm optimization with dynamically varying inertia. Int J Comput Sci Appl 4(3):95–106

    Google Scholar 

  24. Li G-D, Masuda S, Yamaguchi D, Nagai M (2009) The optimal GNN-PID control system using particle swarm optimization algorithm. Int J Innov Comput Inf Control 5(10):3457–3470

    Google Scholar 

  25. de A Araujo R (2010) Swarm-based translation-invariant morphological prediction method for financial time series forecasting. Inf Sci 180(24):4784–4805

    Article  MATH  MathSciNet  Google Scholar 

  26. Li X, Wang J (2007) A steganographic method based upon JPEG and particle swarm optimization algorithm. Inf Sci 177(15):3099–3109

    Article  Google Scholar 

  27. Syam MI, Hamdan A (2006) An efficient method for solving Bratu equations. Appl Math Comput 176:704–713

    Article  MATH  MathSciNet  Google Scholar 

  28. Syam MI (2007) The modified Broyden-variational method for solving nonlinear elliptic differential equations. Chaos Solitons Fractals 32:392–404

    Article  MATH  MathSciNet  Google Scholar 

  29. Gidas B, Ni W, Nirenberg L (1979) Symmetry and related properties via the maximum principle. Commun Math Phys 68:209–243

    Article  MATH  MathSciNet  Google Scholar 

  30. Bratu G (1914) Sur les équations intégrales non linéaires. Bull Soc Math France 43:113–142

    MathSciNet  Google Scholar 

  31. Gelfand IM (1963) Some problems in the theory of quasi-linear equations. Trans Am Math Soc Ser 2:295–381

    Google Scholar 

  32. Jacobsen J, Schmitt K (2002) The Liouville–Bratu–Gelfand problem for radial operators. J Differ Equ 184:283–298

    Article  MATH  MathSciNet  Google Scholar 

  33. Buckmire R (2003) On exact and numerical solutions of the one-dimensional planar Bratu problem. http://faculty.oxy.edu/ron/research/bratu/bratu.pdf

  34. Frank-Kamenetski DA (1955) Diffusion and heat exchange in chemical kinetics. Princeton University Press, Princeton, NJ

    Google Scholar 

  35. Wan YQ, Guo Q, Pan N (2004) Thermo-electro-hydrodynamic model for electrospinning process. Int J Nonlinear Sci Numer Simul 5(1):5–8

    Article  Google Scholar 

  36. Jalilian R (2010) Non-polynomial spline method for solving Bratu’s problem. Comput Phys Commun 181:1868–1872

    Article  MATH  MathSciNet  Google Scholar 

  37. Boyd JP (2011) One-point pseudospectral collocation for the one-dimensional Bratu equation. Appl Math Comput 217:5553–5565

    Article  MATH  MathSciNet  Google Scholar 

  38. Abbasbandy S, Hashemi MS, Liu C-S (2011) The Lie-group shooting method for solving the Bratu equation. Commun Nonlinear Sci Numer Simul 16:4238–4249

    Article  MATH  MathSciNet  Google Scholar 

  39. Nocedal J, Wright SJ (1999) Numerical optimization. Springer Series in Operations Research, Springer, Berlin

    Book  MATH  Google Scholar 

  40. Fletcher R (1987) Practical methods of optimization. Wiley, New York

    MATH  Google Scholar 

  41. Schittkowski K (1985) NLQPL: a FORTRAN-subroutine solving constrained nonlinear programming problems. Ann Oper Res 5:485–500

    Article  MathSciNet  Google Scholar 

  42. Sivasubramani S, Swarup KS (2011) Sequential quadratic programming based differential evolution algorithm for optimal power flow problem. IET Gener Transm Distrib 5(11):1149–1154

    Article  Google Scholar 

  43. Aleem SHEA, Zobaa AF, Abdel Aziz MM (2012) Optimal C-type passive filter based on minimization of the voltage harmonic distortion for nonlinear loads. IEEE Trans Ind Electron 59(1):281–289

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering, COMSATS Institute of Information Technology, Attock Campus, Attock, Pakistan

    Muhammad Asif Zahoor Raja

  2. Pakistan Institute of Engineering and Applied Science, Nilore, Islamabad, Pakistan

    Siraj-ul-Islam Ahmad

  3. Mohammad Ali Jinnah University, Islamabad, Pakistan

    Raza Samar

  4. Department of Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan

    Siraj-ul-Islam Ahmad

Authors
  1. Muhammad Asif Zahoor Raja
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  2. Siraj-ul-Islam Ahmad
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  3. Raza Samar
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Corresponding author

Correspondence to Muhammad Asif Zahoor Raja.

Appendix

Appendix

One set of design parameter of DE-ANN optimized with PSO, SQP and PSO-SQP are tabulated in Tables 10, 11 and 12 in case of Bratu equation given in problems 1, 2 and 3, respectively. The weights given in the appendix are provided in term of real number up to 14 decimal points in order to exactly reproduced the results given in the main body of manuscript and avoid unnecessary rounding of error problems.

Table 10 Parameters obtained for the DE-ANN for the case μ = 0.5
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Table 11 Parameters obtained for the DE-ANN for the case μ = 1.0
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Table 12 Parameters obtained for the DE-ANN for the case μ = 2.0
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Raja, M.A.Z., Ahmad, SuI. & Samar, R. Solution of the 2-dimensional Bratu problem using neural network, swarm intelligence and sequential quadratic programming. Neural Comput & Applic 25, 1723–1739 (2014). https://doi.org/10.1007/s00521-014-1664-3

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