Abstract
In the present study, stochastic numerical computing approach is developed by applying artificial neural networks (ANNs) to compute the solution of Lane–Emden type boundary value problems arising in thermodynamic studies of the spherical gas cloud model. ANNs are used in an unsupervised manner to construct the energy function of the system model. Strength of efficient local optimization procedures based on active-set (AS), interior-point (IP) and sequential quadratic programming (SQP) algorithms is used to optimize the energy functions. The performance of all three design methodologies ANN-AS, ANN-IP and ANN-SQP is evaluated on different nonlinear singular systems. The effectiveness of the proposed schemes in terms of accuracy and convergence is established from the results of statistical indicators.
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Parand K, Shahini M, Dehghan M (2009) Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type. J Comput Phys 228(23):8830–8840
Parand K, Dehghan M, Rezaei AR, Ghaderi SM (2010) An approximation algorithm for the solution of the nonlinear Lane–Emden type equations arising in astrophysics using Hermite functions collocation method. Comput Phys Commun 181(6):1096–1108
Parand K, Rezaei AR, Taghavi A (2010) Lagrangian method for solving Lane–Emden type equation arising in astrophysics on semi-infinite domains. Acta Astronaut 67:673–680
Mukherjee S, Roy B, Chaterjee PK (2011) Solution of Lane–Emden equation by differential transform method. Int J Nonlinear Sci 12(4):478
Horedt GP (1986) Seven-digit tables of Lane–Emden functions. Astrophys Space Sci 126:357
Bender CM, Milton KA, Pinsky SS, Simmons LM Jr (1989) A new perturbative approach to nonlinear problems. J Math Phys 30:1447
Shawagfeh NT (1993) Nonperturbative approximate solution for Lane–Emden equation. J Math Phys 34:4364
Wazwaz AM (2001) A new algorithm for solving differential equations of Lane–Emden type. Appl Math Comput 118:287–310
Mandelzweig VB, Tabakin F (2001) Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. Comput Phys Commun 141:268
He JH (2003) Variational approach to the Lane–Emden equation. Appl Math Comput 143:539
Liao SJ (2003) A new algorithm for solving singular IVPs of Lane–Emden type. Appl Math Comput 142:1
Singh OP, Pandey RK, Singh VK (2009) An analytic algorithm of Lane–Emden type equations arising in astrophysics using modified homotopy analysis method. Comput Phys Commun 180(7):1116–1124
Yousefi SA (2006) Legendre wavelets method for solving differential equations of Lane–Emden type. Appl Math Comput 181(2):1417
Chowdhury MSH, Hashim I (2007) Solutions of a class of singular second-order IVPs by homotopy-perturbation method. Phys Lett A 365(5):439
Yildirim A, Özis T (2007) Solutions of singular IVPs of Lane–Emden type by homotopy perturbation method. Phys Lett A 369:70–76
Marzban HR, Tabrizidooz HR, Razzaghi M (2008) Hybrid functions for nonlinear initial-value problems with applications to Lane–Emden type equations. Phys Lett A 372(37):5883–5886
Ramos JI (2008) Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method. Chaos Solitons Fractals 38(2):400–408
VanGorder RA, Vajravelu K (2008) Analytic and numerical solutions to the Lane–Emden equation. Phys Lett A 372:6060–6065
Dehghan M, Shakeri F (2008) Approximate solution of a differential equation arising in astrophysics using the variational iteration method. New Astron 13(1):53–59
Kumar N, Pandey RK, Cattani C (2011) Solution of Lane–Emden type equations Bernstein operational matrix of integration. ISRN Astron Astrophys 2011(2011):351747. doi:10.5402/2011/351747
Pandey RK, Kumar N (2012) Solution of Lane–Emden type equations using Bernstein operational matrix of differentiation. New Astron 17(3):303–308
Iqbal S, Javed A (2011) Application of optimal homotopy asymptotic method for the analytic solution of singular Lane–Emden type equation. Appl Math Comput 217(19):7753–7761
Eslahchi MR, Dehghan M, Ahmadi-Asl S (2012) The general shifted Jacobi matrix method for solving the general high order linear differential difference equations with variable coefficients. Appl Math Model 36:3387
Boubaker K, VanGorder RA (2012) Application of the BPES to Lane–Emden equations governing polytropic and isothermal gas spheres. New Astron 17:565–569
Rismani AM, Monfared H (2012) Numerical solution of singular IVPs of Lane–Emden type using a modified Legendre-spectral method. Appl Math Model 36(10):4830–4836
Pandey RK, Kumar N, Bhardwaj A, Dutta G (2012) Solution of Lane–Emden type equations using Legendre operational matrix of differentiation. Appl Math Comput 218(14):7629–7637
Dehghan M, Aryanmehr S, Eslahch MR (2013) A technique for the numerical solution of initial-value problems based on a class of Birkhoff-type interpolation method. J Comput Appl Math 244:125–139
Lakestani M, Dehghan M (2013) Four techniques based on the B-spline expansion and the collocation approach for the numerical solution of the Lane–Emden equation. Math Methods Appl Sci 36(16):2243–2253
Căruntu B, Bota C (2013) Approximate polynomial solutions of the nonlinear Lane–Emden type equations arising in astrophysics using the squared remainder minimization method. Comput Phys Commun 184:1643–1648
Kumar M, Yadav N (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
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
Shirvany Y, Hayati M, Moradian R (2008) Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks. Commun Nonlinear Sci Numer Simul 13(10):2132–2145
Hayati M, Karami B (2007) Feedforward neural network for solving partial differential equations. J Appl Sci 7(19):2812–2817
Raja MAZ, Samar R (2014) Numerical treatment of nonlinear MHD Jeffery–Hamel problems using stochastic algorithms. Comput Fluids 91:28–46
Khan JA, Raja MAZ, Rashidi MM, Syam MI, Wazwaz AM (2015) Nature-inspired computing approach for solving non-linear singular Emden–Fowler problem arising in electromagnetic theory. Connect Sci 27(4):377–396
Raja MAZ, Samar R, Alaidarous ES, Shivanian E (2016) Bio-inspired computing platform for reliable solution of Bratu-type equations arising in the modeling of electrically conducting solids. Appl Math Model 40(11–12):5964–5977
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
Raja MAZ, Samar R, Haroon T, Shah SM (2015) Unsupervised neural network model optimized with evolutionary computations for solving variants of nonlinear MHD Jeffery–Hamel problem. Appl Math Mech 36(12):1611–1638
Raja MAZ, Khan JA, Haroon T (2015) Stochastic numerical treatment for thin film flow of third grade fluid using unsupervised neural networks. J Taiwan Inst Chem Eng 48:26–39
Raja MAZ, Shah FH, Khan AA, Khan NA (2016) Design of bio-inspired computational intelligence technique for solving steady thin film flow of Johnson–Segalman fluid on vertical cylinder for drainage problems. J Taiwan Inst Chem Eng 60:59–75
Raja MAZ (2013) Unsupervised neural networks for solving Troesch’s problem. Chin Phys B 23(1):018903
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 25(7–8):1585–1601
Khan JA, Raja MAZ, Syam MI, Tanoli SAK, Awan SE (2015) Design and application of nature inspired computing approach for nonlinear stiff oscillatory problems. Neural Comput Appl 26(7):1763–1780
Raja MAZ (2014) Stochastic numerical treatment for solving Troesch’s problem. Inf Sci 279:860–873
Raja MAZ, Khan MAR, Mahmood T, Farooq U, Chaudhary NI (2016) Design of bio-inspired computing technique for nanofluidics based on nonlinear Jeffery–Hamel flow equations. Can J Phys 94(999):1–16
Raja MAZ, Farooq U, Chaudhary NI, Wazwaz AM (2016) Stochastic numerical solver for nanofluidic problems containing multi-walled carbon nanotubes. Appl Soft Comput 38:561–586
Raja MAZ, Khan JA, Siddiqui AM, Behloul D, Haroon T, Samar R (2015) Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation. Appl Soft Comput 26:244–256
Raja MAZ (2014) Numerical treatment for boundary value problems of pantograph functional differential equation using computational intelligence algorithms. Appl Soft Comput 24:806–821
Arqub OA, Abo-Hammour Z (2014) Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm. Inf Sci 279:396–415
Abu Arqub O, Abo-Hammour Z, Momani S, Shawagfeh N (2012) Solving singular two-point boundary value problems using continuous genetic algorithm. In: Abstract and applied analysis, vol 2012. Hindawi Publishing Corporation
Abo-Hammour Z, Abu Arqub O, Momani S, Shawagfeh N (2014) Optimization solution of Troesch’s and Bratu’s problems of ordinary type using novel continuous genetic algorithm. Discrete Dyn Nat Soc 2014(2014):401696. doi:10.1155/2014/401696
Mall S, Chakraverty S (2015) Numerical solution of nonlinear singular initial value problems of Emden–Fowler type using Chebyshev Neural Network method. Neurocomputing 149:975–982
Mall S, Chakraverty S (2016) Application of Legendre Neural Network for solving ordinary differential equations. Appl Soft Comput 43:347–356
Raja MAZ, Manzar MA, Samar R (2015) An efficient computational intelligence approach for solving fractional order Riccati equations using ANN and SQP. Appl Math Model 39(10):3075–3093
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
Karmarkar N (1984) A new polynomial time algorithm for linear programming. Combinatorica 4:373–395
Wright SJ (1997) Primal-dual interior-point methods. SIAM, Philadelphia. ISBN 0-89871-382-X
Yana W, Wenb L, Lic W, Chunga CY, Wong KP (2011) Decomposition–coordination interior point method and its application to multi-area optimal reactive power flow. Int J Electrc Power Energy Syst 33(1):55–60
Duvvuru N, Swarup KS (2011) A hybrid interior point assisted differential evolution algorithm for economic dispatch. IEEE Trans Power Syst 26(2):541–549
Hager WW, Zhang H (2006) A new active set algorithm for box constrained optimization. SIAM J Optim 17(02):526–557
Sloan SW (2005) A steepest edge active set algorithm for solving sparse linear programming problems. Int J Numer Methods Eng 26(12):2671–2685
Judice JJ, Sherali HD, Ribeiro IM, Faustino AM (2007) Complementarity active set algorithm for mathematical programming problems with equilibrium constraints. J Optim Theory Appl 134(3):467–481
Wright S, Nocedal J (1999) Numerical optimization. Springer Sci 35:67–68
Ahmad I, Mukhtar A (2015) Stochastic approach for the solution of multi-pantograph differential equation arising in cell-growth model. Appl Math Comput 261:360
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Ahmad, I., Raja, M.A.Z., Bilal, M. et al. Neural network methods to solve the Lane–Emden type equations arising in thermodynamic studies of the spherical gas cloud model. Neural Comput & Applic 28 (Suppl 1), 929–944 (2017). https://doi.org/10.1007/s00521-016-2400-y
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DOI: https://doi.org/10.1007/s00521-016-2400-y