Bio-inspired computing platform for reliable solution of Bratu-type equations arising in the modeling of electrically conducting solids

Abstract In this study, a bio-inspired computing approach is developed to solve Bratu-type equations arising in modeling of electrically conducting solids and various other physical phenomena. We employ feed-forward artificial neural networks (ANN) optimized with genetic algorithm (GA) and the active-set method (ASM). The mathematical formulation consists of an ANN with an unsupervised error, which is minimized by tuning weights of the network. The evolutionary technique based on GAs is used as a tool for global search of the weights in conjunction with the ASM for rapid local convergence. The designed methodology is applied to solve a number of initial and boundary value problems based on Bratu equations. Monte Carlo simulations and their statistical analyses are used to validate accuracy, convergence and effectiveness of the scheme. Comparison of results is made with exact solutions, the fully explicit Runge–Kutta numerical method, and other reported solutions of analytical and numerical solvers to establish correctness of the designed scheme.

[1]  Sennur Somali,et al.  Taylor's Decomposition on two points for one-dimensional Bratu problem , 2010 .

[2]  M. A. Manzar,et al.  An efficient computational intelligence approach for solving fractional order Riccati equations using ANN and SQP , 2015 .

[3]  Raja Muhammad Asif Zahoor,et al.  Numerical treatment for nonlinear MHD Jeffery-Hamel problem using neural networks optimized with interior point algorithm , 2014, Neurocomputing.

[4]  Muhammad Asif Zahoor Raja,et al.  Solution of the one-dimensional Bratu equation arising in the fuel ignition model using ANN optimised with PSO and SQP , 2014 .

[5]  Raja Muhammad Asif Zahoor,et al.  Reliable numerical treatment of nonlinear singular Flierl-Petviashivili equations for unbounded domain using ANN, GAs, and SQP , 2016, Appl. Soft Comput..

[6]  Scott W. Sloan,et al.  A steepest edge active set algorithm for solving sparse linear programming problems , 1988 .

[7]  Junaid Ali Khan,et al.  Solution of Fractional Order System of Bagley-Torvik Equation Using Evolutionary Computational Intelligence , 2011 .

[8]  A. Serghini Mounim,et al.  From the fitting techniques to accurate schemes for the Liouville‐Bratu‐Gelfand problem , 2006 .

[9]  Alaeddin Malek,et al.  Solving initial-boundary value problems for systems of partial differential equations using neural networks and optimization techniques , 2009, J. Frankl. Inst..

[10]  M. Raja.  Stochastic numerical treatment for solving Troesch’s problem , 2014 .

[11]  D. Frank-Kamenetskii,et al.  Diffusion and heat exchange in chemical kinetics , 1955 .

[12]  Muhammad Asif Zahoor Raja,et al.  Numerical treatment for boundary value problems of Pantograph functional differential equation using computational intelligence algorithms , 2014 .

[13]  Abdul-Majid Wazwaz,et al.  Nature-inspired computing approach for solving non-linear singular Emden–Fowler problem arising in electromagnetic theory , 2015, Connect. Sci..

[14]  Hanif D. Sherali,et al.  Complementarity Active-Set Algorithm for Mathematical Programming Problems with Equilibrium Constraints , 2007 .

[15]  Raja Muhammad Asif Zahoor,et al.  Design of stochastic solvers based on genetic algorithms for solving nonlinear equations , 2014, Neural Computing and Applications.

[16]  R. Jalilian,et al.  Non-polynomial spline method for solving Bratu's problem , 2010, Comput. Phys. Commun..

[17]  Abdul-Majid Wazwaz,et al.  A variational approach to a BVP arising in the modelling of electrically conducting solids , 2012 .

[18]  Mehmet Pakdemirli,et al.  New perturbation-iteration solutions for Bratu-type equations , 2010, Comput. Math. Appl..

[19]  B. Batiha,et al.  Numerical Solution of Bratu-type Equations by the Variational Iteration Method , 2010 .

[20]  Muhammad Aslam Noor,et al.  Variational Iteration Method for Solving Initial and Boundary Value Problems of Bratu-type , 2008 .

[21]  Raja Muhammad Asif Zahoor,et al.  Numerical treatment for solving one-dimensional Bratu problem using neural networks , 2012, Neural Computing and Applications.

[22]  M. Raja,et al.  Design of bio-inspired computational intelligence technique for solving steady thin film flow of Johnson–Segalman fluid on vertical cylinder for drainage problems , 2016 .

[23]  Raja Muhammad Asif Zahoor,et al.  Application of three unsupervised neural network models to singular nonlinear BVP of transformed 2D Bratu equation , 2014, Neural Computing and Applications.

[24]  Hikmet Caglar,et al.  B-spline method for solving Bratu's problem , 2010, Int. J. Comput. Math..

[25]  Raja Muhammad Asif Zahoor,et al.  Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation , 2015, Appl. Soft Comput..

[26]  Raja Muhammad Asif Zahoor,et al.  Neural network optimized with evolutionary computing technique for solving the 2-dimensional Bratu problem , 2012, Neural Computing and Applications.

[27]  Raza Samar,et al.  Numerical treatment of nonlinear MHD Jeffery–Hamel problems using stochastic algorithms , 2014 .

[28]  Muhammed I. Syam,et al.  The modified Broyden-variational method for solving nonlinear elliptic differential equations , 2007 .

[29]  Zheng Shijie,et al.  Genetic algorithm based wireless vibration control of multiple modal for a beam by using photostrictive actuators , 2014 .

[30]  Ron Buckmire,et al.  Application of a Mickens finite‐difference scheme to the cylindrical Bratu‐Gelfand problem , 2004 .

[31]  Raja Muhammad Asif Zahoor,et al.  Design and application of nature inspired computing approach for nonlinear stiff oscillatory problems , 2015, Neural Computing and Applications.

[32]  Abdul-Majid Wazwaz,et al.  Stochastic numerical solver for nanofluidic problems containing multi-walled carbon nanotubes , 2016, Appl. Soft Comput..

[33]  Syed Muslim Shah,et al.  Unsupervised neural network model optimized with evolutionary computations for solving variants of nonlinear MHD Jeffery-Hamel problem , 2015 .

[34]  Raja Muhammad Asif Zahoor,et al.  A new stochastic approach for solution of Riccati differential equation of fractional order , 2010, Annals of Mathematics and Artificial Intelligence.

[35]  G. Hariharan,et al.  An Efficient Wavelet Method for Initial Value Problems of Bratu-Type Arising in Engineering , 2013 .

[36]  K. Schmitt,et al.  The Liouville–Bratu–Gelfand Problem for Radial Operators , 2002 .

[37]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[38]  William W. Hager,et al.  A New Active Set Algorithm for Box Constrained Optimization , 2006, SIAM J. Optim..

[39]  Muhammad Asif Zahoor Raja,et al.  Unsupervised neural networks for solving Troesch's problem , 2013 .

[40]  Reza Pourgholi,et al.  Solving an inverse heat conduction problem using genetic algorithm: Sequential and multi-core parallelization approach , 2014 .

[41]  D. Parisi,et al.  Solving differential equations with unsupervised neural networks , 2003 .

[42]  Junaid Ali Khan,et al.  Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations , 2011 .

[43]  Abdul-Majid Wazwaz,et al.  Adomian decomposition method for a reliable treatment of the Bratu-type equations , 2005, Appl. Math. Comput..

[44]  Suheil A. Khuri,et al.  A new approach to Bratu's problem , 2004, Appl. Math. Comput..

[45]  Junaid Ali Khan,et al.  Stochastic numerical treatment for thin film flow of third grade fluid using unsupervised neural networks , 2015 .

[46]  G. Bratu.  Sur les équations intégrales non linéaires , 1913 .

[47]  Saeid Abbasbandy,et al.  The Lie-group shooting method for solving the Bratu equation , 2011 .

[48]  Raja Muhammad Asif Zahoor,et al.  Solution of the 2-dimensional Bratu problem using neural network, swarm intelligence and sequential quadratic programming , 2014, Neural Computing and Applications.

[49]  M. M. Toufigh,et al.  The solution of unconfined seepage problem using Natural Element Method (NEM) coupled with Genetic Algorithm (GA) , 2013 .

[50]  S. Khuri,et al.  An Algorithm for Solving Boundary Value Problems , 2000 .

[51]  Muhammed I. Syam,et al.  An efficient method for solving Bratu equations , 2006, Appl. Math. Comput..