Unsupervised neural network model optimized with evolutionary computations for solving variants of nonlinear MHD Jeffery-Hamel problem

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• 作者：M. A. Z. Raja ; R. Samar ; T. Haroon ; S. M. Shah
• 关键词：Jeffery ; Hamel problem ; neural network ; genetic algorithm (GA) ; nonlinear ordinary differential equation (ODE) ; hybrid technique ; sequential quadratic programming ; O361 ; 65L10 ; 76M35 ; 65C20 ; 65C05
• 刊名：Applied Mathematics and Mechanics
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：36
• 期：12
• 页码：1611-1638
• 全文大小：1,142 KB
• 参考文献：[1]Jeffery, G. B. The two-dimensional steady motion of a viscous fluid. Philosophical Magazine, 6, 455鈥?65 (1915)CrossRef
  [2]Hamel, G. Spiralfrmige bewgungen zher flssigkeiten. Philosophical Magazine, 25, 34鈥?0 (1916)MATH
  [3]Reza, M. S. Channel Entrance Flow, Ph.D. dissertation, The University of Western Ontario, Ontario (1997)
  [4]Hamadiche, M., Scott, J., and Jeandel, D. Temporal stability of Jeffery-Hamel flow. Philosophical Magazine, 268, 71鈥?8 (1994)MATH
  [5]Makinde, O. D. and Mhone, P. Y. Hermite-Pad茅 approximation approach to MHD Jeffery-Hamel flows. Philosophical Magazine, 181, 966鈥?72 (2006)MATH
  [6]Schlichting, H. Boundary-Layer Theory, McGraw-Hill Press, New York (2000)MATH CrossRef
  [7]McAlpine, A. and Drazin, P. G. On the spatio-temporal development of small perturbations of Jeffery-Hamel flows. Fluid Dynamics Research, 22, 123鈥?38 (1998)MATH MathSciNet CrossRef
  [8]Axford, W. I. The magnetohydrodynamic Jeffrey-Hamel problem for a weakly conducting fluid. Quarterly Journal of Mechanics and Applied Mathematics, 14, 335鈥?51 (1961)MATH MathSciNet CrossRef
  [9]Esmaili, Q., Ramiar, A., Alizadeh, E., and Ganji, D. D. Anapproximation of the analytical solution of the Jeffery-Hamel flow by decomposition method. Physics Letters A, 372, 3434 (2008)MATH CrossRef
  [10]Makinde, O. D. Effect of arbitrary magnetic Reynolds number on MHD flows in convergentdivergent channels. International Journal of Numerical Methods for Heat and Fluid Flow, 18, 697鈥?07 (2008)MATH MathSciNet CrossRef
  [11]He, J. H. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics, 35, 37鈥?3 (2000)MATH MathSciNet CrossRef
  [12]Reza, H., Sadegh, P., and Saeed, D. MHD flow of an incompressible viscous fluid through convergent or divergent channels in presence of a high magnetic field. Journal of Applied Mathematics, 2012, 157067 (2012)
  [13]Domairry, G., Mohsenzadeh, A., and Famouri, M. The application of homotopy analysis method to solve nonlinear differential equation governing Jeffery-Hamel flow. Communications in Nonlinear Science and Numerical Simulation, 14, 85鈥?5 (2008)MathSciNet CrossRef
  [14]He, J. H. Variational iteration method: a kind of non-linear analytical technique: some examples. International Journal of Non-Linear Mechanics, 34, 699鈥?08 (1999)MATH CrossRef
  [15]He, J. H. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics, 35, 37鈥?3 (2000)MATH MathSciNet CrossRef
  [16]Sheikholeslami, M., Ganji, D. D., Ashorynejad, H. R., and Rokni, H. B. Analytical investigation of Jeffery-Hamel flow with high magnetic field and nanoparticle by Adomian decomposition method. Applied Mathematics and Mechanics (English Edition), 33(1), 25鈥?6 (2012) DOI 10.1007/s10483012-1531-7MATH MathSciNet CrossRef
  [17]Bararnia, H., Ganji, Z. Z., Ganji, D. D., and Moghimi, S. M. Numerical and analytical approaches to MHD Jeffery-Hamel flow in a porous channel. International Journal of Numerical Methods for Heat and Fluid Flow, 22, 491鈥?02 (2012)CrossRef
  [18]Abbasbandy, S. and Shivanian, E. Exact analytical solution of the MHD Jeffery-Hamel flow problem. Meccanica, 47, 1379鈥?389 (2012)MATH MathSciNet CrossRef
  [19]Motsa, S. S., Sibanda, P., Awad, F. G., and Shateyi, S. A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem. Computers and Fluids, 39, 1219鈥?225 (2010)MATH MathSciNet CrossRef
  [20]Khan, J. A., Raja, M. A. Z., and Qureshi, I. M. Stochastic computational approach for complex nonlinear ordinary differential equations. Chinese Physics Letter, 28, 020206 (2011)CrossRef
  [21]Zongben, X., Mingwei, D., and Deyu, M. Fast and efficient strategies for model selection of support vector machines. IEEE Transactions on Systems, Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions, 39, 1292鈥?307 (2009)CrossRef
  [22]Meada, A. J. and Fernandez, A. A. The numerical solution of linear ordinary differential equation by feed forward neural network. Mathematical and Computer Modelling, 19, 1鈥?5 (1994)CrossRef
  [23]Khan, J. A., Raja, M. A. Z., and Qureshi, I. M. Novel approach for van der Pol oscillator on the continuous time domain. Chinese Physics Letters, 28, 110205 (2011)CrossRef
  [24]Khan, J. A., Raja, M. A. Z., and Qureshi, I. M. Numerical treatment of nonlinear Emden-Fowler equation using Stochastic technique. Annals of Mathematics and Artificial Intelligence, 63, 185鈥?07 (2011)MathSciNet CrossRef
  [25]Raja, M. A. Z. Solution of the one-dimensional Bratu equation arising in the fuel ignition model using ANN optimised with PSO and SQP. Connection Science, 26, 195鈥?14 (2014)CrossRef
  [26]Raja, M. A. Z. and Ahmad, S. I. Numerical treatment for solving one-dimensional Bratu problem using neural networks. Neural Computing and Applications, 24, 549鈥?61 (2014)CrossRef
  [27]Raja, M. A. Z., Ahmad, S. I., and Raza, S. Neural network optimized with evolutionary computing technique for solving the 2-dimensional Bratu problem. Neural Computing and Applications, 23, 2199鈥?210 (2013)CrossRef
  [28]Raja, M. A. Z., Samar, R., and Rashidi, M. M. Application of three unsupervised neural network models to singular nonlinear BVP of transformed 2D Bratu equation. Neural Computing and Applications, 25, 1585鈥?601 (2014)CrossRef
  [29]Raja, M. A. Z., Ahmad, S. I., and Raza, S. Solution of the 2-dimensional Bratu problem using neural network, swarm intelligence and sequential quadratic programming. Neural Computing and Applications, 25, 1723鈥?739 (2014)CrossRef
  [30]Raja, M. A. Z., Sabir, Z., Mahmood, N., Eman, S. A., and Khan, A. I. Design of Stochastic solvers based on variants of genetic algorithms for solving nonlinear equations. Neural Computing and Applications, 26, 1鈥?3 (2015)CrossRef
  [31]Raja, M. A. Z. and Samar, R. Numerical treatment for nonlinear MHD Jeffery-Hamel problem using neural networks optimized with interior point algorithm. Neurocomputing, 124, 178鈥?93 (2014)CrossRef
  [32]Raja, M. A. Z. and Samar, R. Numerical treatment of nonlinear MHD Jeffery-Hamel problems using stochastic algorithms. Computers and Fluids, 91, 28鈥?6 (2014)MathSciNet CrossRef
  [33]Raja, M. A. Z., Khan, J. A., and Haroon, T. Stochastic numerical treatment for thin film flow of third grade fluid using unsupervised neural networks. Journal of Chemical Institute of Taiwan, 48, 26鈥?9 (2014)CrossRef
  [34]Raja, M. A. Z. Stochastic numerical techniques for solving Troesch鈥檚 Problem. Information Sciences, 279, 860鈥?73 (2014)MathSciNet CrossRef
  [35]Raja, M. A. Z. Unsupervised neural networks for solving Troesch鈥檚 problem. Chinese Physics B, 23, 018903 (2014)CrossRef
  [36]Raja, M. A. Z. Numerical treatment for boundaryvalue problems of Pantograph functional differential equation using computational intelligence algorithms. Applied Soft Computing, 24, 806鈥?21 (2014)CrossRef
  [37]Raja, M. A. Z., Khan, J. A., Behloul, D., Tahira, H., Siddiqui, A. M., and Samar, R. Exactly satisfying initial conditions neural network models for mumerical treatment of first Painlev茅 equation. Applied Soft Computing, 26, 244鈥?56 (2015)CrossRef
  [38]Raja, M. A. Z., Khan, J. A., Ahmad, S. I., and Qureshi, I. M. A new stochastic technique for Painlev茅 equation-I using neural network optimized with swarm intelligence. Computational Intelligence and Neuroscience, 2012, 1鈥?0 (2012)CrossRef
  [39]Raja, M. A. Z., Khan, J. A., and Qureshi, I. M. Swarm intelligent optimized neural networks for solving fractional differential equations. International Journal of Innovative Computing, Information and Control, 7, 6301鈥?318 (2011)
  [40]Raja, M. A. Z., Khan, J. A., and Qureshi, I. M. Heuristic computational approach using swarm intelligence in solving fractional differential equations. Computers and Fluids, 91, 28鈥?6 (2014)MathSciNet CrossRef
  [41]Raja, M. A. Z., Khan, J. A., and Qureshi, I. M. A new Stochastic approach for solution of Riccati differential equation of fractional order. Annals of Mathematics and Artificial Intelligence, 60, 229鈥?50 (2010)MATH MathSciNet CrossRef
  [42]Raja, M. A. Z., Khan, J. A., and Qureshi, I.M. Solution of fractional order system of Bagley-Torvik equation using Evolutionary computational intelligence. Mathematical Problems in Engineering, 2011, 675075 (2011)MathSciNet CrossRef
  [43]Rajai, M. A. Z., Manzar, M. A., and Samar, R. An efficient computational intelligence approach for solving fractional order Riccati equations using ANN and SQP. Applied Mathematical Modeling, 39, 3075鈥?093 (2015)CrossRef
  [44]Joneidi, A. A., Domairry, G., and Babaelahi, M. Three analytical methods applied to JefferyHamel flow. Communications in Nonlinear Science and Numerical Simulation, 15, 3423鈥?434 (2010)CrossRef
  [45]Moghimia, S. M., Ganji, D. D., Bararnia, H., Hosseini, M., and Jalaal, M. homotopy perturbation method for nonlinear MHD Jeffery-Hamel problem. Computers and Mathematics with Applications, 61, 2213鈥?216 (2011)MathSciNet CrossRef
  [46]Ganji, Z. Z., Ganji, D. D., and Esmaeilpour, M. Study on nonlinear Jeffery-Hamel flow by He鈥檚 semi-analytical methods and comparison with numerical results. Computers and Mathematics with Applications, 58, 2107鈥?116 (2009)MATH MathSciNet CrossRef
  [47]Moghimi, S. M., Domairry, G., Soheil, S., Ghasemi, E., and Bararnia, H. Application of homotopy analysis method to solve MHD Jeffery-Hamel flows in non-parallel walls. Advances in Engineering Software, 42, 108鈥?13 (2011)MATH CrossRef
  [48]Esmaeilpour, M. and Ganji, D. D. Solution of the Jeffery-Hamel flow problem by optimal homotopy asymptotic method. Computers and Mathematics with Applications, 59, 3405鈥?411 (2010)MATH MathSciNet CrossRef
  [49]Beidokhti, R. S. and Malek, A. Solving initial-boundary value problems for systems of partial differential equations using neural networks and optimization techniques. Journal of the Franklin Institute, 346, 898鈥?13 (2009)MATH CrossRef
  [50]Parisia, D. R., Mariani, M. C., and Laborde, M. A. Solving differential equations with unsupervised neural networks. Chemical Engineering and Processing: Process Intensification, 42, 715鈥?21 (2003)CrossRef
  [51]Nocedal, J. and Wright, S. J. Numerical Optimization. Springer Series in Operations Research, Springer Verlag, Berlin (1999)
  [52]Fletcher, R. Practical Methods of Optimization, John Wiley and Sons, New York (1987)MATH
  [53]Schittkowski, K. NLQPL: a FORTRAN-subroutine solving constrained nonlinear programming problems. Annals of Operations Research, 5, 485鈥?00 (1985)MathSciNet CrossRef
  [54]Zhu, Z. B., Jian, J. B., and Cong, Z. An SQP algorithm for mathematical programs with nonlinear complementarity constraints. Applied Mathematics and Mechanics (English Edition), 30(5), 659鈥?68 (2009) DOI 10.1007/s10483-009-0512-xMATH MathSciNet CrossRef
  [55]Qian, Q. J. and Sun, D. J. Numerical method for optimum motion of undulatory swimming plate in fluid flow. Applied Mathematics and Mechanics (English Edition), 32(3), 339鈥?48 (2011) DOI 10.1007/s10483-011-1419-xMATH MathSciNet CrossRef
  [56]Holland, H. Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, University of Michigan Press, Ann Arbor (1975)
  [57]Wang, G. M., Wang, X. J., Wan, Z. P., and Jia, S. H. An adaptive genetic algorithm for solving bilevel linear programming problem. Applied Mathematics and Mechanics (English Edition), 28(12), 1605鈥?612 (2007) DOI 10.1007/s10483-007-1207-1MATH MathSciNet CrossRef
  [58]Mousavi, S. S., Hooman, K., and Mousavi, S. J. Genetic algorithm optimization for finned channel performance. Applied Mathematics and Mechanics (English Edition), 28(12), 1597鈥?604 (2007) DOI 10.1007/s10483-007-1206-zMATH CrossRef
  [59]Wang, L., Wang, T. G., and Luo, Y. Improved non-dominated sorting genetic algorithm (NSGA)-II in multi-objective optimization studies of wind turbine blades. Applied Mathematics and Mechanics (English Edition), 32(6), 739鈥?48 (2011) DOI 10.1007/s10483-011-1453-xMATH MathSciNet CrossRef
  [60]Bai, Y. L., Ma, X. Y., and Meng, X. Lift enhancement of airfoil and tip flow control for wind turbine. Applied Mathematics and Mechanics (English Edition), 32(7), 825鈥?36 (2011) DOI 10.1007/s10483-011-1462-8MATH MathSciNet CrossRef
  
• 作者单位：M. A. Z. Raja (1)
  R. Samar (2)
  T. Haroon (3)
  S. M. Shah (2)
  
  1. Department of Electrical Engineering, COMSATS Institute of Information Technology, Attock Campus, Attock, 43600, Pakistan
  2. Department of Electrical Engineering, Mohammad Ali Jinnah University, Islamabad, 44000, Pakistan
  3. Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, 44000, Pakistan
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Applications of Mathematics
  Mechanics
  Mathematical Modeling and IndustrialMathematics
  Chinese Library of Science
  
• 出版者：Shanghai University, in co-publication with Springer
• ISSN：1573-2754

文摘

A heuristic technique is developed for a nonlinear magnetohydrodynamics (MHD) Jeffery-Hamel problem with the help of the feed-forward artificial neural network (ANN) optimized with the genetic algorithm (GA) and the sequential quadratic programming (SQP) method. The two-dimensional (2D) MHD Jeffery-Hamel problem is transformed into a higher order boundary value problem (BVP) of ordinary differential equations (ODEs). The mathematical model of the transformed BVP is formulated with the ANN in an unsupervised manner. The training of the weights of the ANN is carried out with the evolutionary calculation based on the GA hybridized with the SQP method for the rapid local convergence. The proposed scheme is evaluated on the variants of the Jeffery-Hamel flow by varying the Reynold number, the Hartmann number, and the angles of the walls. A large number of simulations are performed with an extensive analysis to validate the accuracy, convergence, and effectiveness of the scheme. The comparison of the standard numerical solution and the analytic solution establishes the correctness of the proposed designed methodologies. Keywords Jeffery-Hamel problem neural network genetic algorithm (GA) nonlinear ordinary differential equation (ODE) hybrid technique sequential quadratic programming