Design of stochastic solvers based on genetic algorithms for solving nonlinear equations

详细信息    查看全文

• 作者：Muhammad Asif Zahoor Raja (1)
  Zulqurnain Sabir (2)
  Nasir Mehmood (2)
  Eman S. Al-Aidarous (3)
  Junaid Ali Khan (4)
  
  1. Department of Electrical Engineering ; COMSATS Institute of Information Technology ; Attock Campus ; Attock ; Pakistan
  2. Department of Mathematics ; Preston University Kohat ; Islamabad Campus ; Islamabad ; Pakistan
  3. Department of Mathematics ; King Abdulaziz University ; Jeddah ; 21589 ; Kingdom of Saudi Arabia
  4. Hamdard Institute of Information Technology ; Hamdard University ; Islamabad Campus ; Islamabad ; Pakistan
  
• 关键词：Genetic algorithm ; Iterative techniques ; Predictor鈥揷orrector method ; Convergence analysis ; Nonlinear equations
• 刊名：Neural Computing & Applications
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：26
• 期：1
• 页码：1-23
• 全文大小：3,512 KB
• 参考文献：1. Chun C (2006) Construction of Newton-like iteration methods for solving nonlinear equations. Numeriche Mathematik 104:297鈥?15 CrossRef
  2. Kumar M, Singh AK, Srivastava A (2013) Various Newton-type iterative methods for solving nonlinear equations. J Egypt Math Soc 21(3):334鈥?39
  3. Darvishi MT, Barati A (2007) A third-order Newton-type method to solve systems of nonlinear equations. Appl Math Comput 187(2):630鈥?35 CrossRef
  4. Noor MA (2007) A new family of iterative methods for nonlinear equations. Appl Math Comput 190:553鈥?58 CrossRef
  5. Noor MA (2010) Some iterative methods for solving nonlinear equations using homotopy perturbation technique. International Journal of Computational Mathematics 87:141鈥?49 3" target="_blank" title="It opens in new window">CrossRef
  6. Noor MA, Khan WA, Noor KI, Al-Said E (2011) Higher-order iterative methods free from second derivative for solving nonlinear equations. International Journal of the Physical Sciences 6(8):1887鈥?893
  7. Thukral R (2011) Eighth-order iterative methods without derivatives for solving nonlinear equations. ISRN Applied Mathematics 2011:1鈥?2 3787" target="_blank" title="It opens in new window">CrossRef
  8. Traub JF (1964) Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs, p 310
  9. Raja, MAZ, Khan JA, and Qureshi IM (2013) Numerical treatment for Painlev茅 equation i using neural networks and stochastic solvers. In: Jordanov I, Jain LC (eds) Innovation in intelligent machines-3. Studies in computational intelligence, vol 442. Springer, Berlin, pp 103鈥?17
  10. Raja MAZ, Khan JA, Ahmad SI, and Qureshi IM (2012) Solution of the Painlev茅 equation-I using neural network optimized with swarm intelligence. Comput Intell Neurosci, Article ID. 721867, pp 1鈥?0
  11. Raja MAZ (2014) Unsupervised neural networks for solving Troesch鈥檚 problem. Chin Phys B 23(1):018903 3/1/018903" target="_blank" title="It opens in new window">CrossRef
  12. Raja MAZ (2014) Stochastic numerical techniques for solving Troesch鈥檚 Problem. Accepted in Information Sciences 279:860鈥?73. doi:10.1016/j.ins.2014.04.036 36" target="_blank" title="It opens in new window">CrossRef
  13. Raja MAZ. Solution of one-dimension Bratu equation arising in fuel ignition model using ANN optimized with PSO and SQP. Connection Science, 17-04-2014 doi:10.1080/09540091.2014.907555
  14. Raja MAZ, Ahmad SI (2014) Numerical treatment for solving one-dimensional Bratu problem using neural networks. Neural Comput Appl 24(3鈥?):549鈥?61. doi:10.1007/s00521-012-1261-2 CrossRef
  15. 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鈥?):2199鈥?210 CrossRef
  16. Raja MAZ, Samar R, and Rashidi MM (2014) Application of three Unsupervised Neural Network Models to singular nonlinear BVP of transformed 2D Bratu equation. Neural Comput Appl. doi:10.1007/s00521-014-1641-x
  17. Khan JA, Raja MAZ, Qureshi IM (2011) Numerical treatment of nonlinear Emden鈥揊owler equation using stochastic technique. Ann Math Artif Intell 63(2):185鈥?07 CrossRef
  18. Khan JA, Raja MAZ, Qureshi IM (2011) Hybrid evolutionary computational approach: application to van der Pol oscillator. Int J Phys Sci 6(31):7247鈥?261. doi:10.5897/IJPS11.922
  19. Khan JA, Raja MAZ and Qureshi IM. Novel approach for van der Pol oscillator on the continuous time domain. Chin Phys Lett 28:110205. doi:10.1088/0256-307X/28/11/110205
  20. Raja MAZ, Khan JA, and Qureshi IM (2011) Solution of fractional order system of Bagley-Torvik equation using evolutionary computational intelligence. Math Probl Eng 2011, Article ID. 765075, 18聽pp
  21. 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鈥?):229鈥?50 CrossRef
  22. Raja MAZ, Samar R (2014) Numerical treatment for nonlinear MHD Jeffery鈥揌amel problem using neural networks optimized with interior point algorithm. Neurocomputing 124:178鈥?93. doi:10.1016/j.neucom.2013.07.013 3.07.013" target="_blank" title="It opens in new window">CrossRef
  23. Raja MAZ, Samar R (2014) Numerical treatment of nonlinear MHD Jeffery鈥揌amel problems using stochastic algorithms. Comput Fluids 91:28鈥?6 3.12.005" target="_blank" title="It opens in new window">CrossRef
  24. Noor MA, Khan WA, Hussain A (2007) A new modified Halley method without second derivatives for nonlinear equation. Appl Math Comput 189.2:1268鈥?273 CrossRef
  25. Traub JF (1964) Iterative methods for solution of equations. Prentice-Hall, Englewood Cliffs
  26. Cordero A, Hueso JL, Martinez E, Torregrosa JR (2012) Steffensen type method for solving nonlinear equations. J Comput Appl Math 236(12):3058鈥?064 3" target="_blank" title="It opens in new window">CrossRef
  27. Noor MA, Inayat Noor K (2006) Fifth-order iterative methods for solving nonlinear equations. Appl Math Comput. doi:10.1016/j.amc.2006.10.007 .
  28. Kou J, Li Y, Wang X (2006) A family of fifth-order iterations composed of Newton and third-order methods. Appl Math Comput. doi:10.1016/j.amc.2006.07.150
  29. Kou J, Li Y (2006) Improvements of Chebyshev鈥揌alley methods with fifth-order convergence. Appl Math Comput. doi:10.1016/j.amc.2006.09.097
  30. Holland JH (1975) Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. University of Michigan Press, Ann Arbor
  31. Goldberg DE, Holland JH (1988) Genetic algorithms and machine learning. Mach Learn 3(2):95鈥?9 3/A:1022602019183" target="_blank" title="It opens in new window">CrossRef
  32. Kumar S et al (2013) Hybrid evolutionary techniques in feed forward neural network with distributed error for classification of handwritten Hindi 鈥楽WARS鈥? Connection Science 25(4):197鈥?15 3.869556" target="_blank" title="It opens in new window">CrossRef
  33. Song A, Zhang M (2012) Genetic programming for detecting target motions. Connection Science 24(2鈥?):117鈥?41 3" target="_blank" title="It opens in new window">CrossRef
  34. Pini G, Tuci E (2008) On the design of neuro-controllers for individual and social learning behaviour in autonomous robots: an evolutionary approach. Connection science 20(2鈥?):211鈥?30 CrossRef
  35. Wang J-B, Jun L (2011) Double screen frequency selective surface structure optimized by genetic algorithm. Acta Phys. Sin. 60(5):057304
  36. Petkovi膰 MS (2013) A note on the priority of optimal multipoint methods for solving nonlinear equations. Appl Math Comput 219(10):5249鈥?252 CrossRef
  37. Herceg D, Herceg D (2013) Means based modifications of Newton鈥檚 method for solving nonlinear equations. Appl Math Comput 219(11):6126鈥?133 CrossRef
  38. Chicharro F et al (2013) Complex dynamics of derivative-free methods for nonlinear equations. Appl Math Comput 219(12):7023鈥?035 CrossRef
  39. Ardelean G (2013) A new third-order Newton-type iterative method for solving nonlinear equations. Appl Math Comput 219(18):9856鈥?864
  40. Soleymani Fazlollah (2013) An efficient twelfth-order iterative method for finding all the solutions of nonlinear equations. J Comput Methods Sci Eng 13(3):309鈥?20
  
• 刊物类别：Computer Science
• 刊物主题：Simulation and Modeling
  
• 出版者：Springer London
• ISSN：1433-3058

文摘

In the present study, a novel intelligent computing approach is developed for solving nonlinear equations using evolutionary computational technique mainly based on variants of genetic algorithms (GA). The mathematical model of the equation is formulated by defining an error function. Optimization of fitness function is carried out with the competency of GA used as a tool for viable global search methodology. Comprehensive numerical experimentation has been performed on number of benchmark nonlinear algebraic and transcendental equations to validate the accuracy, convergence and robustness of the designed scheme. Comparative studies have also been made with available standard solution to establish the correctness of the proposed scheme. Reliability and effectiveness of the design approaches are validated based on results of statistical parameters.