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Application of three unsupervised neural network models to singular nonlinear BVP of transformed 2D Bratu equation

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Abstract

In this paper, numerical techniques are developed for solving two-dimensional Bratu equations using different neural network models optimized with the sequential quadratic programming technique. The original two-dimensional problem is transformed into an equivalent singular, nonlinear boundary value problem of ordinary differential equations. Three neural network models are developed for the transformed problem based on unsupervised error using log-sigmoid, radial basis and tan-sigmoid functions. Optimal weights for each model are trained with the help of the sequential quadratic programming algorithm. Three test cases of the equation are solved using the proposed schemes. Statistical analysis based on a large number of independent runs is carried out to validate the models in terms of accuracy, convergence and computational complexity.

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Authors and Affiliations

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

    Muhammad Asif Zahoor Raja

  2. Mohammad Ali Jinnah University, Islamabad, Pakistan

    Raza Samar

  3. Mechanical Engineering Department, University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai, Peoples Republic of China

    Mohammad Mehdi Rashidi

  4. Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, Iran

    Mohammad Mehdi Rashidi

Authors
  1. Muhammad Asif Zahoor Raja
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  2. Raza Samar
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  3. Mohammad Mehdi Rashidi
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Corresponding author

Correspondence to Muhammad Asif Zahoor Raja.

Appendix

Appendix

The derived solutions by each neural networks model in case of all three BVPs of Bratu-type equations are given in Tables 8, 9 and 10, respectively. The solutions are presented with 14 decimal points for the unknown weights to exactly reproduce the results presented in the body of manuscript and to avoid rounding of error problem.

Table 8 Derive solutions by neural network models in case of problem 1
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Table 9 Derive solutions by neural network models in case of problem 2
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Table 10 Derive solutions by neural network models in case of Problem 3
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Raja, M.A.Z., Samar, R. & Rashidi, M.M. Application of three unsupervised neural network models to singular nonlinear BVP of transformed 2D Bratu equation. Neural Comput & Applic 25, 1585–1601 (2014). https://doi.org/10.1007/s00521-014-1641-x

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  • DOI: https://doi.org/10.1007/s00521-014-1641-x

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