Numerical Treatment for Painlevé Equation I Using Neural Networks and Stochastic Solvers
In this chapter, a new stochastic numerical treatment is presented for solving Painlevé I equation. The mathematical model of the equation is formulated with feed-forward artificial neural networks. Linear combination of the networks defines the unsupervised error for the equation. The error is reduced subject to the availability of appropriate weights of networks. Training of weights is done with genetic algorithm, simulating annealing and pattern search algorithms hybridized with interior point algorithm for rapid local search. The reliability and effectiveness is validated with the help of statistical analysis. Comparison of results is made with standard approximate analytic solvers of the equation. It is found that the proposed results are in a good agreement with their corresponding numerical solutions.
KeywordsGenetic Algorithm Fractional Differential Equation Pattern Search Homotopy Perturbation Method Variational Iteration Method
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