Numerical Treatment for Painlevé Equation I Using Neural Networks and Stochastic Solvers

2013 ◽  
pp. 103-117 ◽  
Author(s):  
Muhammad Asif Zahoor Raja ◽  
Junaid Ali Khan ◽  
Siraj-ul-Islam Ahmad ◽  
Ijaz Mansoor Qureshi
Keyword(s):  
Neural Networks ◽  
Painlevé Equation ◽  
Numerical Treatment ◽  
Painleve Equation

Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation

2015 ◽  
Vol 26 ◽  
pp. 244-256 ◽  
Author(s):  
Muhammad Asif Zahoor Raja ◽  
Junaid Ali Khan ◽  
A.M. Siddiqui ◽  
D. Behloul ◽  
T. Haroon ◽  
...  
Keyword(s):  
Neural Network ◽  
Initial Conditions ◽  
Network Models ◽  
Painlevé Equation ◽  
Numerical Treatment ◽  
Neural Network Models ◽  
Painleve Equation

scholarly journals A New Stochastic Technique for Painlevé Equation-I Using Neural Network Optimized with Swarm Intelligence

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Muhammad Asif Zahoor Raja ◽  
Junaid Ali Khan ◽  
Siraj-ul-Islam Ahmad ◽  
Ijaz Mansoor Qureshi
Keyword(s):  
Neural Networks ◽  
Particle Swarm Optimization ◽  
Particle Swarm ◽  
Variational Iteration Method ◽  
Painlevé Equation ◽  
Active Set ◽  
Swarm Optimization ◽  
Computational Intelligence Technique ◽  
Active Set Algorithm ◽  
Painleve Equation

A methodology for solution of Painlevé equation-I is presented using computational intelligence technique based on neural networks and particle swarm optimization hybridized with active set algorithm. The mathematical model of the equation is developed with the help of linear combination of feed-forward artificial neural networks that define the unsupervised error of the model. This error is minimized subject to the availability of appropriate weights of the networks. The learning of the weights is carried out using particle swarm optimization algorithm used as a tool for viable global search method, hybridized with active set algorithm for rapid local convergence. The accuracy, convergence rate, and computational complexity of the scheme are analyzed based on large number of independents runs and their comprehensive statistical analysis. The comparative studies of the results obtained are made with MATHEMATICA solutions, as well as, with variational iteration method and homotopy perturbation method.

Numerical treatment of nonlinear singular Flierl–Petviashivili systems using neural networks models

2017 ◽  
Vol 31 (7) ◽  
pp. 2371-2394 ◽  
Author(s):  
Muhammad Asif Zahoor Raja ◽  
Junaid Ali Khan ◽  
Aneela Zameer ◽  
Najeeb Alam Khan ◽  
Muhammad Anwaar Manzar
Keyword(s):  
Neural Networks ◽  
Numerical Treatment

scholarly journals On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2095
Author(s):  
Artyom V. Yurov ◽  
Valerian A. Yurov
Keyword(s):  
Jordan Algebra ◽  
Painlevé Equation ◽  
Bäcklund Transformations ◽  
Jordan Product ◽  
Completely Integrable ◽  
Matrix Generalization ◽  
Backlund Transformations ◽  
Painleve Equation ◽  
Second Painlevé Equation ◽  
Multiple Field

We demonstrate the way to derive the second Painlevé equation P2 and its Bäcklund transformations from the deformations of the Nonlinear Schrödinger equation (NLS), all the while preserving the strict invariance with respect to the Schlesinger transformations. The proposed algorithm allows for a construction of Jordan algebra-based completely integrable multiple-field generalizations of P2 while also producing the corresponding Bäcklund transformations. We suggest calling such models the JP-systems. For example, a Jordan algebra JMat(N,N) with the Jordan product in the form of a semi-anticommutator is shown to generate an integrable matrix generalization of P2, whereas the VN algebra produces a different JP-system that serves as a generalization of the Sokolov’s form of a vectorial NLS.

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