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A Novel Method to Solve nonlinear Klein-Gordon Equation Arising in Quantum Field Theory Based on Bessel Functions and Jacobian Free Newton-Krylov Sub-Space Methods

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Communications in Theoretical Physics 2018年第6期69卷 637-644页

作者： K.ParAND M.Nikarya Department of Computer Sciences Shahid Beheshti University

The Klein-Cordon equation arises in many scientific areas of quantum mechanics and quantum field theory. In this paper a novel method based on spectral method and Jacobian free Newton method composed by generalized minimum residual （JFNGMRes） method with adaptive preconditioner will be introduced to solve nonlinear Klein-Gordon equation. In this work the nonlinear Klein-Gordon equation has been converted to a nonlinear system of algebraic equations using collocation method based on Bessel functions without any linearization, discretization and getting help of any other methods. Finally, by using JFNGMRes, solution of the nonlinear algebraic system will be achieved. To illustrate the reliability and efficiency of the proposed method, we solve some examples of the Klein-Gordon equation and compare our results with other methods.

关键词： nonlinear partial differential equation spectral collocation methods Jacobian free Newton-GMRes adaptive preconditioning Klein-Gordon equations nonlinear algebraic systems

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A Back Propagation-Type Neural Network Architecture for Solving the Complete n ×n nonlinear algebraic System of Equations

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Advances in Pure Mathematics 2016年第6期6卷 455-480页

作者： Konstantinos Goulianas Athanasios Margaris Ioannis Refanidis Konstantinos Diamantaras Theofilos Papadimitriou TEI of Thessaloniki Department of Informatics Thessaloniki Greece TEI of Larissa Department of Computer Science and Engineering Larissa Greece Department of Applied Informatics University of Macedonia Thessaloniki Greece Department of Economics Democritus University of Thrace Komotini Greece

The objective of this research is the presentation of a neural network capable of solving complete nonlinear algebraic systems of n equations with n unknowns. The proposed neural solver uses the classical back propagation algorithm with the identity function as the output function, and supports the feature of the adaptive learning rate for the neurons of the second hidden layer. The paper presents the fundamental theory associated with this approach as well as a set of experimental results that evaluate the performance and accuracy of the proposed method against other methods found in the literature.

关键词： nonlinear algebraic systems Neural Networks Back Propagation Numerical Analysis Computational Methods

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