Elegant scheme for one-way wave propagation in Kerr media
Department of Mathematics, University of Karachi, Karachi, 75270, Pakistan
2 Department of Humanities and Social Sciences, Bahria University, Karachi, 75260, Pakistan
* e-mail: email@example.com
Accepted: 17 November 2019
Published online: 3 February 2020
In this article, an intelligent computational scheme is presented for the dynamical solution of fractional-order linear and nonlinear (Kerr nonlinearity) Helmholtz equations with boundary conditions, which have a significant impact on electromagnetics, hydrodynamics and acoustic phenomena. The propagation of electromagnetic waves in Kerr media describes the linearly polarized electric field in an isotropic material, for a range of important phenomena in nonlinear optics and other areas. Here, the generalized Helmholtz equation of dimensions is considered with fractional-order derivative. The fractional Laplace transform is utilized, and parameter is linearized by series expansion. Ultimately, the inverse fractional Laplace transform of this expansion reduces the fractional-order derivative to integer-order derivative. The new attained structure of partial differential equation becomes equivalent to the so-called fractional-order Helmholtz equation. The contribution also provides an artificial neural network-based technique, designed with a metaheuristic algorithm, to analyze the governing model numerically and graphically. Taking sigmoidal function as an activation function, the unsupervised error function is formulated, which is then optimized with the aid of the firefly algorithm (FFA). By using FFA, the minimum search path of the error function is tracked with the convergent values of the network weights. Besides, the validity and accuracy of the adapted technique are ascertained by calculating the error norms that ensure the convergence of the approximation. Accordingly, the achieved facts and figures accentuate comprehensively the novel implication of this attempt.
© Società Italiana di Fisica (SIF) and Springer-Verlag GmbH Germany, part of Springer Nature, 2020