Fractional Differential Equations

Fractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is to prove the stability and convergence of the fractional Euler method, the fractional Adams method and the high order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical experiments are also presented, which verify the theoretical analysis.

The finite difference methods for fractional ordinary differential equations

The authors study the prototypical, genuinely nonlinear, equation; u{sub t} + a(u{sup m}){sub x} + (u{sup n}){sub xxx} = {mu}(u{sup k}){sub xx}, a, {mu} = consts., which encompasses a wide variety of dissipative-dispersive interactions. The parametric surface k = (m + n)/2 separates diffusion dominated from dissipation dominated phenomena. On this surface dissipative and dispersive effects are in detailed balance for all amplitudes. In particular, the m = n + 2 = k + 1 subclass can be transformed into a form free of convection and dissipation making it accessible to theoretical studies. Both bounded and unbounded oscillations are found and certain exact solutions are presented. When a = (2{mu}3/){sup 2} the map yields a linear equation; rational, periodic and aperiodic solutions are constructed.

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On a class of nonlinear dispersive-dissipative interactions

In the present study, a novel fractional Meyer neuro-evolution-based intelligent computing solver (FMNEICS) is presented for numerical treatment of doubly singular multi-fractional Lane–Emden system (DSMF-LES) using combined heuristics of Meyer wavelet neural networks (MWNN) optimized with global search efficacy of genetic algorithms (GAs) and sequential quadratic programming (SQP), i.e., MWNN-GASQP. The design of novel FMNEICS for DSMF-LES is presented after derivation from standard Lane–Emden equation, and the singular points and shape factors along with fractional-order terms are analyzed. The MWNN modeling strength is used to represent the system model DSMF-LES in the mean-squared error-based merit function and optimization of the networks is carried out with integrated optimization ability of GASQP. The verification, validation, and perfection of the FMNEICS for three different cases of DSMF-LES are established through comparative studies from reference solutions on convergence, robustness, accuracy, and stability measures. Moreover, the observations through the statistical analysis further authenticate the worth of proposed fractional MWNN-GASQP-based stochastic solver.

FMNEICS: fractional Meyer neuro-evolution-based intelligent computing solver for doubly singular multi-fractional order Lane–Emden system

This research paper investigates the analytical and numerical solutions of the generalized formula of Hirota-Satsuma coupled KdV system which is also known as the generalized KdV equation that is derived by R. Hirota and J. Satsuma. The modified Khater method and B-spline scheme are used to earn abundant of computational and approximate solutions on this model. This equation characterizes an interaction of two long undulations with diverse dispersion kinsmen. The comparison between our obtained computational and numerical solutions to clarify the convergence of solutions is explained and discussed. For more explanation of the model’s physical properties, some of the obtained solutions are sketched in different types, and the comparison between the computational and numerical solutions are explained by showing the values of absolute error between them. The performance of both used method is effective, powerful, and shows its ability to apply to many nonlinear evolution equations.

Abundant numerical and analytical solutions of the generalized formula of Hirota-Satsuma coupled KdV system

Shapes and dynamics of dual-mode Hirota–Satsuma coupled KdV equations: Exact traveling wave solutions and analysis

Dark and singular optical solutions with dual-mode nonlinear Schrödinger's equation and Kerr-law nonlinearity

In this paper, we introduced a new dual-mode nonlinear Schrodinger (DMNLS) equation with nonlinearity Kerr of types square-root law and dual-power law. The new model consists of three parameters defined as dissipative, nonlinearity and the phase velocity. Also, this model describes propagations of two simultaneously directional waves instead of single wave as in the standard Schrodinger model. We determined the necessary conditions on the dissipative nonlinearity parameters that produce soliton solutions of DMNLS. Finally, a graphical analysis regarding the effect of the phase velocity on the shapes of the obtained dual-waves is accomplished.

Multiplicative of dual-waves generated upon increasing the phase velocity parameter embedded in dual-mode Schrödinger with nonlinearity Kerr laws

The latent potentialities and applications of fractional calculus present a mathematical challenge to establish its theoretical framework. One of these challenges is to have a compact and self-contained fractional power series representation that has a wider application scope and allows studying analytical properties. In this letter, we introduce a new more general form of fractional power series expansion, based on the Caputo sense of fractional derivative, with corresponding convergence property. In order to show the functionality of the proposed expansion, we apply the corresponding iterative fractional power series scheme to solve several fractional (integro-)differential equations.

Theory and applications of a more general form for fractional power series expansion

In this work, we present a new two-mode version to the complex Hirota equation. This new model arises in many applications in the field of optics, communications and other engineering sciences. It describes the propagation of two-symmetric solitary waves moving simultaneously with dependent phase-velocity parameter. The extended tanh-coth expansion method and the polynomial-function method are used to extract novel new cusp-type solutions to the Hirota model. A comprehensive graphical analysis is conducted to show physical aspects of this new type of nonlinear equations. The findings of this work are from mathematical point of view and we propose some benefits regarding the concept of two-mode where it could be visualized and practically applied.

Imad Jaradat

Papers

Shapes and dynamics of dual-mode Hirota–Satsuma coupled KdV equations: Exact traveling wave solutions and analysis

Dark and singular optical solutions with dual-mode nonlinear Schrödinger's equation and Kerr-law nonlinearity

Multiplicative of dual-waves generated upon increasing the phase velocity parameter embedded in dual-mode Schrödinger with nonlinearity Kerr laws

Theory and applications of a more general form for fractional power series expansion

Heart-cusp and bell-shaped-cusp optical solitons for an extended two-mode version of the complex Hirota model: application in optics