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The modified exponential function method is used to obtain some new analytical solutions of the nonlinear Gerdjikov–Ivanov equation with the M-fractional operator. The novel obtained solutions are expressed in hyperbolic, trigonometric, and exponential function forms. Moreover, the instability modulation and gain spectra of the Gerdjikov–Ivanov equation are also analyzed. Constraints conditions are utilized to verify the existence of the solutions. Presented solutions are novel, satisfy and verify the M-fractional Gerdjikov–Ivanov equation. [ABSTRACT FROM AUTHOR]

In this work, we investigate the perturbed optical solitons to the Gerdjikov-Ivanov equation consisting of group velocity dispersion and quintic nonlinearity coefficients, which communicate the propagation of pulses in nonlinear optics. The various kinds of solitons in single and combined forms like dark, singular, dark-singular, bright-dark are derived by Fan-extended sub equation method. Moreover, the singular periodic, triangular type solutions are also obtained. And, we also discuss the stability analysis of the studied nonlinear model with the concept of linear stability, we prove that the governing model is stable. Parametric conditions on physical parameters to ensure the existence criteria of optical solitons are also listed. We also plot 3D profiles for the physical behavior of the obtained solutions by selecting the suitable values of the parameters. [ABSTRACT FROM AUTHOR]

In this paper, we have obtained approximate solution of the nonlinear (3+1)-dimensional breaking soliton Equation (BSE) by using Homotopy Perturbation Method (HPM) and then formed a chart including numerical conclusion for this equation by drawing graphics of the the obtained solution by means of programming language Mathematica. [ABSTRACT FROM AUTHOR]

The current study provides the solutions of the mathematical model based on the coronavirus including the effects of vaccination and quarantine. The numerical stochastic process relying on Levenberg-Marquardt backpropagation technique (L-MB) neural networks (NN), i.e., L-MBNNs, is presented to solve the model. The entire dynamics of the proposed model depends upon the human population, which is represented by N and is further divided into multiple subgroups. The detail of these subgroups is presented in the form of susceptible population (S), exposed population (E), and infected people (I). Likewise, Q represents the quarantined and R shows the recovered or deceased individuals. Those who have been immunized are symbolized by V. All these categories make the model SEIQRV, that is based on a system of nonlinear differential equations. The statistics that is used to provide the numerical solutions of the SEIQRV model is 76% for training, 10% for testing and 14% for authorization. The correctness of the L-MBNNs is tested by using the comparison of the proposed and reference solutions (Adam method). The statistical representations are provided in order to check the reliability, competence and validity of L-MBNNs using the procedures of error histograms (EH), state transitions (ST), regression and correlation. [ABSTRACT FROM AUTHOR]

To solve fractional partial differential equations (FPDEs) under various physical conditions, this study developed a novel method known as the Hermite wavelet method employing the functional integration matrix. The method that has been suggested is based on the Hermite wavelet collocation process. To determine the solution of the fractional differential equations, the Caputo fractional derivative operator of order α ∈ (0 , 1 ] is used. With the use of appropriate grid points, this method converts FPDEs into a system of nonlinear algebraic equations. We achieve a solution by solving these nonlinear algebraic equations by the Newton–Raphson method. Tables and graphs show that the suggested method produces superior results. We provide various illustrative examples to establish the effectiveness of the suggested concept, and the outcomes support the applicability of the suggested strategy. Obtained results are numerically expressed in terms of absolute errors. Finally, convergence analyses are discussed as some theorem with proof. [ABSTRACT FROM AUTHOR]

This paper constructs dark, bright, combined dark-bright, singular, combined singular soliton optical and singular periodic solutions to the conformable space-time fractional second order nonlinear Schrödinger equation with group velocity dispersion coefficient and spatiotemporal dispersion by using the extended sinh-Gordon equation expansion method. The constraint conditions for the existence of valid soliton solutions are given. The reported results in this study may be helpful in explaining the physical meaning of the studied model. [ABSTRACT FROM AUTHOR]

This study uses the extended sinh-Gordon equation expansion method in constructing various optical soliton solutions to the resonant nonlinear Schrödinger equation with both spatio-temporal and inter-modal dispersions. Resonant nonlinear Schrödinger equation expresses the propagation dynamics of optical solitons and Madelung fluids. Dark, bright, combined dark–bright and singular optical solitons are successfully constructed. Under the choice of suitable values of parameters, the two-dimensional, three-dimensional and the contour graphs to some of the acquired results are plotted. The reported results may be useful in explaining the physical meaning of the studied model. [ABSTRACT FROM AUTHOR]

In this paper, the rational sine–cosine and rational sinh–cosh methods are applied in extracting some properties of nonlinear Phi-four and Gross–Pitaevskii equations. The singular periodic wave solutions, dark soliton solutions and hyperbolic function solutions are reported. The solitary waves are observed from the traveling waves under the values of the parameters. Modulation instability analysis is also observed in various simulations. We also plot to observe the wave distributions of parameters of stability in 2D and 3D visuals via package program. [ABSTRACT FROM AUTHOR]

In this paper, the rational sine–cosine and rational sinh–cosh methods are applied in extracting some properties of nonlinear Phi-four and Gross–Pitaevskii equations. The singular periodic wave solutions, dark soliton solutions and hyperbolic function solutions are reported. The solitary waves are observed from the traveling waves under the values of the parameters. Modulation instability analysis is also observed in various simulations. We also plot to observe the wave distributions of parameters of stability in 2D and 3D visuals via package program. [ABSTRACT FROM AUTHOR]

In this paper, we study on the numerical solution of fractional nonlinear system of equations representing the one-dimensional Cauchy problem arising in thermoelasticity. The proposed technique is graceful amalgamations of Laplace transform technique with q -homotopy analysis scheme and fractional derivative defined with Atangana–Baleanu (AB) operator. The fixed-point hypothesis is considered in order to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional order model. In order to illustrate and validate the efficiency of the future technique, we consider three different cases and analyzed the projected model in terms of fractional order. Moreover, the physical behavior of the obtained solution has been captured in terms of plots for diverse fractional order, and the numerical simulation is demonstrated to ensure the exactness. The obtained results elucidate that the proposed scheme is easy to implement, highly methodical as well as accurate to analyze the behavior of coupled nonlinear differential equations of arbitrary order arisen in the connected areas of science and engineering. [ABSTRACT FROM AUTHOR]

In the present investigation, the q-homotopy analysis transform method (q-HATM) is applied to find approximated analytical solution for the system of fractional differential equations describing the unsteady flow of a polytropic gas. Numerical simulation has been conducted to prove that the proposed technique is reliable and accurate, and the outcomes are revealed using plots and tables. The comparison between the obtained solutions and the exact solutions shows that the proposed method is efficient and effective in solving nonlinear complex problems. Moreover, the proposed algorithm controls and manipulates the obtained series solution in a huge acceptable region in an extreme manner and it provides us a simple procedure to control and adjust the convergence region of the series solution. [ABSTRACT FROM AUTHOR]