Analytical approximate solutions of time-fractional integro-differential equations using a new iterative technique

In this manuscript, a new iterative technique is proposed to obtain the solutions of linear and nonlinear time-fractional integro-differential equations. The suggested algorithm is a modification of the homotopy analysis method. The deformation equations obtained in this case are easily integrable and the calculations involved in the algorithm are much simpler than the standard homotopy analysis method. The method is illustrated with the help of different numerical test applications. The numerical and graphical results explicitly reveal the potential and accuracy of the proposed technique.Publisher's Versio

Application of homotopy analysis method to the solution of ninth order boundary value problems in AFTI-F16 fighters

This paper is devoted to the study of ninth order boundary value problems that emerge in the mathematical modeling of AFTI-F16 fighter. An augmented longitudinal and lateral dynamics of the AFTI-F16 fighter is described by ninth order differential equations, containing unknown parameters which can be determined using automated system identification algorithms. The solution of the boundary value problem is obtained in terms of a convergent series using homotopy analysis method (HAM). The method is effectively applied on numerical examples and the results are compared with those given in the literature, revealing that the presented method gives better approximations to the exact solution

A study of variation in dynamical behavior of fractional complex Ginzburg-Landau model for different fractional operators

The fractional complex Ginzburg–Landau model is widely used in the description of wave propagation through optical transmission lines. It has many useful applications in the fields of telecommunications and nonlinear optics. This paper investigates the fractional effects of the complex Ginzburg–Landau model with quadratic–cubic, anti–cubic and generalized anti–cubic laws of nonlinearity by using generalized projective Riccati equation method. The variation in the traveling wave behavior of the governing model is examined for beta, conformable and M-truncated derivatives. Some constraint conditions are carried out during mathematical analysis, which are further used for evaluating the traveling wave solutions. The analytic solutions of the considered model are determined in terms of hyperbolic and trigonometric function solutions. Consequently, dark, bright, kink, bell-shaped and singular solitons are extracted. The reported solutions are presented using 2D and 3D graphs. These graphs are showing the fractional effects for different values of fractional parameter. The evolution of the wave profiles shows that the retrieved solitons become similar for all three definitions of fractional derivatives as the fractional parameter approaches unity

A Comparative Study of Time Fractional Nonlinear Drinfeld–Sokolov–Wilson System via Modified Auxiliary Equation Method

The time-fractional nonlinear Drinfeld–Sokolov–Wilson system, which has significance in the study of traveling waves, shallow water waves, water dispersion, and fluid mechanics, is examined in the presented work. Analytic exact solutions of the system are produced using the modified auxiliary equation method. The fractional implications on the model are examined under β-fractional derivative and a new fractional local derivative. Extracted solutions include rational, trigonometric, and hyperbolic functions with dark, periodic, and kink solitons. Additionally, by specifying values for fractional parameters, graphs are utilized to comprehend the fractional effects on the obtained solutions

Solitary wave solutions to Gardner equation using improved tan(Ω(Υ)2) \left(\frac{\Omega(\Upsilon)}{2}\right) -expansion method

In this study, the improved tan$\left(\frac{\Omega(\Upsilon)}{2}\right)$-expansion method is used to construct a variety of precise soliton and other solitary wave solutions of the Gardner equation. Gardner equation is extensively utilized in plasma physics, quantum field theory, solid-state physics and fluid dynamics. It is the simplest model for the description of water waves with dual power law nonlinearity. Hyperbolic, exponential, rational and trigonometric traveling wave solutions are obtained. The retrieved solutions include kink solitons, bright solitons, dark-bright solitons and periodic wave solutions. The efficacy of this method is determined by the comparison of the newly obtained results with already reported results

Analytical Solutions of the Fractional Complex Ginzburg-Landau Model Using Generalized Exponential Rational Function Method with Two Different Nonlinearities

The complex Ginzburg-Landau model appears in the mathematical description of wave propagation in nonlinear optics. In this paper, the fractional complex Ginzburg-Landau model is investigated using the generalized exponential rational function method. The Kerr law and parabolic law are considered to discuss the nonlinearity of the proposed model. The fractional effects are also included using a novel local fractional derivative of order α. Many novel solutions containing trigonometric functions, hyperbolic functions, and exponential functions are acquired using the generalized exponential rational function method. The 3D-surface graphs, 2D-contour graphs, density graphs, and 2D-line graphs of some retrieved solutions are plotted using Maple software. A variety of exact traveling wave solutions are reported including dark, bright, and kink soliton solutions. The nature of the optical solitons is demonstrated through the graphical representations of the acquired solutions for variation in the fractional order of derivative. It is hoped that the acquired solutions will aid in understanding the dynamics of the various physical phenomena and dynamical processes governed by the considered model

Numerical simulations of the soliton dynamics for a nonlinear biological model: Modulation instability analysis.

This article deals with studying the dynamical behavior of the DNA model proposed by Peyrard and Bishop. The proposed model is investigated using the unified method (UM). Unified method successfully extracts solutions in the form of polynomial and rational functions. The solitary wave solutions and soliton solutions are constructed. An investigation of modulation instability is also presented in this paper. 3D and 2D plots are presented to exhibit the physical behavior of some of the obtained solutions

Solitary wave solution of (2+1)-dimensional Chaffee–Infante equation using the modified Khater method

Most of the nonlinear phenomena are described by partial differential equation in natural and applied sciences such as fluid dynamics, plasma physics, solid state physics, optical fibers, acoustics, biology and mathematical finance. The solutions of a wide range of nonlinear evolution equations exhibit the wave behavior corresponding to the underlying physical systems. In particular, solitary wave solutions and soliton solutions are of great interest for researchers owing to many applications in different areas of science. The behavior of gas diffusion in a homogeneous medium is described by the (2+1)-dimensional Chaffee–Infante equation. In the present study, the modified Khater method is used to solve (2+1)-dimensional Chaffee–Infante equation because it provides various forms of solitons. The bright, dark, and periodic traveling wave patterns are produced by choosing different values of parameters. The solutions are presented graphically through 2D, 3D and contour graphs. The obtained solutions are demonstrated that the modified Khater method is a more useful tool than the existing techniques for solving such nonlinear problems

Analytical solitary wave solutions of a time-fractional thin-film ferroelectric material equation involving beta-derivative using modified auxiliary equation method

The main objective of current paper is to examine the impacts of fractional parameters on the dynamic response of soliton waves in a nonlinear time-fractional thin-film ferroelectric materials equation (TFFEME). To achieve such a goal, the TFFEME is first rehabilitated into ordinary differential equation using a complex wave transformation. Solitary wave solutions of the governing equation, representing the dynamics of waves in the material and plays a vital rule in many branches of physics and hydrodynamics, are then constructed through applying the modified auxiliary equation method (MAEM). The extracted solutions are demonstrated using definition of the beta derivative to understand their dynamical behavior. The hyperbolic, periodic and trigonometric function solutions are used to derive the analytical solutions for the given model. As a result, dark, bright, periodic and solitary wave solitons are obtained. The fractional impact of the above derivative on the physical phenomena is observed. The 2D and 3D graphs are also shown to confirm the behavior of analytical wave solutions

Bright, dark, periodic and kink solitary wave solutions of evolutionary Zoomeron equation

The modified auxiliary equation (MAE) approach and the generalized projective Riccati equation (GPRE) method are used for the first time to solve Zoomeron problem which provided different types of exact traveling wave solutions, including some new dynamical behaviors. The governing model covers unique examples of solitons with distinct properties that emerge in a variety of physical situations, including laser physics, fluid dynamics and nonlinear optics. The achieved exact traveling wave solutions include solitary wave, periodic wave, bright, dark peakon, and kink-type wave solutions. Earned results are given as hyperbolic and trigonometric functions. Moreover, the dynamical features of obtained results are demonstrated through interesting plots