Soliton solutions and sensitive analysis to nonlinear wave model arising in optics

The aim of the present analysis is to apply the Adomian decomposition method and He's variational method for the approximate analytical solution of a nonlinear ordinary fractional differential equation. The solutions obtained by the above two methods have been numerically evaluated and presented in the form of tables and also compared with the exact solution. It was found that the results obtained by the above two methods are in excellent agreement with the exact solution. Finally, a surface plot of the approximate solutions of the fractional differential equation by the above two methods is drawn for 0⩽t⩽2 and 1<α⩽2.

Analytical solutions of the space–time fractional Kundu–Eckhaus equation by using modified extended direct algebraic method

Realizing the behavior of the solution in the asymptotic situations is essential for repetitive applications in the control theory and modeling of the real-world systems. This study discusses a robust and definitive attitude to find the interval approximate asymptotic solutions of fractional differential equations (FDEs) with the Atangana-Baleanu (A-B) derivative. In fact, such critical tasks require to observe precisely the behavior of the noninterval case at first. In this regard, we initially shed light on the noninterval cases and analyze the behavior of the approximate asymptotic solutions, and then, we introduce the A-B derivative for FDEs under interval arithmetic and develop a new and reliable approximation approach for fractional interval differential equations with the interval A-B derivative to get the interval approximate asymptotic solutions. We exploit Laplace transforms to get the asymptotic approximate solution based on the interval asymptotic A-B fractional derivatives under interval arithmetic. The techniques developed here provide essential tools for finding interval approximation asymptotic solutions under interval fractional derivatives with nonsingular Mittag-Leffler kernels. Two cases arising in the real-world systems are modeled under interval notion and given to interpret the behavior of the interval approximate asymptotic solutions under different conditions as well as to validate this new approach. This study highlights the importance of the asymptotic solutions for FDEs regardless of interval or noninterval parameters.

Higher-order time-fractional Sasa–Satsuma equation: Various optical soliton solutions in optical fiber

In this study, the stochastic Chen–Lee–Liu equation is considered numerically and analytically which is forced by the multiplicative noise in the Itô sense. The Chen–Lee–Liu equation is a special type of Schrödinger’s equation which has applications in optical fibers and photonic crystal fibers. The stochastic Crank–Nicolson scheme is formed to obtain the computational results. The numerical scheme is analyzed under the mean square sense and Von-Neumann criteria to show consistence and stability, respectively. Meanwhile, stochastic optical soliton solutions are attained by using two techniques, namely, the modified auxiliary equation method and the generalized projective Riccati equation method. These methods provide us with the different types of optical soliton solutions such as hyperbolic, trigonometric, mixed trigonometric, and rational forms. Mainly, the comparison of computational results with newly constructed optical soliton solution is shown graphically. To compare these results, initial conditions and boundary conditions are constructed by selecting some soliton solutions. The 3D and line graphs are drawn by choosing different values of parameters. Additionally, the sensitivity analysis is observed for the different initial values.

This paper is devoted to studying on the locations of zeros of related integral operators and the solutions of some ordinary and fractional differential equations. We generalize Sturm and Picone's theorems and Leighton and Levin's criteria. Moreover, we share some oscillation and disconjugacy criteria for the solutions of ordinary second‐order Sturm–Liouville and fractional differential equations. Finally, we introduce some properties of the solutions of fractional differential equations.

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.

The solutions of fractional differential equations (FDEs) have a natural singularity at the initial point. The accuracy of their numerical solutions is lower than the accuracy of the numerical solutions of FDEs whose solutions are differentiable functions. In the present paper we propose a method for improving the accuracy of the numerical solutions of ordinary linear FDEs with constant coefficients which uses the fractional Taylor polynomials of the solutions. The numerical solutions of the two-term and three-term FDEs are studied in the paper.

Some new soliton solutions to the (3 + 1)-dimensional generalized KdV-ZK equation via enhanced modified extended tanh-expansion approach

&lt;abstract&gt;&lt;p&gt;In the past, the existence and uniqueness of the solutions of fractional differential equations have been investigated by many researchers theoretically in various approaches in the literature. In this paper, there is no discussion of the existence of solutions for the nonlinear differential equations with fractal fractional operators. The objective of this work is to present novel contraction approaches, notably the $ \varpropto $-$ \psi $-contraction $ \varpropto $-type of the $ \tilde{\texttt{F}} $-contraction, within the context of $ \hat{F} $-metric and orbital metric spaces. The aim of this study is to illustrate certain fixed point theorems that offer a new and direct approach to establish the existence and uniqueness of the solution to the general partial differential equations by employing the fractal fractional operators.&lt;/p&gt;&lt;/abstract&gt;

This paper introduces a new concept of dual periodic boundary conditions. One of these conditions relates to the endpoints of an interval, while the other one involves two nonlocal positions within the interval. Under these boundary conditions, we establish the criteria for the existence of solutions for a fractional differential equation with p-Laplacian operator by using Mawhin's continuation theorem. An example is constructed for illustrating the obtained results.

This work is devoted to the noncontinuability to infinity of solutions for superlinear fractional differential equations of arbitrary order, considering the Riemann–Liouville derivative. As an application, we establish the asymptotic behavior of nonoscillatory solutions of such equations.

Discrete fractional Bihari inequality and uniqueness theorem of solutions of nabla fractional difference equations with non-Lipschitz nonlinearities

3D cubic and power-law nonlinear Schrödinger equations with different coefficients of dispersion/diffraction and their optical soliton solutions in parity-time symmetric potentials

This paper presents a numerical scheme for the solutions of Fractional Differential Equations (FDEs) of order α, 1 &amp;lt; α &amp;lt; 2 which have been expressed in terms of Caputo Fractional Derivative (FD). In this scheme, the properties of the Caputo derivative are used to reduce an FDE into a Volterra type integral equation. The entire domain is divided into several small domains, and the distribution of the unknown function over the domain is expressed in terms of the function values and its slopes at the node points. These approximations are then substituted into the Volterra type integral equation to reduce it to algebraic equations. Since the method enforces the continuity of variables at the node points, it provides a solution that is continuous and with a slope that is also continuous over the entire domain. The method is used to solve a simple FDE using two different types of polynomials, namely cubic order and fractional order. Results obtained using both types of polynomials agree well with the analytical results. However, the fractional order polynomials give more accurate results than the cubic order polynomials do. This suggests that for the numerical solutions of FDEs fractional order polynomials may be more suitable than the integer order polynomials. A series of numerical studies suggests that the algorithm is stable.