An efficient scheme for nonlinear shock wave model in a fractal domain under Caputo fractional operator.

This paper introduces a refined approach for obtaining the analytical solution of the nonlinear shock wave model incorporating fractal derivatives. The Fractal Yang Variational Iteration Strategy (FYVIS) is utilized to obtain the approximate solution of a fractal model in the form of a series under Caputo fractional operator. The suggested method is the composition of the fractal Yang transform and the variational iteration approach. By using the two-scale fractal theory, we transform the fractal model into its traditional problem and then apply the yang transform to generate a recurrence relation. The variational iteration approach is now suitable to handle this recurrence relation without imposing any hypotheses or restrictions on variables. The derived results by the proposed scheme are shown in terms of series solution. Numerical calculations verify the accuracy and consistency of the suggested approach, demonstrating its excellent performance. The dynamic behavior of fractal components is explored by evaluating absolute errors and presenting two-dimensional diagrams across the fractal domain. This investigation underscores that the suggested technique offers an efficient and user-friendly solution for solving the nonlinear shock wave model involving fractal derivatives.

Analytical study of one dimensional time fractional Schrödinger problems arising in quantum mechanics.

This work presents the analytical study of one dimensional time-fractional nonlinear Schrödinger equation arising in quantum mechanics. In present research, we establish an idea of the Sumudu transform residual power series method (ST-RPSM) to generate the numerical solution of nonlinear Schrödinger models with the fractional derivatives. The proposed idea is the composition of Sumudu transform (ST) and the residual power series method (RPSM). The fractional derivatives are taken in Caputo sense. The proposed technique is unique since it requires no assumptions or variable constraints. The ST-RPSM obtains its results through a series of successive iterations, and the resulting form rapidly converges to the exact solution. The results obtained via ST-RPSM show that this scheme is authentic, effective, and simple for nonlinear fractional models. Some graphical structures are displayed at different levels of fractional orders using Mathematica Software.

Analyzing the dynamical sensitivity and soliton solutions of time-fractional Schrödinger model with Beta derivative.

In physical domains, Beta derivatives are necessary to comprehend wave propagation across various nonlinear models. In this research work, the modified Sardar sub-equation approach is employed to find the soliton solutions of (1+1)-dimensional time-fractional coupled nonlinear Schrödinger model with Beta fractional derivative. These models are fundamental in real-world applications such as control systems, processing of signals, and fiber optic networks. By using this strategy, we are able to obtain various unique optical solutions, including combo, dark, bright, periodic, singular, and rational wave solutions. In addition, We address the sensitivity analysis of the proposed model to investigate the truth that it is extremely sensitive. These studies are novel and have not been performed before in relation to the nonlinear dynamic features of these solutions. We show these behaviors in 2-D, contour 3-D structures across the associated physical characteristics. Our results demonstrate that the proposed approach offers useful results for producing solutions of nonlinear fractional models in application of mathematics and wave propagation in fiber optics.

The dynamical perspective of soliton solutions, bifurcation, chaotic and sensitivity analysis to the (3+1)-dimensional Boussinesq model.

In this study, we examine multiple perspectives on soliton solutions to the (3+1)-dimensional Boussinesq model by applying the unified Riccati equation expansion (UREE) approach. The Boussinesq model examines wave propagation in shallow water, which is derived from the fluid dynamics of a dynamical system. The UREE approach allows us to derive a range of distinct solutions, such as single, periodic, dark, and rational wave solutions. Furthermore, we present the bifurcation, chaotic, and sensitivity analysis of the proposed model. We use planar dynamical system theory to analyze the structure and characteristics of the system's phase portraits. The current study depends on a dynamic structure that has novel and unexplored results for this model. In addition, we display the behaviors of associated physical models in 3-dimensional, density, and 2-dimensional graphical structures. Our findings demonstrate that the UREE technique is a valuable mathematical tool in engineering and applied mathematics for studying wave propagation in nonlinear evolution equations.

Study of multi-dimensional problems arising in wave propagation using a hybrid scheme.

Many scientific phenomena are linked to wave problems. This paper presents an effective and suitable technique for generating approximation solutions to multi-dimensional problems associated with wave propagation. We adopt a new iterative strategy to reduce the numerical work with minimum time efficiency compared to existing techniques such as the variational iteration method (VIM) and homotopy analysis method (HAM) have some limitations and constraints within the development of recurrence relation. To overcome this drawback, we present a Sawi integral transform ( S T) for constructing a suitable recurrence relation. This recurrence relation is solved to determine the coefficients of the homotopy perturbation strategy (HPS) that leads to the convergence series of the precise solution. This strategy derives the results in algebraic form that are independent of any discretization. To demonstrate the performance of this scheme, several mathematical frameworks and visual depictions are shown.

A robust approach for computing solutions of fractional-order two-dimensional Helmholtz equation.

The Helmholtz equation plays a crucial role in the study of wave propagation, underwater acoustics, and the behavior of waves in the ocean environment. The Helmholtz equation is also used to describe propagation through ocean waves, such as sound waves or electromagnetic waves. This paper presents the Elzaki transform residual power series method ([Formula: see text]T-RPSM) for the analytical treatment of fractional-order Helmholtz equation. To develop this scheme, we combine Elzaki transform ([Formula: see text]T) with residual power series method (RPSM). The fractional derivatives are described in Caputo sense. The [Formula: see text]T is capable of handling the fractional order and turning the problem into a recurrence form, which is the novelty of our paper. We implement RPSM in such a way that this recurrence relation generates the results in the form of an iterative series. Two numerical applications are considered to demonstrate the efficiency and authenticity of this scheme. The obtained series are determined very quickly and converge to the exact solution only after a few iterations. Graphical plots and absolute error are shown to observe the authenticity of this suggested approach.

Soliton solutions of fractional extended nonlinear Schrödinger equation arising in plasma physics and nonlinear optical fiber.

In this research, we study traveling wave solutions to the fractional extended nonlinear SchrÖdinger equation (NLSE), and the effects of the third-order dispersion parameter. This equation is used to simulate the propagation of femtosecond, plasma physic and in nonlinear optical fiber. To accomplish this goal, we use the extended simple equation approach and the improved F-expansion method to secure a variety of distinct solutions in the form of dark, singular, periodic, rational, and exponential waves. Also, the stability of the outcomes is effectively examined. Several graphs have been sketched under appropriate parametric values to reinforce some reported findings. Computational work along with a graphical demonstration confirms the exactness of the proposed methods. The issue has not previously been investigated by taking into account the impact of the third order dispersion parameter. The main objective of this study is to obtain the different kinds of traveling wave solutions of fractional extended NLSE which are absent in the literature which justify the novelty of this study. We believe that these novel solutions hold a prominent place in the fields of nonlinear sciences and optical engineering because these solutions will enables a through understanding of the development and dynamic nature of such models. The obtained results indicate the reliability, efficiency, and capability of the implemented technique to determine wide-spectral stable traveling wave solutions to nonlinear equations emerging in various branches of scientific, technological, and engineering domains.