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The purpose of this study is to investigate the nonlinear Schrödinger equation (NLS) incorporating spatiotemporal dispersion and other dispersive effects. The goal is to derive various soliton solutions, including bright, dark, singular, periodic and exponential solitons, to enhance the understanding of soliton propagation dynamics in nonlinear metamaterials (MMs) and contribute new findings to the field of nonlinear optics.

The research uses a range of powerful mathematical approaches to solve the NLS. The proposed methodologies are applied systematically to derive a variety of optical soliton solutions, each demonstrating unique optical behaviors and characteristics. The approach ensures that both the theoretical framework and practical implications of the solutions are thoroughly explored.

The study successfully derives several types of soliton solutions using the aforementioned mathematical approaches. Key findings include bright optical envelope solitons, dark optical envelope solitons, periodic solutions, singular solutions and exponential solutions. These results offer new insights into the behavior of ultrashort solitons in nonlinear MMs, potentially aiding further research and applications in nonlinear wave studies.

This study makes an original contribution to nonlinear optics by deriving new soliton solutions for the NLS with spatiotemporal dispersion. The diversity of solutions, including bright, dark, periodic, singular and exponential solitons, adds substantial value to the existing body of knowledge. The use of distinct and reliable methodologies to obtain these solutions underscores the novelty and potential applications of the research in advancing optical technologies. The originality lies in the novel approaches used to obtain these diverse soliton solutions and their potential impact on the study and application of nonlinear waves in MMs.

The purpose of this study is to produce families of exact soliton solutions (2+1)-dimensional Korteweg-de Vries (KdV) equation, that describes shallow water waves, using an ansätze approach.

This article aims to introduce a recently developed ansätze for creating soliton and travelling wave solutions to nonlinear nonintegrable partial differential equations, especially those with physical significance.

A recently developed ansätze solution was used to successfully construct soliton solutions to the (2 + 1)-dimensional KdV equation. This straightforward method is an alternative to the Painleve test analysis, yielding similar results. The strategy demonstrated the existence of a single soliton solution, also known as a localized wave or bright soliton, as well as singular solutions or kink solitons.

The ansätze solution used to construct soliton solutions to the (2 + 1)-dimensional KdV equation is novel. New soliton solutions were also obtained.

The purpose of this study is to implement a newly introduced numerical scheme for the numerical solution of a class of nonlinear fractional Bratu-type boundary value problems (BVPs).

This strategy is based on a generalization of the variational iteration method (VIM). This proposed generalized VIM (GVIM) is particularly suitable for tackling BVPs.

This scheme yields accurate solutions for a class of nonlinear fractional Bratu-type BVPs, for which the errors are uniformly distributed across a given domain. A proof of convergence is included. The numerical results confirm that this approach overcomes the deficiency of the VIM and other methods that exist in the literature in the sense that the solution does not deteriorate as the authors move away from the initial starting point.

The method introduced is based on original research that produces new knowledge. To the best of the authors’ knowledge, this is the first time that this GVIM is applied to fractional BVPs.

The purpose of this paper is to find approximate solutions for a general class of fractional order boundary value problems that arise in engineering applications.

A newly developed semi-analytical scheme will be applied to find approximate solutions for fractional order boundary value problems. The technique is regarded as an extension of the well-established variation iteration method, which was originally proposed for initial value problems, to cover a class of boundary value problems.

It has been demonstrated that the method yields approximations that are extremely accurate and have uniform distributions of error throughout their domain. The numerical examples confirm the method’s validity and relatively fast convergence.

The generalized variational iteration method that is presented in this study is a novel strategy that can handle fractional boundary value problem more effectively than the classical variational iteration method, which was designed for initial value problems.

The aim of this study is to offer a contemporary approach for getting optical soliton and traveling wave solutions for the Date–Jimbo–Kashiwara–Miwa equation.

The approach is based on a recently constructed ansätze strategy. This method is an alternative to the Painleve test analysis, producing results similarly, but in a more practical, straightforward manner.

The approach proved the existence of both singular and optical soliton solutions. The method and its application show how much better and simpler this new strategy is than current ones. The most significant benefit is that it may be used to solve a wide range of partial differential equations that are encountered in practical applications.

The approach has been developed recently, and this is the first time that this method is applied successfully to extract soliton solutions to the Date–Jimbo–Kashiwara–Miwa equation.

The purpose of this paper is to apply an efficient semi-analytical method for the approximate solution of Lienard’s equation of fractional order.

A Laplace decomposition method (LDM) is implemented for the nonlinear fractional Lienard’s equation that is complemented with initial conditions. The nonlinear term is decomposed and then a recursive algorithm is constructed for the determination of the proposed infinite series solution.

A number of examples are tested to explicate the efficiency of the proposed technique. The results confirm that this approach is convergent and highly accurate by using only few iterations of the proposed scheme.

The approach is original and is of value because it is the first time that this approach is used successfully to tackle fractional differential equations, which are of great interest for authors in the recent years.

The purpose of this study is to obtain an analytical solution for a nonlinear system of the COVID-19 model for susceptible, exposed, infected, isolated and recovered.

The Laplace decomposition method and the differential transformation method are used.

The obtained analytical results are useful on two fronts: first, they would contribute to a better understanding of the dynamic spread of the COVID-19 disease and help prepare effective measures for prevention and control. Second, researchers would benefit from these results in modifying the model to study the effect of other parameters such as partial closure, awareness and vaccination of isolated groups on controlling the pandemic.

The approach presented is novel in its implementation of the nonlinear system of the COVID-19 model

Ordinary differential equations (ODEs) are widely used in the engineering curriculum. They model a spectrum of interesting physical problems that arise in engineering disciplines. Studies of different types of ODEs are determined by engineering applications. Various techniques are used to solve practical differential equations problems. This paper aims to present a computational tool or a computer-assisted technique aimed at tackling ODEs. This method is usually not taught and/or not accessible to undergraduate students. The aim of this strategy is to help the readers to develop an effective and relatively novel problem-solving skill. Because of the drudgery of hand computations involved, the method requires the need to use computers packages. In this work, the successive differentiation method (SDM) for solving linear and nonlinear and homogeneous or non-homogeneous ODEs is presented. The algorithm uses the successive differentiation of any given ODE to determine the values of the function’s derivatives at a single point, mostly x = 0. The obtained values are used to construct the Taylor series of the solution of the examined ODE. The algorithm does not require any new assumption, hence handles the problem in a direct manner. The power of the method is emphasized by testing a variety of models with distinct orders, with constant and variable coefficients. Most of the symbolic and numerical computations can be carried out using computer algebra systems.

This study presents a computational tool or a computer-assisted technique aimed at tackling ODEs. This method is usually not taught and/or not accessible to undergraduate students. The aim of this strategy is to help the readers to develop an effective and relatively novel problem-solving skill. Because of the drudgery of hand computations involved, the method requires the need to use computers packages.

This method is applied to a variety of well-known equations, such as the Bernoulli equation, the Riccati equation, the Abel equation and the second-order Euler equation, some with constant and variable coefficients. SDM handles linear and nonlinear and homogeneous or nonhomogeneous ODEs in a direct manner without any need to restrictive conditions. The method works effectively to the Volterra integral equations, as will be discussed in a coming work.

The method can be extended to a wide range of engineering problems that are modeled by differential equations. The method is simple and novel and highly accurate.

Fractional Fokker-Planck equation (FFPE) and time fractional coupled Boussinesq-Burger equations (TFCBBEs) play important roles in the fields of solute transport, fluid dynamics, respectively. Although there are many methods for solving the approximate solution, simple and effective methods are more preferred. This paper aims to utilize Laplace Adomian decomposition method (LADM) to construct approximate solutions for these two types of equations and gives some examples of numerical calculations, which can prove the validity of LADM by comparing the error between the calculated results and the exact solution.

This paper analyzes and investigates the time-space fractional partial differential equations based on the LADM method in the sense of Caputo fractional derivative, which is a combination of the Laplace transform and the Adomian decomposition method. LADM method was first proposed by Khuri in 2001. Many partial differential equations which can describe the physical phenomena are solved by applying LADM and it has been used extensively to solve approximate solutions of partial differential and fractional partial differential equations.

This paper obtained an approximate solution to the FFPE and TFCBBEs by using the LADM. A number of numerical examples and graphs are used to compare the errors between the results and the exact solutions. The results show that LADM is a simple and effective mathematical technique to construct the approximate solutions of nonlinear time-space fractional equations in this work.

This paper verifies the effectiveness of this method by using the LADM to solve the FFPE and TFCBBEs. In addition, these two equations are very meaningful, and this paper will be helpful in the study of atmospheric diffusion, shallow water waves and other areas. And this paper also generalizes the drift and diffusion terms of the FFPE equation to the general form, which provides a great convenience for our future studies.