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The purpose of this paper is to find out some new rational non-traveling wave solutions and to study localized structures for (2+1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation.

Along with some special transformations, the Lie group method and the rational function method are applied to the (2+1)-dimensional AKNS equation.

Some new non-traveling wave solutions are obtained, including generalized rational solutions with two arbitrary functions of time variable.

As a typical nonlinear evolution equation, some dynamical behaviors are also discussed.

With the help of the Lie group method, special transformations and the rational function method, new non-traveling wave solutions are derived for the AKNS equation by Maple software. These results are much useful for investigating some new localized structures and the interaction of waves in high-dimensional models, and enrich dynamical features of solutions for the higher dimensional systems.

The aim is to present in this paper an effective strategy in dealing with a semi‐infinite interval by using a suitable mapping that transforms a semi‐infinite interval to a finite interval.

The authors introduce a new orthogonal system of rational functions induced by general Jacobi polynomials with the parameters alpha and beta. It is more flexible in applications. In particular, alpha and beta could be regulated, so that the systems are mutually orthogonal in certain weighted Hilbert spaces.

This approach is applied for solving a non‐linear system two‐point boundary value problem (BVP) on semi‐infinite interval, describing the flow and diffusion of chemically reactive species over a nonlinearly stretching sheet immersed in a porous medium. The new approach reduces the solution of a problem to the solution of a system of algebraic equations.

The paper presents an effective strategy in dealing with a semi‐infinite interval by using a suitable mapping that transforms a semi‐infinite interval to a finite interval.

The purpose of this paper is to discuss the homoclinic breathe-wave solutions and the singular periodic solutions for (2 + 1)-dimensional generalized shallow water wave (GSWW) equation.

The Hirota bilinear method, the Lie symmetry method and the non-Lie symmetry method are applied to the (2 + 1)D GSWW equation.

A reduced (1 + 1)D potential KdV equation can be derived, and its soliton solutions are also presented.

As a typical nonlinear evolution equation, some dynamical behaviors are also discussed.

These results are very useful for investigating some localized geometry structures of dynamical behaviors and enriching dynamical features of solutions for the higher dimensional systems.

The purpose of this paper is to implement the variational iteration method and the homotopy perturbation method to give a rational approximation solution of the foam drainage equation with time‐ and space‐fractional derivatives.

The fractional derivatives are described in the Caputo sense. In these schemes, the solution takes the form of a convergent series with easily computable components.

Numerical examples are given to demonstrate the effectiveness of the present methods.

Results show that the proposed schemes are very effective and convenient for solving linear and nonlinear fractional differential equations with high accuracy.

The purpose of this paper is to propose a stable high-order absorbing boundary condition (ABC) based on new continued fraction for scalar wave propagation in 2D and 3D unbounded layers.

The ABC is obtained based on continued fraction (CF) expansion of the frequency-domain dynamic stiffness coefficient (DtN kernel) on the artificial boundary of a truncated infinite domain. The CF which has been used to the thin layer method in [69] will be applied to the DtN method to develop a time-domain high-order ABC for the transient scalar wave propagation in 2D. Furthermore, a new stable composite-CF is proposed in this study for 3D unbounded layers by nesting the above CF for 2D layer and another CF.

The ABS has been transformed from frequency to time domain by using the auxiliary variable technique. The high-order time-domain ABC can couple seamlessly with the finite element method. The instability of the ABC-FEM coupled system is discussed and cured.

This manuscript establishes a stable high-order time-domain ABC for the scalar wave equation in 2D and 3D unbounded layers, which is based on the new continued fraction. The high-order time-domain ABC can couple seamlessly with the finite element method. The instability of the coupled system is discussed and cured.

The purpose of this paper is to analyze the stability of aeroelastic systems using aeroelastic frequency response function (FRF).

The proposed technique determines the instability boundary of an aeroelastic system based on condition number (CN) of aeroelastic FRF matrix or directly from FRFs data.

Stability margins of typical section and hingeless helicopter rotor blade in the subsonic flow regimes (quasi‐steady and unsteady models) are determined using proposed techniques as two case studies.

The paper introduces a technique which is applicable not only when aerodynamic and structure analytical models are available but also when there are experimental models for structure and/or aerodynamics, such as impulse response functions data or FRFs data. In other words, the main advantage of the proposed method, besides its simplicity and low memory requirement, is its ability to utilize experimental data.

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.

The purpose of this paper is to propose an approximate method for solving a fractional population growth model in a closed system. The fractional derivatives are described in the Caputo sense.

The approach is based on the differential transform method. The solutions of a fractional model equation are calculated in the form of convergent series with easily computable components.

The diagonal Padé approximants are effectively used in the analysis to capture the essential behavior of the solution.

Illustrative examples are included to demonstrate the validity and applicability of the technique.

This study aims to investigate the approximate solution of the time fractional time-fractional Newell–Whitehead–Segel (TFNWS) model that reflects the appearance of the stripe patterns in two-dimensional systems. The significant results of plot distribution show that the proposed approach is highly authentic and reliable for the fractional-order models.

The Laplace transform residual power series method (ℒT-RPSM) is the combination of Laplace transform (ℒT) and residual power series method (RPSM). The ℒT is examined to minimize the order of fractional order, whereas the RPSM handles the series solution in the form of convergence. The graphical results of the fractional models are represented through the fractional order α.

The derived results are obtained in a successive series and yield the results toward the exact solution. These successive series confirm the consistency and accuracy of ℒT-RPSM. This study also compares the exact solutions with the graphical solutions to show the performance and authenticity of the visual solutions. The proposed scheme does not require the restriction of variables and produces the numerical results in terms of a series. This strategy is capable to handle the nonlinear terms very easily for the TFNWS model.

This paper presents the original work. This study reveals that ℒT can perform the solution of fractional-order models without any restriction of variables.