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1 – 10 of over 27000Xiao-rong Kang and Xian Daquan
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…
Abstract
Purpose
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.
Design/methodology/approach
Along with some special transformations, the Lie group method and the rational function method are applied to the (2+1)-dimensional AKNS equation.
Findings
Some new non-traveling wave solutions are obtained, including generalized rational solutions with two arbitrary functions of time variable.
Research limitations/implications
As a typical nonlinear evolution equation, some dynamical behaviors are also discussed.
Originality/value
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.
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K. Parand and L. Hosseini
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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
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.
Originality/value
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.
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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…
Abstract
Purpose
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.
Design/methodology/approach
The Hirota bilinear method, the Lie symmetry method and the non-Lie symmetry method are applied to the (2 + 1)D GSWW equation.
Findings
A reduced (1 + 1)D potential KdV equation can be derived, and its soliton solutions are also presented.
Research limitations/implications
As a typical nonlinear evolution equation, some dynamical behaviors are also discussed.
Originality/value
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.
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Ahmet Yıldırım and Hüseyin Koçak
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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
Numerical examples are given to demonstrate the effectiveness of the present methods.
Originality/value
Results show that the proposed schemes are very effective and convenient for solving linear and nonlinear fractional differential equations with high accuracy.
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Huifang Li, Mi Zhao, Lihua Wu, Piguang Wang and Xiuli Du
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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
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.
Originality/value
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.
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A.S. Roknizadeh, A.S. Nobari, M. Mohagheghi and H. Shahverdi
The purpose of this paper is to analyze the stability of aeroelastic systems using aeroelastic frequency response function (FRF).
Abstract
Purpose
The purpose of this paper is to analyze the stability of aeroelastic systems using aeroelastic frequency response function (FRF).
Design/methodology/approach
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.
Findings
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.
Originality/value
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.
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Vedat Suat Erturk, Ahmet Yıldırım, Shaher Momanic and Yasir Khan
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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
The diagonal Padé approximants are effectively used in the analysis to capture the essential behavior of the solution.
Originality/value
Illustrative examples are included to demonstrate the validity and applicability of the technique.
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Xiankang Luo and Muhammad Nadeem
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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
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.
Originality/value
This paper presents the original work. This study reveals that ℒT can perform the solution of fractional-order models without any restriction of variables.
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Velinda Calvert and Mohsen Razzaghi
This paper aims to propose a new numerical method for the solution of the Blasius and magnetohydrodynamic (MHD) Falkner-Skan boundary-layer equations. The Blasius and MHD…
Abstract
Purpose
This paper aims to propose a new numerical method for the solution of the Blasius and magnetohydrodynamic (MHD) Falkner-Skan boundary-layer equations. The Blasius and MHD Falkner-Skan equations are third-order nonlinear boundary value problems on the semi-infinite domain.
Design/methodology/approach
The approach is based upon modified rational Bernoulli functions. The operational matrices of derivative and product of modified rational Bernoulli functions are presented. These matrices together with the collocation method are then utilized to reduce the solution of the Blasius and MHD Falkner-Skan boundary-layer equations to the solution of a system of algebraic equations.
Findings
The method is computationally very attractive and gives very accurate results.
Originality/value
Many problems in science and engineering are set in unbounded domains. One approach to solve these problems is based on rational functions. In this work, a new rational function is used to find solutions of the Blasius and MHD Falkner-Skan boundary-layer equations.
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