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1 – 10 of 53Syed Tauseef Mohyud‐Din, Yasir Khan, Naeem Faraz and Ahmet Yıldırım
The purpose of this paper is to apply exp‐function method to construct generalized solitary and periodic solutions of Fitzhugh‐Nagumo equation, which plays a very important role…
Abstract
Purpose
The purpose of this paper is to apply exp‐function method to construct generalized solitary and periodic solutions of Fitzhugh‐Nagumo equation, which plays a very important role in mathematical physics and engineering sciences.
Design/methodology/approach
The authors apply exp‐function method to construct generalized solitary and periodic solutions of Fitzhugh‐Nagumo equation.
Findings
Numerical results clearly indicate the reliability and efficiency of the proposed exp‐function method. The suggested algorithm is quite efficient and is practically well suited for use in these problems.
Originality/value
In this paper, the authors applied the exp‐function method to obtain solutions of the Fitzhugh‐Nagumo equation and show that the exp‐function method gives more realistic solutions without disturbing the basic physics of the physical problems.
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Keywords
The purpose of this paper is to apply the exp‐function method to construct exact solutions of nonlinear wave equations. The proposed technique is tested on the (2+1) and (3+1…
Abstract
Purpose
The purpose of this paper is to apply the exp‐function method to construct exact solutions of nonlinear wave equations. The proposed technique is tested on the (2+1) and (3+1) dimensional extended shallow water wave equations. These equations play a very important role in mathematical physics and engineering sciences.
Design/methodology/approach
In this paper, the authors apply the exp‐function method to construct exact solutions of nonlinear wave equations.
Findings
In total, four forms of the extended shallow water wave equation have been studied, from the point of view of its exact solutions using computational method. Exp‐function method was employed to achieve the goal set for this work. The applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear wave equations. Finally, it is worthwhile to mention that the proposed method is straightforward, concise, and it is a promising and powerful new method for other nonlinear wave equations in mathematical physics.
Originality/value
The algorithm suggested in the paper is quite efficient and is practically well suited for use in these problems. The method is straightforward and concise, and its applications are promising.
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Zhijuan Jia, Mingsheng Hu, Qiaoling Chen and Suimin Jai
The fractional complex transform is used to convert the fractional differential equation to its differential partner and the exp-function method is to solve the resultant…
Abstract
Purpose
The fractional complex transform is used to convert the fractional differential equation to its differential partner and the exp-function method is to solve the resultant equation. The exact solutions for the equation are successfully established. The paper aims to discuss these issues.
Design/methodology/approach
Use the chain rule of the local fractional derivative and the exp-function method.
Findings
Some new exact solutions for the fractional differential equation are successfully established, and the process of the solution is extremely simple and remarkably accessible.
Originality/value
The fractional complex transform is used to convert the fractional differential equation to its differential partner and the exp-function method is to solve the resultant equation.
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Mehdi Dehghan, Jalil Manafian Heris and Abbas Saadatmandi
The purpose of this paper is to use He's Exp‐function method (EFM) to construct solitary and soliton solutions of the nonlinear evolution equation.
Abstract
Purpose
The purpose of this paper is to use He's Exp‐function method (EFM) to construct solitary and soliton solutions of the nonlinear evolution equation.
Design/methodology/approach
This technique is straightforward and simple to use and is a powerful method to overcome some difficulties in the nonlinear problems.
Findings
This method is developed for searching exact traveling wave solutions of the nonlinear partial differential equations. The EFM presents a wider applicability for handling nonlinear wave equations.
Originality/value
The paper shows that EFM, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations. Application of EFM to Fitzhugh‐Nagumo equation illustrates its effectiveness.
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Keywords
Xiao-rong Kang, Xian Daquan and Zhengde Dai
– The purpose of this paper is to find new non-traveling wave solutions and study its localized structure of Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation.
Abstract
Purpose
The purpose of this paper is to find new non-traveling wave solutions and study its localized structure of Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation.
Design/methodology/approach
The authors apply the Lie group method twice and combine with the Exp-function method and Riccati equation mapping method to the (2+1)-dimensional CDGKS equation.
Findings
The authors have obtained some new non-traveling wave solutions with two arbitrary functions of time variable.
Research limitations/implications
As non-linear evolution equations is characterized by rich dynamical behavior, the authors just found some of them and others still to be found.
Originality/value
These results may help the authors to investigate some new localized structure and the interaction of waves in high-dimensional models. The new non-traveling wave solutions with two arbitrary functions of time variable are obtained for CDGKS equation using Lie group approach twice and combining with the Exp-function method and Riccati equation mapping method by the aid of Maple.
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This paper aims to review some effective methods for fully fourth-order nonlinear integral boundary value problems with fractal derivatives.
Abstract
Purpose
This paper aims to review some effective methods for fully fourth-order nonlinear integral boundary value problems with fractal derivatives.
Design/methodology/approach
Boundary value problems arise everywhere in engineering, hence two-scale thermodynamics and fractal calculus have been introduced. Some analytical methods are reviewed, mainly including the variational iteration method, the Ritz method, the homotopy perturbation method, the variational principle and the Taylor series method. An example is given to show the simple solution process and the high accuracy of the solution.
Findings
An elemental and heuristic explanation of fractal calculus is given, and the main solution process and merits of each reviewed method are elucidated. The fractal boundary value problem in a fractal space can be approximately converted into a classical one by the two-scale transform.
Originality/value
This paper can be served as a paradigm for various practical applications.
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A three-dimensional (3D) unsteady potential flow might admit a variational principle. The purpose of this paper is to adopt a semi-inverse method to search for the variational…
Abstract
Purpose
A three-dimensional (3D) unsteady potential flow might admit a variational principle. The purpose of this paper is to adopt a semi-inverse method to search for the variational formulation from the governing equations.
Design/methodology/approach
A suitable trial functional with a possible unknown function is constructed, and the identification of the unknown function is given in detail. The Lagrange multiplier method is used to establish a generalized variational principle, but in vain.
Findings
Some new variational principles are obtained, and the semi-inverse method can easily overcome the Lagrange crisis.
Practical implications
The semi-inverse method sheds a promising light on variational theory, and it can replace the Lagrange multiplier method for the establishment of a generalized variational principle. It can be used for the establishment of a variational principle for fractal and fractional calculus.
Originality/value
This paper establishes some new variational principles for the 3D unsteady flow and suggests an effective method to eliminate the Lagrange crisis.
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Keywords
Xuejuan Li and Ji-Huan He
The purpose of this paper is to develop an effective numerical algorithm for a gas-melt two-phase flow and use it to simulate a polymer melt filling process. Moreover, the…
Abstract
Purpose
The purpose of this paper is to develop an effective numerical algorithm for a gas-melt two-phase flow and use it to simulate a polymer melt filling process. Moreover, the suggested algorithm can deal with the moving interface and discontinuities of unknowns across the interface.
Design/methodology/approach
The algebraic sub-grid scales-variational multi-scale (ASGS-VMS) finite element method is used to solve the polymer melt filling process. Meanwhile, the time is discretized using the Crank–Nicolson-based split fractional step algorithm to reduce the computational time. The improved level set method is used to capture the melt front interface, and the related equations are discretized by the second-order Taylor–Galerkin scheme in space and the third-order total variation diminishing Runge–Kutta scheme in time.
Findings
The numerical method is validated by the benchmark problem. Moreover, the viscoelastic polymer melt filling process is investigated in a rectangular cavity. The front interface, pressure field and flow-induced stresses of polymer melt during the filling process are predicted. Overall, this paper presents a VMS method for polymer injection molding. The present numerical method is extremely suitable for two free surface problems.
Originality/value
For the first time ever, the ASGS-VMS finite element method is performed for the two-phase flow of polymer melt filling process, and an effective numerical method is designed to catch the moving surface.
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It is extremely difficult to establish a variational principle for plasma. Kalaawy obtained a variational principle by using the semi-inverse method in 2016, and Li and He…
Abstract
Purpose
It is extremely difficult to establish a variational principle for plasma. Kalaawy obtained a variational principle by using the semi-inverse method in 2016, and Li and He suggested a modification in 2017. This paper aims to search for a generalized variational formulation with a free parameter.
Design/methodology/approach
The semi-inverse method is used by suitable construction of a trial functional with some free parameters.
Findings
A modification of Li-He’s variational principle with a free parameter is obtained.
Originality/value
This paper suggests a new approach to construction of a trial-functional with some free parameters.
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Ji-Huan He and Habibolla Latifizadeh
The purpose of this paper is to suggest a general numerical algorithm for nonlinear problems by the variational iteration method (VIM).
Abstract
Purpose
The purpose of this paper is to suggest a general numerical algorithm for nonlinear problems by the variational iteration method (VIM).
Design/methodology/approach
Firstly, the Laplace transform technique is used to reconstruct the variational iteration algorithm-II. Secondly, its convergence is strictly proved. Thirdly, the numerical steps for the algorithm is given. Finally, some examples are given to show the solution process and the effectiveness of the method.
Findings
No variational theory is needed to construct the numerical algorithm, and the incorporation of the Laplace method into the VIM makes the solution process much simpler.
Originality/value
A universal iteration formulation is suggested for nonlinear problems. The VIM cleans up the numerical road to differential equations.
Details