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1 – 10 of 13The purpose of this paper is to apply the fractional sub-equation method to research on coupled fractional variant Boussinesq equation and fractional approximate long water wave…
Abstract
Purpose
The purpose of this paper is to apply the fractional sub-equation method to research on coupled fractional variant Boussinesq equation and fractional approximate long water wave equation.
Design/methodology/approach
The algorithm is implemented with the aid of fractional Ricatti equation and the symbol computational system Mathematica.
Findings
New travelling wave solutions, which include generalized hyperbolic function solutions, generalized trigonometric function solutions and rational solutions, for these two equations are obtained.
Originality/value
The algorithm is demonstrated to be direct and precise, and can be used for many other nonlinear fractional partial differential equations. The fractional derivatives described in this paper are in the Jumarie's modified Riemann-Liouville sense.
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Tarikul Islam and Armina Akter
Fractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to…
Abstract
Purpose
Fractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to their importance in the nature of real world. In this article, the autors suggest a productive technique, called the rational fractional
Design/methodology/approach
The rational fractional
Findings
Achieved fresh and further abundant closed form traveling wave solutions to analyze the inner mechanisms of complex phenomenon in nature world which will bear a significant role in the of research and will be recorded in the literature.
Originality/value
The rational fractional
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This study aims to purpose the idea of a new hybrid approach to examine the approximate solution of the fourth-order partial differential equations (PDEs) with time fractional…
Abstract
Purpose
This study aims to purpose the idea of a new hybrid approach to examine the approximate solution of the fourth-order partial differential equations (PDEs) with time fractional derivative that governs the behaviour of a vibrating beam. The authors have also demonstrated the physical representations of the problem in different fractional order.
Design/methodology/approach
Mohand transform is a new technique that the authors use to reduce the order of fractional problems, and then the homotopy perturbation method can be used to handle the further series solution in the form of convergence. The formulation of Mohand transform and the homotopy perturbation method is known as Mohand homotopy perturbation transform (MHPT). The fractional order in this paper is considered in the Caputo sense.
Findings
The results are formulated in the shape of iterative series and predict the solution close to the exact solution. This successive iteration demonstrates the authenticity and reliability of this scheme.
Research limitations/implications
This paper presents the significance of MHPT such that, firstly, Mohand transform is coupled with homotopy perturbation method and, secondly, the fractional order a is used to show the physical behaviour of the graphical solution.
Practical implications
This study presents the consistency and authenticity of the graphical solution with the exact solutions.
Social implications
This study demonstrates that Mohand transform is capable to handle the fractional order problem without any constraints and assumptions.
Originality/value
A new integral transform has been introduced without any restriction of variables that produces the results in a series form and confirms the validity of the proposed algorithm by graphical illustrations.
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The purpose of this paper is to apply an efficient hybrid computational numerical technique, namely, q-homotopy analysis Sumudu transform method (q-HASTM) and residual power…
Abstract
Purpose
The purpose of this paper is to apply an efficient hybrid computational numerical technique, namely, q-homotopy analysis Sumudu transform method (q-HASTM) and residual power series method (RPSM) for finding the analytical solution of the non-linear time-fractional Hirota–Satsuma coupled KdV (HS-cKdV) equations.
Design/methodology/approach
The proposed technique q-HASTM is the graceful amalgamations of q-homotopy analysis method with Sumudu transform via Caputo fractional derivative, whereas RPSM depend on generalized formula of Taylors series along with residual error function.
Findings
To illustrate and validate the efficiency of the proposed technique, the authors analyzed the projected non-linear coupled equations in terms of fractional order. Moreover, the physical behavior of the attained solution has been captured in terms of plots and by examining the L2 and L∞ error norm for diverse value of fractional order.
Originality/value
The authors implemented two technique, q-HASTM and RPSM to obtain the solution of non-linear time-fractional HS-cKdV equations. The obtained results and comparison between q-HASTM and RPSM, shows that the proposed methods provide the solution of non-linear models in form of a convergent series, without using any restrictive assumption. Also, the proposed algorithm is easy to implement and highly efficient to analyze the behavior of non-linear coupled fractional differential equation arisen in various area of science and engineering.
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Bo Tang, Xuemin Wang, Leilei Wei and Xindong Zhang
This paper aims to apply fractional variational iteration method using He's polynomials (FVIMHP) to obtain exact solutions for variable-coefficient fractional heat-like and…
Abstract
Purpose
This paper aims to apply fractional variational iteration method using He's polynomials (FVIMHP) to obtain exact solutions for variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions.
Design/methodology/approach
The approach is based on FVIMHP. The authors choose as some examples to illustrate the validity and the advantages of the method.
Findings
The results reveal that the FVIMHP method provides a very effective, convenient and powerful mathematical tool for solving fractional differential equations.
Originality/value
The variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions are solved first. Illustrative examples are included to demonstrate the validity and applicability of the method.
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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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Sheng Zhang, Bin Cai and Bo Xu
– The purpose of this paper is to analytically solve the (2+1)-dimensional nonlinear time fractional biological population model in the Caputo sense.
Abstract
Purpose
The purpose of this paper is to analytically solve the (2+1)-dimensional nonlinear time fractional biological population model in the Caputo sense.
Design/methodology/approach
The paper uses the variable separation method and the properties of Gamma function to construct exact solutions of the time fractional biological population model.
Findings
New variable separation solutions are obtained, from which some known solutions are recovered as special cases.
Originality/value
Solving fractional biological population model by the variable separation method and the properties of Gamma function is original. It is shown that the method presented in this paper can be also used for some other nonlinear fractional partial differential equations arising in sciences and engineering.
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Jian-Gen Liu, Yi-Ying Feng and Hong-Yi Zhang
The purpose of this paper is to construct the algebraic traveling wave solutions of the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsve (KdV-Z-K) equation, which can be…
Abstract
Purpose
The purpose of this paper is to construct the algebraic traveling wave solutions of the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsve (KdV-Z-K) equation, which can be usually used to express shallow water wave phenomena.
Design/methodology/approach
The authors apply the planar dynamical systems and invariant algebraic cure approach to find the algebraic traveling wave solutions and rational solutions of the (3 + 1)-dimensional modified KdV-Z-K equation. Also, the planar dynamical systems and invariant algebraic cure approach is applied to considered equation for finding algebraic traveling wave solutions.
Findings
As a result, the authors can find that the integral constant is zero and non-zero, the algebraic traveling wave solutions have different evolutionary processes. These results help to better reveal the evolutionary mechanism of shallow water wave phenomena and find internal connections.
Research limitations/implications
The paper presents that the implemented methods as a powerful mathematical tool deal with (3 + 1)-dimensional modified KdV-Z-K equation by using the planar dynamical systems and invariant algebraic cure.
Practical implications
By considering important characteristics of algebraic traveling wave solutions, one can understand the evolutionary mechanism of shallow water wave phenomena and find internal connections.
Originality/value
To the best of the authors’ knowledge, the algebraic traveling wave solutions have not been reported in other places. Finally, the algebraic traveling wave solutions nonlinear dynamics behavior was shown.
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The purpose of this paper is to introduce an innovative strategy for the approximate solution of the heat flow problems in two- and three-dimensional spaces. This new strategy is…
Abstract
Purpose
The purpose of this paper is to introduce an innovative strategy for the approximate solution of the heat flow problems in two- and three-dimensional spaces. This new strategy is very easy to implement and handles the restrictive variable that may ruin the physical nature of the problem.
Design/methodology/approach
This study combines Sawi transform (ST) and the homotopy perturbation method (HPM) to formulate the idea of Sawi homotopy perturbation transform method (SHPTM). First, this study implements ST to handle the recurrence relation and then incorporates HPM to derive the series solutions of this recurrence relation. ST has the advantage in that it does not require any assumptions or hypothesis for the evaluation of series solutions.
Findings
This strategy finds the results very accurate and close to the precise solution. The graphical observations and the surface solution demonstrate that SHPTM is a reliable and powerful scheme for finding the approximate solution of heat flow problems.
Originality/value
The study presents an original work. This study develops SHPTM for the approximate solution of two- and three-dimensional heat flow problems. The obtained results and graphical representation demonstrate that SHPTM is a very authentic and reliable approach.
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Ismail Onder, Melih Cinar, A. Secer and Mustafa Bayram
The soliton solutions are obtained by using extended rational sin/cos and sinh-cosh method. The methods are powerful and have ease of use. Applying wave transformation to the…
Abstract
Purpose
The soliton solutions are obtained by using extended rational sin/cos and sinh-cosh method. The methods are powerful and have ease of use. Applying wave transformation to the nonlinear partial differential equations (NLPDEs) and the considered equation turns into a nonlinear differential equation (NODE). According to the methods, the solution sets of the NODE are supposed to the form of the rational terms as sinh/cosh and sin/cos and the trial solutions are substituted into the NODE. Collecting the same power of the trigonometric functions, a set of algebraic equations is derived.
Design/methodology/approach
The main purpose of this paper is to obtain soliton solutions of the modified equal width (MEW) equation. MEW is a form of regularized-long-wave (RLW) equation that represents one-dimensional wave propagation in nonlinear media with dispersion processes. This is also used to simulate the undular bore in a long shallow water canal.
Findings
Thus, the solution of the main PDE is reduced to the solution of a set of algebraic equations. In this paper, the kink, singular and singular periodic solitons have been successfully obtained.
Originality/value
Illustrative plots of the solutions have been presented for physical interpretation of the obtained solutions. The methods are powerful and might be used to solve a broad class of differential equations in real-life problems.
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