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Exact Traveling Wave Solutions of the Generalized Fractional Differential mBBM Equation
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作者 Yuting Zhong Renzhi Lu Heng Su 《Advances in Pure Mathematics》 2023年第3期167-173,共7页
By using the fractional complex transform and the bifurcation theory to the generalized fractional differential mBBM equation, we first transform this fractional equation into a plane dynamic system, and then find its... By using the fractional complex transform and the bifurcation theory to the generalized fractional differential mBBM equation, we first transform this fractional equation into a plane dynamic system, and then find its equilibrium points and first integral. Based on this, the phase portraits of the corresponding plane dynamic system are given. According to the phase diagram characteristics of the dynamic system, the periodic solution corresponds to the limit cycle or periodic closed orbit. Therefore, according to the phase portraits and the properties of elliptic functions, we obtain exact explicit parametric expressions of smooth periodic wave solutions. This method can also be applied to other fractional equations. 展开更多
关键词 A Generalized fractional Differential mbbm equation Traveling Wave Solution Phase Portrait
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THE QUASI-BOUNDARY VALUE METHOD FOR IDENTIFYING THE INITIAL VALUE OF THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION 被引量:3
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作者 Fan YANG Yan ZHANG +1 位作者 Xiao LIU Xiaoxiao LI 《Acta Mathematica Scientia》 SCIE CSCD 2020年第3期641-658,共18页
In this article,we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation.This problem is ill-posed and the quasi-boundary value method is proposed to deal wi... In this article,we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation.This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem.We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule.Some numerical results in one-dimensional case and two-dimensional case show that our method is efficient and stable. 展开更多
关键词 space-time fractional diffusion equation Ill-posed problem quasi-boundary value method identifying the initial value
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Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation
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作者 Khaled A. Gepreel Mohamed S. Mohameda 《Chinese Physics B》 SCIE EI CAS CSCD 2013年第1期33-38,共6页
The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Klein-Gordon equation. The numerical... The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Klein-Gordon equation. The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space-time fractional derivatives Klein- Gordon equation. This method introduces a promising tool for solving many space-time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations. 展开更多
关键词 homotopy analysis method nonlinear space-time fractional Klein-Gordon equation Caputo derivative
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The Exact Solution of the Space-Time Fractional Modified Kdv-Zakharov-Kuznetsov Equation
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作者 Qiuyan Jin Tiecheng Xia Jinbo Wang 《Journal of Applied Mathematics and Physics》 2017年第4期844-852,共9页
In this paper, we get many new analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov Kuznetsov (mKDV-ZK) equation by means of a new approach namely method of undetermined coefficients based... In this paper, we get many new analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov Kuznetsov (mKDV-ZK) equation by means of a new approach namely method of undetermined coefficients based on a fractional complex transform. These solutions have physics meanings in natural sciences. This method can be used to other nonlinear fractional differential equations. 展开更多
关键词 Analytical Solutions the space-time fractional MODIFIED KDV-ZK equation Nonlinear fractional Differential equation MODIFIED Riemann-Liouville Derivative
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Bright and dark soliton solutions for some nonlinear fractional differential equations 被引量:6
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作者 Ozkan Guner Ahmet Bekir 《Chinese Physics B》 SCIE EI CAS CSCD 2016年第3期52-59,共8页
In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified... In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified Benjamin-Bona- Mahoney (mBBM) equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional deriva- tives are described in the modified Riemann-Liouville sense. 展开更多
关键词 exact solutions ansatz method space-time fractional modified Benjamin-Bona-Mahoney equa-tion time fractional mKdV equation
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Exact solutions of nonlinear fractional differential equations by (G'/G)-expansion method 被引量:6
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作者 Ahmet Bekir zkan Güner 《Chinese Physics B》 SCIE EI CAS CSCD 2013年第11期140-145,共6页
In this paper, we use the fractional complex transform and the (G'/G)-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is prop... In this paper, we use the fractional complex transform and the (G'/G)-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie's modified Riemann-Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations. 展开更多
关键词 (G'/G)-expansion method time-fractional Burgers equation fractional-order biological popula-tion model space-time fractional Whitham-Broer-Kaup equations
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Singular and non-topological soliton solutions for nonlinear fractional differential equations 被引量:4
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作者 Ozkan Guner 《Chinese Physics B》 SCIE EI CAS CSCD 2015年第10期10-15,共6页
In this article, the fractional derivatives are described in the modified Riemann-Liouville sense. We propose a new approach, namely an ansatz method, for solving fractional differential equations(FDEs) based on a f... In this article, the fractional derivatives are described in the modified Riemann-Liouville sense. We propose a new approach, namely an ansatz method, for solving fractional differential equations(FDEs) based on a fractional complex transform and apply it to solve nonlinear space-time fractional equations. As a result, the non-topological as well as the singular soliton solutions are obtained. This method can be suitable and more powerful for solving other kinds of nonlinear fractional FDEs arising in mathematical physics. 展开更多
关键词 SOLITONS ansatz method the space-time fractional Boussinesq equation the space-time fractional(2+l)-dimensional breaking soliton equations
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Space time fractional KdV Burgers equation for dust acoustic shock waves in dusty plasma with non-thermal ions 被引量:2
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作者 Emad K.El-Shewy Abeer A.Mahmoud +2 位作者 Ashraf M.Tawfik Essam M.Abulwafa Ahmed Elgarayhi 《Chinese Physics B》 SCIE EI CAS CSCD 2014年第7期316-322,共7页
The KdV-Burgers equation for dust acoustic waves in unmagnetized plasma having electrons, singly charged non- thermal ions, and hot and cold dust species is derived using the reductive perturbation method. The Boltzma... The KdV-Burgers equation for dust acoustic waves in unmagnetized plasma having electrons, singly charged non- thermal ions, and hot and cold dust species is derived using the reductive perturbation method. The Boltzmann distribution is used for electrons in the presence of the cold (hot) dust viscosity coefficients. The semi-inverse method and Agrawal variational technique are applied to formulate the space-time fractional KdV-Burgers equation which is solved using the fractional sub-equation method. The effect of the fractional parameter on the behavior of the dust acoustic shock waves in the dusty plasma is investigated. 展开更多
关键词 dust-acoustic waves reductive perturbation method modified Riemann-Liouville fractionalderivative space-time fractional KdV-Burgers equation
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Some space-time fractional bright–dark solitons and propagation manipulations for a fractional Gross–Pitaevskii equation with an external potential
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作者 Li Li Fajun Yu 《Communications in Theoretical Physics》 SCIE CAS CSCD 2023年第7期86-94,共9页
Some nonautonomous bright–dark solitons(NBDSs)and nonautonomous controllable behaviors in the conformable space-time fractional Gross–Pitaevskii(FGP)equation with some external potentials are derived.We consider the... Some nonautonomous bright–dark solitons(NBDSs)and nonautonomous controllable behaviors in the conformable space-time fractional Gross–Pitaevskii(FGP)equation with some external potentials are derived.We consider the relations between the space-time FGP equation and the fractional nonlinear Schr?dinger equation and analyze the properties of the obtained equation with group velocity dispersion and spatiotemporal dispersion.Then,some constraint conditions of the valid soliton solutions are given.Furthermore,we consider the effect ofαandβin NBDSs of the space-time FGP equation.Some fractional spatial–temporal controlling wave prolong phenomena are considered,and some different propagation dynamics are generated via the different parametersαandβ.We study novel shape bright soliton solution,novel‘h’-shape dark soliton and some interactions of nonautonomous bright–dark solitons.The reported results of some novel interactions are considered,which can explain some models of the electrical and optical fields. 展开更多
关键词 nonautonomous fractional soliton soliton interaction space-time fractional Gross-Pitaevskii equation
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Soliton solutions for nonlinear variable-order fractional Korteweg-de Vries(KdV)equation arising in shallow water waves
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作者 Umair Ali Hijaz Ahmad Hanaa Abu-Zinadah 《Journal of Ocean Engineering and Science》 SCIE 2024年第1期50-58,共9页
Nonlinear fractional differential equations provide suitable models to describe real-world phenomena and many fractional derivatives are varying with time and space.The present study considers the advanced and broad s... Nonlinear fractional differential equations provide suitable models to describe real-world phenomena and many fractional derivatives are varying with time and space.The present study considers the advanced and broad spectrum of the nonlinear(NL)variable-order fractional differential equation(VO-FDE)in sense of VO Caputo fractional derivative(CFD)to describe the physical models.The VO-FDE transforms into an ordinary differential equation(ODE)and then solving by the modified(G/G)-expansion method.For ac-curacy,the space-time VO fractional Korteweg-de Vries(KdV)equation is solved by the proposed method and obtained some new types of periodic wave,singular,and Kink exact solutions.The newly obtained solutions confirmed that the proposed method is well-ordered and capable implement to find a class of NL-VO equations.The VO non-integer performance of the solutions is studied broadly by using 2D and 3D graphical representation.The results revealed that the NL VO-FDEs are highly active,functional and convenient in explaining the problems in scientific physics. 展开更多
关键词 space-time VO fractional KdV equation modified(G′/G)-expansion method VO Caputo fractional derivative generalized Riccati equation
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Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation 被引量:6
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作者 Xianjuan Li Chuanju Xu 《Communications in Computational Physics》 SCIE 2010年第10期1016-1051,共36页
In this paper,we investigate initial boundary value problems of the spacetime fractional diffusion equation and its numerical solutions.Two definitions,i.e.,Riemann-Liouville definition and Caputo one,of the fractiona... In this paper,we investigate initial boundary value problems of the spacetime fractional diffusion equation and its numerical solutions.Two definitions,i.e.,Riemann-Liouville definition and Caputo one,of the fractional derivative are considered in parallel.In both cases,we establish the well-posedness of the weak solution.Moveover,based on the proposed weak formulation,we construct an efficient spectral method for numerical approximations of the weak solution.The main contribution of this work are threefold:First,a theoretical framework for the variational solutions of the space-time fractional diffusion equation is developed.We find suitable functional spaces and norms in which the space-time fractional diffusion problem can be formulated into an elliptic weak problem,and the existence and uniqueness of the weak solution are then proved by using existing theory for elliptic problems.Secondly,we show that in the case of Riemann-Liouville definition,the well-posedness of the space-time fractional diffusion equation does not require any initial conditions.This contrasts with the case of Caputo definition,in which the initial condition has to be integrated into the weak formulation in order to establish the well-posedness.Finally,thanks to the weak formulation,we are able to construct an efficient numerical method for solving the space-time fractional diffusion problem. 展开更多
关键词 space-time fractional diffusion equation existence and uniqueness spectral methods error estimates
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A Posteriori Error Estimates of the Galerkin Spectral Methods for Space-Time Fractional Diffusion Equations 被引量:3
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作者 Huasheng Wang Yanping Chen +1 位作者 Yunqing Huang Wenting Mao 《Advances in Applied Mathematics and Mechanics》 SCIE 2020年第1期87-100,共14页
In this paper,an initial boundary value problem of the space-time fractional diffusion equation is studied.Both temporal and spatial directions for this equation are discreted by the Galerkin spectral methods.And then... In this paper,an initial boundary value problem of the space-time fractional diffusion equation is studied.Both temporal and spatial directions for this equation are discreted by the Galerkin spectral methods.And then based on the discretization scheme,reliable a posteriori error estimates for the spectral approximation are derived.Some numerical examples are presented to verify the validity and applicability of the derived a posteriori error estimator. 展开更多
关键词 Galerkin spectral methods space-time fractional diffusion equations a posteriori error estimates.
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Solving Space-Time Fractional Differential Equations by Using Modified Simple Equation Method 被引量:4
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作者 Melike Kaplan Arzu Akbulut Ahmet Bekir 《Communications in Theoretical Physics》 SCIE CAS CSCD 2016年第5期563-568,共6页
In this article,we establish new and more general traveling wave solutions of space-time fractional Klein–Gordon equation with quadratic nonlinearity and the space-time fractional breaking soliton equations using the... In this article,we establish new and more general traveling wave solutions of space-time fractional Klein–Gordon equation with quadratic nonlinearity and the space-time fractional breaking soliton equations using the modified simple equation method.The proposed method is so powerful and effective to solve nonlinear space-time fractional differential equations by with modified Riemann–Liouville derivative. 展开更多
关键词 symbolic computation exact solution space-time fractional differential equation space-time fractional Klein–Gordon equation the space-time fractional breaking soliton equations
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SITEM for the conformable space-time fractional Boussinesq and(2+1)-dimensional breaking soliton equations 被引量:1
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作者 H.Çerdik Yaslan Ayse Girgin 《Journal of Ocean Engineering and Science》 SCIE 2021年第3期228-236,共9页
In the present paper,new analytical solutions for the space-time fractional Boussinesq and(2+1)-dimensional breaking soliton equations are obtained by using the simplified tan(φ(ξ)2)-expansion method.Here,fractional... In the present paper,new analytical solutions for the space-time fractional Boussinesq and(2+1)-dimensional breaking soliton equations are obtained by using the simplified tan(φ(ξ)2)-expansion method.Here,fractional derivatives are defined in the conformable sense.To show the correctness of the obtained traveling wave solutions,residual error function is defined.It is observed that the new solutions are very close to the exact solutions.The solutions obtained by the presented method have not been reported in former literature. 展开更多
关键词 space-time fractional Boussinesq equation (2+1)-dimensional breaking soliton equation Simplified tan(φ(ξ)2)-expansion method(SITEM) Conformable derivative.
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Infinite Sequence Solutions for Space-Time Fractional Symmetric Regularized Long Wave Equation
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作者 KANG Zhouzheng 《Journal of Partial Differential Equations》 CSCD 2016年第1期48-58,共11页
In this paper, we investigate the space-time fractional symmetric regularized long wave equation. By using the Backlund transformations and nonlinear superposition formulas of solutions to Riccati equation, we present... In this paper, we investigate the space-time fractional symmetric regularized long wave equation. By using the Backlund transformations and nonlinear superposition formulas of solutions to Riccati equation, we present infinite sequence solutions for space-time fractional symmetric regularized long wave equation. This method can be extended to solve other nonlinear fractional partial differential equations. 展开更多
关键词 space-time fractional symmetric regularized long wave equation Backlund transformations nonlinear superposition formulas exact solutions.
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Ulam-Hyers stability and analytical approach for m-dimensional Caputo space-time variable fractional order advection-dispersion equation
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作者 Pratibha Verma Manoj Kumar Anand Shukla 《International Journal of Modeling, Simulation, and Scientific Computing》 EI 2022年第1期131-174,共44页
This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomia... This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomian decomposition method and obtain the exact solution in just one iteration.Moreover,with the help of fixed point theory,we study the existence and uniqueness conditions for the positive solution and prove some new results.Also,obtain the Ulam–Hyers stabilities for the proposed problem.Two gen-eralized examples are considered to show the method’s applicability and compared with other existing numerical methods.The present method performs exceptionally well in terms of efficiency and simplicity.Further,we solved both examples using the two most well-known numerical methods and compared them with the TSADM solution. 展开更多
关键词 Fixed point theorems space-time variable Caputo’s fractional operators advection-dispersion equation Ulam-Hyers stability two-step Adomian decomposition method
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Investigation of adequate closed form travelling wave solution to the space-time fractional non-linear evolution equations
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作者 Mohammad Asif Arefin M.Ayesha Khatun +1 位作者 M.Hafiz Uddin Mustafa Inc 《Journal of Ocean Engineering and Science》 SCIE 2022年第3期292-303,共12页
This work aims to construct exact solutions for the space-time fractional(2+1)-dimensional dispersive longwave(DLW)equation and approximate long water wave equation(ALW)utilizing the twovariable(G′/G,1/G)-expansion m... This work aims to construct exact solutions for the space-time fractional(2+1)-dimensional dispersive longwave(DLW)equation and approximate long water wave equation(ALW)utilizing the twovariable(G′/G,1/G)-expansion method and the modified Riemann-Liouville fractional derivative.The recommended equations play a significant role to describe the travel of the shallow water wave.The fractional complex transform is used to convert fractional differential equations into ordinary differential equations.Several wave solutions have been successfully achieved using the proposed approach and the symbolic computer Maple package.The Maple package program was used to set up and validate all of the computations in this investigation.By choosing particular values of the embedded parameters,we pro-duce multiple periodic solutions,periodic wave solutions,single soliton solutions,kink wave solutions,and more forms of soliton solutions.The achieved solutions might be useful to comprehend nonlinear phenomena.It is worth noting that the implemented method for solving nonlinear fractional partial dif-ferential equations(NLFPDEs)is efficient,and simple to find further and new-fangled solutions in the arena of mathematical physics and coastal engineering. 展开更多
关键词 Riemann-Liouville fractional derivative space-time fractional(2+1)-dimensional dispersive long wave equation Approximate long water wave equation Wave transformation The two-variable(G′/G 1/G)-expansion method
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Dark Soliton Solutions of Space-Time Fractional Sharma–Tasso–Olver and Potential Kadomtsev–Petviashvili Equations
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作者 Ozkan Guner Alper Korkmaz Ahmet Bekir 《Communications in Theoretical Physics》 SCIE CAS CSCD 2017年第2期182-188,共7页
Dark soliton solutions for space-time fractional Sharma–Tasso–Olver and space-time fractional potential Kadomtsev–Petviashvili equations are determined by using the properties of modified Riemann–Liouville derivat... Dark soliton solutions for space-time fractional Sharma–Tasso–Olver and space-time fractional potential Kadomtsev–Petviashvili equations are determined by using the properties of modified Riemann–Liouville derivative and fractional complex transform. After reducing both equations to nonlinear ODEs with constant coefficients, the tanh ansatz is substituted into the resultant nonlinear ODEs. The coefficients of the solutions in the ansatz are calculated by algebraic computer computations. Two different solutions are obtained for the Sharma–Tasso–Olver equation as only one solution for the potential Kadomtsev–Petviashvili equation. The solution profiles are demonstrated in 3D plots in finite domains of time and space. 展开更多
关键词 exact solution modified Riemann–Liouville derivative space-time fractional Sharma–Tasso–Olver equation space-time fractional potential Kadomtsev–Petviashvili equation
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空时分数阶mBBM方程的新精确解(英文) 被引量:2
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作者 康丽 孙峪怀 +1 位作者 廖红梅 熊淑雪 《四川大学学报(自然科学版)》 CAS CSCD 北大核心 2018年第4期673-677,共5页
为了构造空时分数阶mBBM方程的新显示解,本文首先利用分数阶复变换技巧将分数阶偏微分方程转化为常微分方程,然后应用扩展的(G′/G)-展开法求解该常微分方程.新精确解包括分别带有负幂次项的三角函数解,双曲函数解及有理函数解.
关键词 扩展的(G′/G)-展开法 空时分数阶mbbm方程 精确解
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New Exact Solutions of Fractional Zakharov–Kuznetsov and Modified Zakharov–Kuznetsov Equations Using Fractional Sub-Equation Method 被引量:3
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作者 S.Saha Ray S.Sahoo 《Communications in Theoretical Physics》 SCIE CAS CSCD 2015年第1期25-30,共6页
In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics; namely the space-time fractional Zakharov–Kuznetsov(ZK) and modified Zakharov–Kuznetso... In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics; namely the space-time fractional Zakharov–Kuznetsov(ZK) and modified Zakharov–Kuznetsov(m ZK) equations by using fractional sub-equation method. As a result, new types of exact analytical solutions are obtained. The obtained results are shown graphically. Here the fractional derivative is described in the Jumarie's modified Riemann–Liouville sense. 展开更多
关键词 fractional sub-equation method space-time fractional Zakharov-Kuznetsov (ZK) equation space-time fractional modified Zakharov-Kuznetsov (mZK) equation modified Riemann-Liouvillederivative Mittag-leffler function
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