In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov （KdV-KZK） equa- tion are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For thi...In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov （KdV-KZK） equa- tion are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the non- linear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.展开更多
In the present paper, we established a traveling wave solution by using modified Kudryashov method for the space-time fractional nonlinear partial differential equations. The method is used to obtain the exact solutio...In the present paper, we established a traveling wave solution by using modified Kudryashov method for the space-time fractional nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of the space-time fractional nonlinear partial differential equations such as, the space-time fractional coupled equal width wave equation(CEWE) and the space-time fractional coupled modified equal width wave equation(CMEW), which are the important soliton equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform and properties of modified Riemann–Liouville derivative. We plot the exact solutions for these equations at different time levels.展开更多
Our purpose of this paper is to apply the improved Kudryashov method for solving various types of nonlinear fractional partial differential equations. As an application, the time-space fractional Korteweg-de Vries-Bur...Our purpose of this paper is to apply the improved Kudryashov method for solving various types of nonlinear fractional partial differential equations. As an application, the time-space fractional Korteweg-de Vries-Burger (KdV-Burger) equation is solved using this method and we get some new travelling wave solutions. To acquire our purpose a complex transformation has been also used to reduce nonlinear fractional partial differential equations to nonlinear ordinary differential equations of integer order, in the sense of the Jumarie’s modified Riemann-Liouville derivative. Afterwards, the improved Kudryashov method is implemented and we get our required reliable solutions where the results are justified by mathematical software Maple-13.展开更多
We mainly investigate the rational solutions and N-wave resonance solutions for the(3+1)-dimensional Kudryashov–Sinelshchikov equation, which could be used to describe the liquid containing gas bubbles. With appropri...We mainly investigate the rational solutions and N-wave resonance solutions for the(3+1)-dimensional Kudryashov–Sinelshchikov equation, which could be used to describe the liquid containing gas bubbles. With appropriate transformations, two kinds of bilinear forms are derived. Employing the two bilinear equations, dynamical behaviors of nine district solutions for this equation are discussed in detail, including bright rogue wave-type solution, dark rogue wave-type solution, bright W-shaped solution, dark W-shaped rational solution, generalized rational solution and bright-fusion, darkfusion, bright-fission, and dark-fission resonance solutions. In addition, the generalized rational solutions, which depending on two arbitrary parameters, have an interesting structure: splitting from two peaks into three peaks.展开更多
We introduce a new integral scheme namely improved Kudryashov method for solving any nonlinear fractional differential model.Specifically,we apply the approach to the nonlinear space-time fractional model leading the ...We introduce a new integral scheme namely improved Kudryashov method for solving any nonlinear fractional differential model.Specifically,we apply the approach to the nonlinear space-time fractional model leading the wave to spread in electrical transmission lines(s-tfETL),the time fractional complex Schrödinger(tfcS),and the space-time M-fractional Schrödinger-Hirota(s-tM-fSH)models to verify the effectiveness of the proposed approach.The implementing of the introduced new technique based on the models provides us with periodic envelope,exponentially changeable soliton envelope,rational rogue wave,periodic rogue wave,combo periodic-soliton,and combo rational-soliton solutions,which are much interesting phenomena in nonlinear sciences.Thus the results disclose that the proposed technique is very effective and straight-forward,and such solutions of the models are much more fruitful than those from the generalized Kudryashov and the modified Kudryashov methods.展开更多
In this paper, the modified Kudryashov method is employed to find the traveling wave solutions of two well-known space-time fractional partial differential equations, namely the Zakharov Kuznetshov Benjamin Bona Mahon...In this paper, the modified Kudryashov method is employed to find the traveling wave solutions of two well-known space-time fractional partial differential equations, namely the Zakharov Kuznetshov Benjamin Bona Mahony equation and Kolmogorov Petrovskii Piskunov equation, and as a helping tool, the sense of modified Riemann-Liouville derivative is also used. The propagation properties of obtained solutions are investigated where the graphical representations and justifications of the results are done by mathematical software Maple.展开更多
In this article, exact solutions of Wick-type stochastic Kudryashov–Sinelshchikov equation have been obtained by using improved Sub-equation method. We have used Hermite transform for transforming the Wick-type stoch...In this article, exact solutions of Wick-type stochastic Kudryashov–Sinelshchikov equation have been obtained by using improved Sub-equation method. We have used Hermite transform for transforming the Wick-type stochastic Kudryashov–Sinelshchikov equation to deterministic partial differential equation. Also we have applied inverse Hermite transform for obtaining a set of stochastic solutions in the white noise space.展开更多
The paper deals with the obliquely propagating wave solutions of fractional nonlinear evolution equations(NLEEs)arising in science and engineering.The conformable time fractional(2+1)-dimensional extended Zakharov-Kuz...The paper deals with the obliquely propagating wave solutions of fractional nonlinear evolution equations(NLEEs)arising in science and engineering.The conformable time fractional(2+1)-dimensional extended Zakharov-Kuzetsov equation(EZKE),coupled space-time fractional(2+1)-dimensional dispersive long wave equation(DLWE)and space-time fractional(2+1)-dimensional Ablowitz-Kaup-Newell-Segur(AKNS)equation are considered to investigate such physical phenomena.The modified Kudryashov method along with the properties of conformable and modified Riemann-Liouville derivatives is employed to construct the oblique wave solutions of the considered equations.The obtained results may be useful for better understanding the nature of internal oblique propagating wave dynamics in ocean engineering.展开更多
In this article,we establish solitary wave solutions to the Estevez-MansfieldClarkson(EMC)equation and the coupled sine-Gordon equations which are model equations to analyze the formation of shapes in liquid drops,sur...In this article,we establish solitary wave solutions to the Estevez-MansfieldClarkson(EMC)equation and the coupled sine-Gordon equations which are model equations to analyze the formation of shapes in liquid drops,surfaces of negative constant curvature,etc.through contriving the generalized Kudryashov method.The extracted results introduce several types’solitary waves,such as the kink soliton,bell-shape soliton,compacton,singular soliton,peakon and other sort of soliton for distinct valuation of the unknown parameters.The achieved analytic solutions are interpreted in details and their 2D and 3D graphs are sketched.The obtained solutions and the physical structures explain the soliton phenomenon and reproduce the dynamic properties of the front of the travelling wave deformation generated in the dispersive media.It shows that the generalized Kudryashov method is powerful,compatible and might be used in further works to found novel solutions for other types of nonlinear evolution equations ascending in physical science and engineering.展开更多
New diverse enormous soliton solutions to the Gross-Pitaevskii equation,which describes the dynamics of two dark solitons in a polarization condensate under non-resonant pumping,have been constructed for the first tim...New diverse enormous soliton solutions to the Gross-Pitaevskii equation,which describes the dynamics of two dark solitons in a polarization condensate under non-resonant pumping,have been constructed for the first time by using two different schemes.The two schemes utilized are the generalized Kudryashov scheme and the(G'/G)-expansion scheme.Throughout these two suggested schemes we construct new diverse forms solutions that include dark,bright-shaped soliton solutions,combined bright-shaped,dark-shaped soliton solutions,hyperbolic function soliton solutions,singular-shaped soliton solutions and other rational soliton solutions.The two 2D and 3D figure designs have been configured using the Mathematica program.In addition,the Haar wavelet numerical scheme has been applied to construct the identical numerical behavior for all soliton solutions achieved by the two suggested schemes to show the existing similarity between the soliton solutions and numerical solutions.展开更多
In this study,we investigate a variety of exact soliton solutions of general(2+1)-dimensional Bogoyavlensky–Konopelchenko equation via the exp(-Φ(η))-expansion method and modified Kudryashov method.The exact soluti...In this study,we investigate a variety of exact soliton solutions of general(2+1)-dimensional Bogoyavlensky–Konopelchenko equation via the exp(-Φ(η))-expansion method and modified Kudryashov method.The exact solutions are characterized in the form of hyperbolic,trigonometric and rational function solutions using exp(-Φ(η))-expansion method,whereas the solution in the form of hyperbolic function expression is obtained by the modified Kudryashov method.These exact solutions also include kink,bright,dark,singular and periodic soliton solutions.The graphical interpretation of the exact solutions is addressed for specific choices of the parameters appearing in the solutions.展开更多
Exact solutions to conformable time fractional (3+1)-dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integ...Exact solutions to conformable time fractional (3+1)-dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE.The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo–Miwa, the Zakharov–Kuznetsov, and the modified Zakharov–Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.展开更多
The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odin...The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.展开更多
In this article, a special type of fractional differential equations(FDEs) named the density-dependent conformable fractional diffusion-reaction(DDCFDR) equation is studied. Aforementioned equation has a significant r...In this article, a special type of fractional differential equations(FDEs) named the density-dependent conformable fractional diffusion-reaction(DDCFDR) equation is studied. Aforementioned equation has a significant role in the modelling of some phenomena arising in the applied science. The well-organized methods, including the exp(-φ(ε))-expansion and modified Kudryashov methods are exerted to generate the exact solutions of this equation such that some of the solutions are new and have been reported for the first time. Results illustrate that both methods have a great performance in handling the DDCFDR equation.展开更多
文摘In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov （KdV-KZK） equa- tion are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the non- linear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.
文摘In the present paper, we established a traveling wave solution by using modified Kudryashov method for the space-time fractional nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of the space-time fractional nonlinear partial differential equations such as, the space-time fractional coupled equal width wave equation(CEWE) and the space-time fractional coupled modified equal width wave equation(CMEW), which are the important soliton equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform and properties of modified Riemann–Liouville derivative. We plot the exact solutions for these equations at different time levels.
文摘Our purpose of this paper is to apply the improved Kudryashov method for solving various types of nonlinear fractional partial differential equations. As an application, the time-space fractional Korteweg-de Vries-Burger (KdV-Burger) equation is solved using this method and we get some new travelling wave solutions. To acquire our purpose a complex transformation has been also used to reduce nonlinear fractional partial differential equations to nonlinear ordinary differential equations of integer order, in the sense of the Jumarie’s modified Riemann-Liouville derivative. Afterwards, the improved Kudryashov method is implemented and we get our required reliable solutions where the results are justified by mathematical software Maple-13.
基金Project supported by the National Natural Science Foundation of China(Grant No.11675054)the Future Scientist/Outstanding Scholar Training Program of East China Normal University(Grant No.WLKXJ2019-004)+1 种基金the Fund from Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things(Grant No.ZF1213)the Project from the Science and Technology Commission of Shanghai Municipality,China(Grant No.18dz2271000)。
文摘We mainly investigate the rational solutions and N-wave resonance solutions for the(3+1)-dimensional Kudryashov–Sinelshchikov equation, which could be used to describe the liquid containing gas bubbles. With appropriate transformations, two kinds of bilinear forms are derived. Employing the two bilinear equations, dynamical behaviors of nine district solutions for this equation are discussed in detail, including bright rogue wave-type solution, dark rogue wave-type solution, bright W-shaped solution, dark W-shaped rational solution, generalized rational solution and bright-fusion, darkfusion, bright-fission, and dark-fission resonance solutions. In addition, the generalized rational solutions, which depending on two arbitrary parameters, have an interesting structure: splitting from two peaks into three peaks.
文摘We introduce a new integral scheme namely improved Kudryashov method for solving any nonlinear fractional differential model.Specifically,we apply the approach to the nonlinear space-time fractional model leading the wave to spread in electrical transmission lines(s-tfETL),the time fractional complex Schrödinger(tfcS),and the space-time M-fractional Schrödinger-Hirota(s-tM-fSH)models to verify the effectiveness of the proposed approach.The implementing of the introduced new technique based on the models provides us with periodic envelope,exponentially changeable soliton envelope,rational rogue wave,periodic rogue wave,combo periodic-soliton,and combo rational-soliton solutions,which are much interesting phenomena in nonlinear sciences.Thus the results disclose that the proposed technique is very effective and straight-forward,and such solutions of the models are much more fruitful than those from the generalized Kudryashov and the modified Kudryashov methods.
文摘In this paper, the modified Kudryashov method is employed to find the traveling wave solutions of two well-known space-time fractional partial differential equations, namely the Zakharov Kuznetshov Benjamin Bona Mahony equation and Kolmogorov Petrovskii Piskunov equation, and as a helping tool, the sense of modified Riemann-Liouville derivative is also used. The propagation properties of obtained solutions are investigated where the graphical representations and justifications of the results are done by mathematical software Maple.
文摘In this article, exact solutions of Wick-type stochastic Kudryashov–Sinelshchikov equation have been obtained by using improved Sub-equation method. We have used Hermite transform for transforming the Wick-type stochastic Kudryashov–Sinelshchikov equation to deterministic partial differential equation. Also we have applied inverse Hermite transform for obtaining a set of stochastic solutions in the white noise space.
文摘The paper deals with the obliquely propagating wave solutions of fractional nonlinear evolution equations(NLEEs)arising in science and engineering.The conformable time fractional(2+1)-dimensional extended Zakharov-Kuzetsov equation(EZKE),coupled space-time fractional(2+1)-dimensional dispersive long wave equation(DLWE)and space-time fractional(2+1)-dimensional Ablowitz-Kaup-Newell-Segur(AKNS)equation are considered to investigate such physical phenomena.The modified Kudryashov method along with the properties of conformable and modified Riemann-Liouville derivatives is employed to construct the oblique wave solutions of the considered equations.The obtained results may be useful for better understanding the nature of internal oblique propagating wave dynamics in ocean engineering.
基金the Research Grant No.:A-1220/5/52/RU/Science-37/2019-2020 and the authors acknowledge this support.
文摘In this article,we establish solitary wave solutions to the Estevez-MansfieldClarkson(EMC)equation and the coupled sine-Gordon equations which are model equations to analyze the formation of shapes in liquid drops,surfaces of negative constant curvature,etc.through contriving the generalized Kudryashov method.The extracted results introduce several types’solitary waves,such as the kink soliton,bell-shape soliton,compacton,singular soliton,peakon and other sort of soliton for distinct valuation of the unknown parameters.The achieved analytic solutions are interpreted in details and their 2D and 3D graphs are sketched.The obtained solutions and the physical structures explain the soliton phenomenon and reproduce the dynamic properties of the front of the travelling wave deformation generated in the dispersive media.It shows that the generalized Kudryashov method is powerful,compatible and might be used in further works to found novel solutions for other types of nonlinear evolution equations ascending in physical science and engineering.
文摘New diverse enormous soliton solutions to the Gross-Pitaevskii equation,which describes the dynamics of two dark solitons in a polarization condensate under non-resonant pumping,have been constructed for the first time by using two different schemes.The two schemes utilized are the generalized Kudryashov scheme and the(G'/G)-expansion scheme.Throughout these two suggested schemes we construct new diverse forms solutions that include dark,bright-shaped soliton solutions,combined bright-shaped,dark-shaped soliton solutions,hyperbolic function soliton solutions,singular-shaped soliton solutions and other rational soliton solutions.The two 2D and 3D figure designs have been configured using the Mathematica program.In addition,the Haar wavelet numerical scheme has been applied to construct the identical numerical behavior for all soliton solutions achieved by the two suggested schemes to show the existing similarity between the soliton solutions and numerical solutions.
文摘In this study,we investigate a variety of exact soliton solutions of general(2+1)-dimensional Bogoyavlensky–Konopelchenko equation via the exp(-Φ(η))-expansion method and modified Kudryashov method.The exact solutions are characterized in the form of hyperbolic,trigonometric and rational function solutions using exp(-Φ(η))-expansion method,whereas the solution in the form of hyperbolic function expression is obtained by the modified Kudryashov method.These exact solutions also include kink,bright,dark,singular and periodic soliton solutions.The graphical interpretation of the exact solutions is addressed for specific choices of the parameters appearing in the solutions.
文摘Exact solutions to conformable time fractional (3+1)-dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE.The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo–Miwa, the Zakharov–Kuznetsov, and the modified Zakharov–Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.
文摘The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.
文摘In this article, a special type of fractional differential equations(FDEs) named the density-dependent conformable fractional diffusion-reaction(DDCFDR) equation is studied. Aforementioned equation has a significant role in the modelling of some phenomena arising in the applied science. The well-organized methods, including the exp(-φ(ε))-expansion and modified Kudryashov methods are exerted to generate the exact solutions of this equation such that some of the solutions are new and have been reported for the first time. Results illustrate that both methods have a great performance in handling the DDCFDR equation.
