In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation,...In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KdV equations and the time-fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.展开更多
In the present article, He＇s fractional derivative, the ansatz method, the （ C / G）-expansion method, and the exp-function method are used to construct the exact solutions of nonlinear space-time fractional Kadomts...In the present article, He＇s fractional derivative, the ansatz method, the （ C / G）-expansion method, and the exp-function method are used to construct the exact solutions of nonlinear space-time fractional Kadomtsev-Petviashvili- Benjamin-Bona Mahony （KP-BBM）. As a result, different types of exact solutions are obtained. Also we have examined the relation between the solutions obtained from the different methods. These methods are an efficient mathematical tool for solving fractional differential equations （FDEs） and it can be applied to other nonlinear FDEs.展开更多
In this article, we construct abundant exact traveling wave solutions involving free parameters to the generalized Bretherton equation via the improved (G′/G)-expansion method. The traveling wave solutions are presen...In this article, we construct abundant exact traveling wave solutions involving free parameters to the generalized Bretherton equation via the improved (G′/G)-expansion method. The traveling wave solutions are presented in terms of the trigonometric, the hyperbolic, and rational functions. When the parameters take special values, the solitary waves are derived from the traveling waves.展开更多
In this study,we implement the generalized(G/G)-expansion method established by Wang et al.to examine wave solutions to some nonlinear evolution equations.The method,known as the double(G/G,1/G)-expansion method is ...In this study,we implement the generalized(G/G)-expansion method established by Wang et al.to examine wave solutions to some nonlinear evolution equations.The method,known as the double(G/G,1/G)-expansion method is used to establish abundant new and further general exact wave solutions to the(3+1)-dimensional Jimbo-Miwa equation,the(3+1)-dimensional Kadomtsev-Petviashvili equation and symmetric regularized long wave equation.The solutions are extracted in terms of hyperbolic function,trigonometric function and rational function.The solitary wave solutions are constructed from the obtained traveling wave solutions if the parameters received some definite values.Graphs of the solutions are also depicted to describe the phenomena apparently and the shapes of the obtained solutions are singular periodic,anti-kink,singular soliton,singular anti-bell shape,compaction etc.This method is straightforward,compact and reliable and gives huge new closed form traveling wave solutions of nonlinear evolution equations in ocean engineering.展开更多
In this paper,some travelling wave solutions involving parameters of the Modified Zakharov-Kuznetsovequation [Phys.Lett.A 372 (2008) 3400] are investigated.We will show that these solutions are not new travellingwave ...In this paper,some travelling wave solutions involving parameters of the Modified Zakharov-Kuznetsovequation [Phys.Lett.A 372 (2008) 3400] are investigated.We will show that these solutions are not new travellingwave solutions.展开更多
In this article, we construct the exact traveling wave solutions for nonlinear evolution equations in the mathematical physics via the modified Kawahara equation, the nonlinear coupled KdV equations and the classical ...In this article, we construct the exact traveling wave solutions for nonlinear evolution equations in the mathematical physics via the modified Kawahara equation, the nonlinear coupled KdV equations and the classical Boussinesq equations, by using a generalized （G＇/G）-expansion method, where G satisfies the Jacobi elliptic equation. Many exact solutions in terms of Jacobi elliptic functions are obtained.展开更多
The(2+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis is described by the space-time fractional Calogero-Degasperis(CD)and fractional poten-tial Kadomstev-Pe...The(2+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis is described by the space-time fractional Calogero-Degasperis(CD)and fractional poten-tial Kadomstev-Petviashvili(PKP)equation.It can be modeled according to the Hamiltonian structure,the lax pair with the non-isospectral problem,and the pain level property.The proposed equations are widely used in beachfront ocean and coastal engineering to describe the propagation of shallow-water waves,demonstrate the propagation of waves in dissipative and nonlinear media,and reveal the propagation of waves in dissipative and nonlinear media.In this paper,we have established further exact solutions to the nonlinear fractional partial differential equation(NLFPDEs),namely the space-time fractional CD and fractional PKP equations using the modified Rieman-Liouville fractional derivative of Jumarie through the two variable(G/G,1/G)-expansion method.As far as trigonometric,hyperbolic,and rational function so-lutions containing parameters are concerned,solutions are acquired when unique characteristics are as-signed to the parameters.Subsequently,the solitary wave solutions are generated from the solutions of the traveling wave.It is important to observe that this method is a realistic,convenient,well-organized,and ground-breaking strategy for solving various types of NLFPDEs.展开更多
Most fundamental themes in mathematical physics and modern engineering are investigated by the closed form traveling wave solutions of nonlinear evolution equations.In our research,we ascertain abundant new closed for...Most fundamental themes in mathematical physics and modern engineering are investigated by the closed form traveling wave solutions of nonlinear evolution equations.In our research,we ascertain abundant new closed form traveling wave solution of the nonlinear integro-differential equations via Ito equation,integro-differential Sawada-Kotera equation,first integro-differential KP hierarchy equation and second integro-differential KP hierarchy equation by two variable(G/G,1/G)-expansion method with the help of computer package like Mathematica.Some shape of solutions like,bell profile solution,anti-king profile solution,soliton profile solution,periodic profile solution etc.are obtain in this investigation.Trigonometric function solution,hyperbolic function solution and rational function solution are established by using our eminent method and comparing with our results to all of the well-known results which are given in the literature.By means of free parameters,plentiful solitary solutions are derived from the exact traveling wave solutions.The method can be easier and more applicable to investigate such type of nonlinear evolution models.展开更多
Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous.Although a considerable amount of research has been carried ...Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous.Although a considerable amount of research has been carried out in the field of nonlinear differential-difference equations,the majority of the results deal with polynomial types.Limited research has been reported regarding such equations of rational type.In this paper we present an adaptation of the(G /G)-expansion method to solve nonlinear rational differential-difference equations.The procedure is demonstrated using two distinct equations.Our approach allows one to construct three types of exact traveling wave solutions(hyperbolic,trigonometric,and rational) by means of the simplified form of the auxiliary equation method with reduced parameters.Our analysis leads to analytic solutions in terms of topological solitons and singular periodic functions as well.展开更多
In a previous work,Zayed and Al-Nowehy have applied the Riccati equation mapping method combined with the generalized extended(G/G)-expansion method and found new exact solutions of the nonlinear KPP equation.In the ...In a previous work,Zayed and Al-Nowehy have applied the Riccati equation mapping method combined with the generalized extended(G/G)-expansion method and found new exact solutions of the nonlinear KPP equation.In the present article,we propose a different method,namely,a new sub-equation method consists of the Riccati equation mapping method and the(G/G,1/G)-expansion method to find new exact solutions of the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in optical fiber materials.This proposed method is not found elsewhere.Hyperbolic,trigonometric and rational function solutions are given.New solutions of the generalized Riccati equation are presented for the first time which are not reported previously.The solutions of the given nonlinear equation can be applied in ocean engineering for calculating the height of tides in the ocean.展开更多
The generalized sub-ODE method, the rational （G＇/G）-expansion method, the exp-function method and the sine-cosine method are applied for constructing many traveling wave solutions of nonlinear partial differential ...The generalized sub-ODE method, the rational （G＇/G）-expansion method, the exp-function method and the sine-cosine method are applied for constructing many traveling wave solutions of nonlinear partial differential equations （PDEs）. Some illustrative equations are investigated by these methods and many hyperbolic, trigonometric and rational function solutions are found. We apply these methods to obtain the exact solutions for the generalized KdV-mKdV （GKdV-mKdV） equation with higherorder nonlinear terms. The obtained results confirm that the proposed methods are efficient techniques for analytic treatment of a wide variety of nonlinear partial differential equations in mathematical physics. We compare between the results yielding from these methods. Also, a comparison between our new results in this paper and the well-known results are given.展开更多
In this work, the (G,/G)- --expansion method is proposed for constructing more general exact solutions of the (2 + 1)--dimensional Kadomtsev-Petviashvili (KP) equation and its generalized forms. Our work is motivated ...In this work, the (G,/G)- --expansion method is proposed for constructing more general exact solutions of the (2 + 1)--dimensional Kadomtsev-Petviashvili (KP) equation and its generalized forms. Our work is motivated by the fact that the (G,/G)---expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.展开更多
In this paper,we set up dynamic solitary perturb solutions of a unidirectional stochastic longitudinal wave equation in a magneto-electro-elastic annular bar by a feasible,useful,and influential method named the dual(...In this paper,we set up dynamic solitary perturb solutions of a unidirectional stochastic longitudinal wave equation in a magneto-electro-elastic annular bar by a feasible,useful,and influential method named the dual(G’/G,1/G)-expansion method.Computer software,like Mathematica,is used to complete this discussion.The obtained solutions of the proposed equation are classified into trigonometric,hyperbolic,and rational types which play an important role in searching for numerous scientific events.The technique employed here is an extension of the(G’/G)-expansion technique for finding all previously discovered solutions.To illustrate our findings more clearly,we provide 2D and 3D charts of the various recovery methods.We then contrasted our findings with those of past solutions.The graphical illustrations of the acquired solutions are singular periodic solitons and kink solitons which are added at the end of this paper.展开更多
In a recent article [Physics Letters A 372 （2008） 417], Wang et al. proposed a method, which is called the （G′/G）-expansion method, to look for travelling wave solutions of nonlinear evolution equations. The trav...In a recent article [Physics Letters A 372 （2008） 417], Wang et al. proposed a method, which is called the （G′/G）-expansion method, to look for travelling wave solutions of nonlinear evolution equations. The travelling wave solutions involving parameters of the KdV equation, the mKdV equation, the variant Boussinesq equations, and the Hirota-Satsuma equations are obtained by using this method. They think the （G′/G）-expansion method is a new method and more travelling wave solutions of many nonlinear evolution equations can be obtained. In this paper, we will show that the （G′/G）-expansion method is equivalent to the extended tanh function method.展开更多
In this article, we apply five different techniques, namely the （G//G）-expansion method, an auxiliary equation method, the modified simple equation method, the first integral method and the Riccati equation method f...In this article, we apply five different techniques, namely the （G//G）-expansion method, an auxiliary equation method, the modified simple equation method, the first integral method and the Riccati equation method for constructing many new exact solutions with parameters as well as the bright-dark, singular and other soliton solutions of the （2＋1）-dimensional nonlinear cubic-quintic Ginzburg-Landau equation. Comparing the solutions of this nonlinear equation together with each other are presented. Comparing our new results obtained in this article with the well-known results are given too.展开更多
In this paper, by introducing a proper transformation, the （Gr/G）-expansion method is further extended into the nonlinear reaction diffusion equations in mathematical biology whose balancing numbers may be negative ...In this paper, by introducing a proper transformation, the （Gr/G）-expansion method is further extended into the nonlinear reaction diffusion equations in mathematical biology whose balancing numbers may be negative integer. As a result, hyperbolic function solutions and trigonometric function solutions with free parameters are obtained. When the parameters are taken as special values the solitary wave solutions and the periodic wave solutions are also derived from the traveling wave solutions. Moreover, it is observed that the suggested techniques is compatible of such problems.展开更多
The propagation of the optical solitons is usually governed by the nonlinear Schrodinger equations. In this article, the two variable （G＇/G,1/G）-expansion method is employed to construct exact traveling wave soluti...The propagation of the optical solitons is usually governed by the nonlinear Schrodinger equations. In this article, the two variable （G＇/G,1/G）-expansion method is employed to construct exact traveling wave solutions with parameters of two higher order nonlinear Schrodinger equations describing the propagation of femtosecond pulses in nonlinear optical fibers. When the parameters are replaced by special values, the well-known solitary wave solutions of these equations rediscovered from the traveling waves. This method can be thought of as the generalization of well-known original （G＇/G）-expansion method proposed by M. Wang et al. It is shown that the two variable （G＇/G,1/G）-expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics.展开更多
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
In this paper,we construct exact solutions for the (2+1)-dimensional Boiti-Leon-Pempinelle equation byusing the (G'/G)-expansion method,and with the help of Maple.As a result,non-travelling wave solutions with thr...In this paper,we construct exact solutions for the (2+1)-dimensional Boiti-Leon-Pempinelle equation byusing the (G'/G)-expansion method,and with the help of Maple.As a result,non-travelling wave solutions with threearbitrary functions are obtained including hyperbolic function solutions,trigonometric function solutions,and rationalsolutions.This method can be applied to other higher-dimensional nonlinear partial differential equations.展开更多
文摘In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KdV equations and the time-fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.
文摘In the present article, He＇s fractional derivative, the ansatz method, the （ C / G）-expansion method, and the exp-function method are used to construct the exact solutions of nonlinear space-time fractional Kadomtsev-Petviashvili- Benjamin-Bona Mahony （KP-BBM）. As a result, different types of exact solutions are obtained. Also we have examined the relation between the solutions obtained from the different methods. These methods are an efficient mathematical tool for solving fractional differential equations （FDEs） and it can be applied to other nonlinear FDEs.
基金supported by the research grant under the Government of Malaysia
文摘In this article, we construct abundant exact traveling wave solutions involving free parameters to the generalized Bretherton equation via the improved (G′/G)-expansion method. The traveling wave solutions are presented in terms of the trigonometric, the hyperbolic, and rational functions. When the parameters take special values, the solitary waves are derived from the traveling waves.
文摘In this study,we implement the generalized(G/G)-expansion method established by Wang et al.to examine wave solutions to some nonlinear evolution equations.The method,known as the double(G/G,1/G)-expansion method is used to establish abundant new and further general exact wave solutions to the(3+1)-dimensional Jimbo-Miwa equation,the(3+1)-dimensional Kadomtsev-Petviashvili equation and symmetric regularized long wave equation.The solutions are extracted in terms of hyperbolic function,trigonometric function and rational function.The solitary wave solutions are constructed from the obtained traveling wave solutions if the parameters received some definite values.Graphs of the solutions are also depicted to describe the phenomena apparently and the shapes of the obtained solutions are singular periodic,anti-kink,singular soliton,singular anti-bell shape,compaction etc.This method is straightforward,compact and reliable and gives huge new closed form traveling wave solutions of nonlinear evolution equations in ocean engineering.
基金Supported by National Natural Science Foundation of China under Grant No.10671171
文摘In this paper,some travelling wave solutions involving parameters of the Modified Zakharov-Kuznetsovequation [Phys.Lett.A 372 (2008) 3400] are investigated.We will show that these solutions are not new travellingwave solutions.
文摘In this article, we construct the exact traveling wave solutions for nonlinear evolution equations in the mathematical physics via the modified Kawahara equation, the nonlinear coupled KdV equations and the classical Boussinesq equations, by using a generalized （G＇/G）-expansion method, where G satisfies the Jacobi elliptic equation. Many exact solutions in terms of Jacobi elliptic functions are obtained.
文摘The(2+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis is described by the space-time fractional Calogero-Degasperis(CD)and fractional poten-tial Kadomstev-Petviashvili(PKP)equation.It can be modeled according to the Hamiltonian structure,the lax pair with the non-isospectral problem,and the pain level property.The proposed equations are widely used in beachfront ocean and coastal engineering to describe the propagation of shallow-water waves,demonstrate the propagation of waves in dissipative and nonlinear media,and reveal the propagation of waves in dissipative and nonlinear media.In this paper,we have established further exact solutions to the nonlinear fractional partial differential equation(NLFPDEs),namely the space-time fractional CD and fractional PKP equations using the modified Rieman-Liouville fractional derivative of Jumarie through the two variable(G/G,1/G)-expansion method.As far as trigonometric,hyperbolic,and rational function so-lutions containing parameters are concerned,solutions are acquired when unique characteristics are as-signed to the parameters.Subsequently,the solitary wave solutions are generated from the solutions of the traveling wave.It is important to observe that this method is a realistic,convenient,well-organized,and ground-breaking strategy for solving various types of NLFPDEs.
文摘Most fundamental themes in mathematical physics and modern engineering are investigated by the closed form traveling wave solutions of nonlinear evolution equations.In our research,we ascertain abundant new closed form traveling wave solution of the nonlinear integro-differential equations via Ito equation,integro-differential Sawada-Kotera equation,first integro-differential KP hierarchy equation and second integro-differential KP hierarchy equation by two variable(G/G,1/G)-expansion method with the help of computer package like Mathematica.Some shape of solutions like,bell profile solution,anti-king profile solution,soliton profile solution,periodic profile solution etc.are obtain in this investigation.Trigonometric function solution,hyperbolic function solution and rational function solution are established by using our eminent method and comparing with our results to all of the well-known results which are given in the literature.By means of free parameters,plentiful solitary solutions are derived from the exact traveling wave solutions.The method can be easier and more applicable to investigate such type of nonlinear evolution models.
文摘Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous.Although a considerable amount of research has been carried out in the field of nonlinear differential-difference equations,the majority of the results deal with polynomial types.Limited research has been reported regarding such equations of rational type.In this paper we present an adaptation of the(G /G)-expansion method to solve nonlinear rational differential-difference equations.The procedure is demonstrated using two distinct equations.Our approach allows one to construct three types of exact traveling wave solutions(hyperbolic,trigonometric,and rational) by means of the simplified form of the auxiliary equation method with reduced parameters.Our analysis leads to analytic solutions in terms of topological solitons and singular periodic functions as well.
文摘In a previous work,Zayed and Al-Nowehy have applied the Riccati equation mapping method combined with the generalized extended(G/G)-expansion method and found new exact solutions of the nonlinear KPP equation.In the present article,we propose a different method,namely,a new sub-equation method consists of the Riccati equation mapping method and the(G/G,1/G)-expansion method to find new exact solutions of the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in optical fiber materials.This proposed method is not found elsewhere.Hyperbolic,trigonometric and rational function solutions are given.New solutions of the generalized Riccati equation are presented for the first time which are not reported previously.The solutions of the given nonlinear equation can be applied in ocean engineering for calculating the height of tides in the ocean.
文摘The generalized sub-ODE method, the rational （G＇/G）-expansion method, the exp-function method and the sine-cosine method are applied for constructing many traveling wave solutions of nonlinear partial differential equations （PDEs）. Some illustrative equations are investigated by these methods and many hyperbolic, trigonometric and rational function solutions are found. We apply these methods to obtain the exact solutions for the generalized KdV-mKdV （GKdV-mKdV） equation with higherorder nonlinear terms. The obtained results confirm that the proposed methods are efficient techniques for analytic treatment of a wide variety of nonlinear partial differential equations in mathematical physics. We compare between the results yielding from these methods. Also, a comparison between our new results in this paper and the well-known results are given.
文摘In this work, the (G,/G)- --expansion method is proposed for constructing more general exact solutions of the (2 + 1)--dimensional Kadomtsev-Petviashvili (KP) equation and its generalized forms. Our work is motivated by the fact that the (G,/G)---expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.
文摘In this paper,we set up dynamic solitary perturb solutions of a unidirectional stochastic longitudinal wave equation in a magneto-electro-elastic annular bar by a feasible,useful,and influential method named the dual(G’/G,1/G)-expansion method.Computer software,like Mathematica,is used to complete this discussion.The obtained solutions of the proposed equation are classified into trigonometric,hyperbolic,and rational types which play an important role in searching for numerous scientific events.The technique employed here is an extension of the(G’/G)-expansion technique for finding all previously discovered solutions.To illustrate our findings more clearly,we provide 2D and 3D charts of the various recovery methods.We then contrasted our findings with those of past solutions.The graphical illustrations of the acquired solutions are singular periodic solitons and kink solitons which are added at the end of this paper.
基金Supported by National Natural Science Foundation of China under Grant No. 10671172
文摘In a recent article [Physics Letters A 372 （2008） 417], Wang et al. proposed a method, which is called the （G′/G）-expansion method, to look for travelling wave solutions of nonlinear evolution equations. The travelling wave solutions involving parameters of the KdV equation, the mKdV equation, the variant Boussinesq equations, and the Hirota-Satsuma equations are obtained by using this method. They think the （G′/G）-expansion method is a new method and more travelling wave solutions of many nonlinear evolution equations can be obtained. In this paper, we will show that the （G′/G）-expansion method is equivalent to the extended tanh function method.
文摘In this article, we apply five different techniques, namely the （G//G）-expansion method, an auxiliary equation method, the modified simple equation method, the first integral method and the Riccati equation method for constructing many new exact solutions with parameters as well as the bright-dark, singular and other soliton solutions of the （2＋1）-dimensional nonlinear cubic-quintic Ginzburg-Landau equation. Comparing the solutions of this nonlinear equation together with each other are presented. Comparing our new results obtained in this article with the well-known results are given too.
文摘In this paper, by introducing a proper transformation, the （Gr/G）-expansion method is further extended into the nonlinear reaction diffusion equations in mathematical biology whose balancing numbers may be negative integer. As a result, hyperbolic function solutions and trigonometric function solutions with free parameters are obtained. When the parameters are taken as special values the solitary wave solutions and the periodic wave solutions are also derived from the traveling wave solutions. Moreover, it is observed that the suggested techniques is compatible of such problems.
文摘The propagation of the optical solitons is usually governed by the nonlinear Schrodinger equations. In this article, the two variable （G＇/G,1/G）-expansion method is employed to construct exact traveling wave solutions with parameters of two higher order nonlinear Schrodinger equations describing the propagation of femtosecond pulses in nonlinear optical fibers. When the parameters are replaced by special values, the well-known solitary wave solutions of these equations rediscovered from the traveling waves. This method can be thought of as the generalization of well-known original （G＇/G）-expansion method proposed by M. Wang et al. It is shown that the two variable （G＇/G,1/G）-expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics.
文摘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.
基金Supported by the Natural Science Foundation of Shanghai under Grant No.09ZR1410800the Science Foundation of Key Laboratory of Mathematics Mechanization under Grant No.KLMM0806+1 种基金the Shanghai Leading Academic Discipline Project under Grant No.J50101Key Disciplines of Shanghai Municipality under Grant No.S30104
文摘In this paper,we construct exact solutions for the (2+1)-dimensional Boiti-Leon-Pempinelle equation byusing the (G'/G)-expansion method,and with the help of Maple.As a result,non-travelling wave solutions with threearbitrary functions are obtained including hyperbolic function solutions,trigonometric function solutions,and rationalsolutions.This method can be applied to other higher-dimensional nonlinear partial differential equations.
