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
Based on an improved fractional sub-equation method involving Jumarie's mo- dified Riemann-Liouville derivative, we construct analytical solutions of space-time fractional compound KdV-Burgers equation and coupled Bu...Based on an improved fractional sub-equation method involving Jumarie's mo- dified Riemann-Liouville derivative, we construct analytical solutions of space-time fractional compound KdV-Burgers equation and coupled Burgers' equations. These results not only reveal that the method is very effective and simple in studying solu- tions to the fractional partial differential equation, but also include some new exact solutions.展开更多
In this article,(2+1)-dimensional time fractional Bogoyavlensky-Konopelchenko(BK)equation is studied,which describes the interaction of wave propagating along the x axis and y axis.To acquire the exact solutions of BK...In this article,(2+1)-dimensional time fractional Bogoyavlensky-Konopelchenko(BK)equation is studied,which describes the interaction of wave propagating along the x axis and y axis.To acquire the exact solutions of BK equation we employed sub equation method that is predicated on Riccati equation,and for numerical solutions the residual power series method is implemented.Some graphical results that compares the numerical and analytical solutions are given for di erent values of.Also comparative table for the obtained solutions is presented.展开更多
In this present paper, the Fan sub-equation method is used to construct exact traveling wave solutions of the (1 + 1) dimensional Kaup-Kupershmidt equation. Many exact traveling wave solutions are successfully obtaine...In this present paper, the Fan sub-equation method is used to construct exact traveling wave solutions of the (1 + 1) dimensional Kaup-Kupershmidt equation. Many exact traveling wave solutions are successfully obtained, which contain solitary wave solutions, trigonometric function solutions, hyperbolic function solutions and Jacobian elliptic function periodic solutions with double periods.展开更多
In this paper,new four fifth-order fractional nonlinear equations are derived and investigated.The frac-tional terms are defined in the conformable sense and these equations are then solved using two effective methods...In this paper,new four fifth-order fractional nonlinear equations are derived and investigated.The frac-tional terms are defined in the conformable sense and these equations are then solved using two effective methods,namely,the sub-equation and the generalized Kudryashov methods.These methods were tested on the proposed models and succeeded in finding new solutions with different values of parameters.A graphical representation of some results is provided and proves the efficiency and applicability of the proposed methods for providing solutions with known physical behavior.These methods are good candi-dates for solving other similar problems in the future.展开更多
The Kortewegde Vries(KdV)equation represents the propagation of long waves in dispersive media,whereas the cubic nonlinear Schrödinger(CNLS)equation depicts the dynamics of narrow-bandwidth wave packets consistin...The Kortewegde Vries(KdV)equation represents the propagation of long waves in dispersive media,whereas the cubic nonlinear Schrödinger(CNLS)equation depicts the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves.A model that couples these two equations seems in-triguing for simulating the interaction of long and short waves,which is important in many domains of applied sciences and engineering,and such a system has been investigated in recent decades.This work uses a modified Sardar sub-equation procedure to secure the soliton-type solutions of the generalized cubic nonlinear Schrödinger-Korteweg-de Vries system of equations.For various selections of arbitrary parameters in these solutions,the dynamic properties of some acquired solutions are represented graph-ically and analyzed.In particular,the dynamics of the bright solitons,dark solitons,mixed bright-dark solitons,W-shaped solitons,M-shaped solitons,periodic waves,and other soliton-type solutions.Our re-sults demonstrated that the proposed technique is highly efficient and effective for the aforementioned problems,as well as other nonlinear problems that may arise in the fields of mathematical physics and engineering.展开更多
The principal objective of this article is to construct new and further exact soliton solutions of the(2+1)-dimensional Heisenberg ferromagnetic spin chain equation which investigates the nonlinear dynamics of magnets...The principal objective of this article is to construct new and further exact soliton solutions of the(2+1)-dimensional Heisenberg ferromagnetic spin chain equation which investigates the nonlinear dynamics of magnets and explains their ordering in ferromagnetic materials.These solutions are exerted via the new extended FAN sub-equation method.We successfully obtain dark,bright,combined bright-dark,combined dark-singular,periodic,periodic singular,and elliptic wave solutions to this equation which are interesting classes of nonlinear excitation presenting spin dynamics in classical and semi-classical continuum Heisenberg systems.3D figures are illustrated under an appropriate selection of parameters.The applied technique is suitable to be used in gaining the exact solutions of most nonlinear partial/fractional differential equations which appear in complex phenomena.展开更多
In this article,two different methods,namely sub-equation method and residual power series method,have been used to obtain new exact and approximate solutions of the generalized Hirota-Satsuma system of equations,whic...In this article,two different methods,namely sub-equation method and residual power series method,have been used to obtain new exact and approximate solutions of the generalized Hirota-Satsuma system of equations,which is a coupled KdV model.The fractional derivative is taken in the conformable sense.Each of the obtained exact solutions were checked by substituting them into the corresponding system with the help of Maple symbolic computation package.The results indicate that both methods are easy to implement,effective and reliable.They are therefore ready to apply for various partial fractional differential equations.展开更多
This paper studied new class of integral equation called the Korteweg-de Vries-Kadomtsev-Petviashvili(KdV-KP)equation.This equation consist of the well-known fifth-order KdV equation in the context of the Kadomtsev-Pe...This paper studied new class of integral equation called the Korteweg-de Vries-Kadomtsev-Petviashvili(KdV-KP)equation.This equation consist of the well-known fifth-order KdV equation in the context of the Kadomtsev-Petviashvili equation.The newly gathered class of sixth-order KdV-KP equation is studied using the sub-equation method to obtain several soliton-type solutions which consist of trigonometric,hyperbolic,and rational solutions.The application of the sub-equation approach in this work draws attention to the outstanding characteristics of the suggested method and its ability to handle completely integrable equations.Furthermore,the obtained solutions have not been reported in the previous literature and might have significant impact on future research.展开更多
This paper studies the new families of exact traveling wave solutions with the modified nonlinear Schrodinger equation,which models the propagation of rogue waves in ocean engineering.The extended Fan sub-equation met...This paper studies the new families of exact traveling wave solutions with the modified nonlinear Schrodinger equation,which models the propagation of rogue waves in ocean engineering.The extended Fan sub-equation method with five parameters is used to find exact traveling wave solutions.It has been observed that the equation exhibits a collection of traveling wave solutions for limiting values of parameters.This method is beneficial for solving nonlinear partial differential equations,because it is not only useful for finding the new exact traveling wave solutions,but also gives us the solutions obtained previously by the usage of other techniques(Riccati equation,or first-kind elliptic equation,or the generalized Riccati equation as mapping equation,or auxiliary ordinary differential equation method)in a combined approach.Moreover,by means of the concept of linear stability,we prove that the governing model is stable.3 D figures are plotted for showing the physical behavior of the obtained solutions for the different values of unknown parameters with constraint conditions.展开更多
In this paper,we applied the sub-equation method to obtain a new exact solution set for the extended version of the time-fractional Kadomtsev-Petviashvili equation,namely Burgers-Kadomtsev-Petviashvili equation(Burger...In this paper,we applied the sub-equation method to obtain a new exact solution set for the extended version of the time-fractional Kadomtsev-Petviashvili equation,namely Burgers-Kadomtsev-Petviashvili equation(Burgers-K-P)that arises in shallow water waves.Furthermore,using the residual power series method(RPSM),approximate solutions of the equation were obtained with the help of the Mathematica symbolic computation package.We also presented a few graphical illustrations for some surfaces.The fractional derivatives were considered in the conformable sense.All of the obtained solutions were replaced back in the governing equation to check and ensure the reliability of the method.The numerical outcomes confirmed that both methods are simple,robust and effective to achieve exact and approximate solutions of nonlinear fractional differential equations.展开更多
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.
基金partially supported by the Natural Science Foundation of China(No.11271008)
文摘Based on an improved fractional sub-equation method involving Jumarie's mo- dified Riemann-Liouville derivative, we construct analytical solutions of space-time fractional compound KdV-Burgers equation and coupled Burgers' equations. These results not only reveal that the method is very effective and simple in studying solu- tions to the fractional partial differential equation, but also include some new exact solutions.
文摘In this article,(2+1)-dimensional time fractional Bogoyavlensky-Konopelchenko(BK)equation is studied,which describes the interaction of wave propagating along the x axis and y axis.To acquire the exact solutions of BK equation we employed sub equation method that is predicated on Riccati equation,and for numerical solutions the residual power series method is implemented.Some graphical results that compares the numerical and analytical solutions are given for di erent values of.Also comparative table for the obtained solutions is presented.
文摘In this present paper, the Fan sub-equation method is used to construct exact traveling wave solutions of the (1 + 1) dimensional Kaup-Kupershmidt equation. Many exact traveling wave solutions are successfully obtained, which contain solitary wave solutions, trigonometric function solutions, hyperbolic function solutions and Jacobian elliptic function periodic solutions with double periods.
文摘In this paper,new four fifth-order fractional nonlinear equations are derived and investigated.The frac-tional terms are defined in the conformable sense and these equations are then solved using two effective methods,namely,the sub-equation and the generalized Kudryashov methods.These methods were tested on the proposed models and succeeded in finding new solutions with different values of parameters.A graphical representation of some results is provided and proves the efficiency and applicability of the proposed methods for providing solutions with known physical behavior.These methods are good candi-dates for solving other similar problems in the future.
文摘The Kortewegde Vries(KdV)equation represents the propagation of long waves in dispersive media,whereas the cubic nonlinear Schrödinger(CNLS)equation depicts the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves.A model that couples these two equations seems in-triguing for simulating the interaction of long and short waves,which is important in many domains of applied sciences and engineering,and such a system has been investigated in recent decades.This work uses a modified Sardar sub-equation procedure to secure the soliton-type solutions of the generalized cubic nonlinear Schrödinger-Korteweg-de Vries system of equations.For various selections of arbitrary parameters in these solutions,the dynamic properties of some acquired solutions are represented graph-ically and analyzed.In particular,the dynamics of the bright solitons,dark solitons,mixed bright-dark solitons,W-shaped solitons,M-shaped solitons,periodic waves,and other soliton-type solutions.Our re-sults demonstrated that the proposed technique is highly efficient and effective for the aforementioned problems,as well as other nonlinear problems that may arise in the fields of mathematical physics and engineering.
基金the Basic Science Research Unit,Scientific Research Deanship at Majmaah University,project number RGP-2019-4。
文摘The principal objective of this article is to construct new and further exact soliton solutions of the(2+1)-dimensional Heisenberg ferromagnetic spin chain equation which investigates the nonlinear dynamics of magnets and explains their ordering in ferromagnetic materials.These solutions are exerted via the new extended FAN sub-equation method.We successfully obtain dark,bright,combined bright-dark,combined dark-singular,periodic,periodic singular,and elliptic wave solutions to this equation which are interesting classes of nonlinear excitation presenting spin dynamics in classical and semi-classical continuum Heisenberg systems.3D figures are illustrated under an appropriate selection of parameters.The applied technique is suitable to be used in gaining the exact solutions of most nonlinear partial/fractional differential equations which appear in complex phenomena.
文摘In this article,two different methods,namely sub-equation method and residual power series method,have been used to obtain new exact and approximate solutions of the generalized Hirota-Satsuma system of equations,which is a coupled KdV model.The fractional derivative is taken in the conformable sense.Each of the obtained exact solutions were checked by substituting them into the corresponding system with the help of Maple symbolic computation package.The results indicate that both methods are easy to implement,effective and reliable.They are therefore ready to apply for various partial fractional differential equations.
文摘This paper studied new class of integral equation called the Korteweg-de Vries-Kadomtsev-Petviashvili(KdV-KP)equation.This equation consist of the well-known fifth-order KdV equation in the context of the Kadomtsev-Petviashvili equation.The newly gathered class of sixth-order KdV-KP equation is studied using the sub-equation method to obtain several soliton-type solutions which consist of trigonometric,hyperbolic,and rational solutions.The application of the sub-equation approach in this work draws attention to the outstanding characteristics of the suggested method and its ability to handle completely integrable equations.Furthermore,the obtained solutions have not been reported in the previous literature and might have significant impact on future research.
文摘This paper studies the new families of exact traveling wave solutions with the modified nonlinear Schrodinger equation,which models the propagation of rogue waves in ocean engineering.The extended Fan sub-equation method with five parameters is used to find exact traveling wave solutions.It has been observed that the equation exhibits a collection of traveling wave solutions for limiting values of parameters.This method is beneficial for solving nonlinear partial differential equations,because it is not only useful for finding the new exact traveling wave solutions,but also gives us the solutions obtained previously by the usage of other techniques(Riccati equation,or first-kind elliptic equation,or the generalized Riccati equation as mapping equation,or auxiliary ordinary differential equation method)in a combined approach.Moreover,by means of the concept of linear stability,we prove that the governing model is stable.3 D figures are plotted for showing the physical behavior of the obtained solutions for the different values of unknown parameters with constraint conditions.
文摘In this paper,we applied the sub-equation method to obtain a new exact solution set for the extended version of the time-fractional Kadomtsev-Petviashvili equation,namely Burgers-Kadomtsev-Petviashvili equation(Burgers-K-P)that arises in shallow water waves.Furthermore,using the residual power series method(RPSM),approximate solutions of the equation were obtained with the help of the Mathematica symbolic computation package.We also presented a few graphical illustrations for some surfaces.The fractional derivatives were considered in the conformable sense.All of the obtained solutions were replaced back in the governing equation to check and ensure the reliability of the method.The numerical outcomes confirmed that both methods are simple,robust and effective to achieve exact and approximate solutions of nonlinear fractional differential equations.