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
In this study, we will introduce the modified (G'/G<sup>2</sup>)-expansion method to explore some of the exact traveling wave solutions of some nonlinear partial differential equations namely, Phi-4 eq...In this study, we will introduce the modified (G'/G<sup>2</sup>)-expansion method to explore some of the exact traveling wave solutions of some nonlinear partial differential equations namely, Phi-4 equation, Joseph-Egri (TRLW) equation, and Calogro-Degasperis (CD) equation. As a result, we have obtained solutions for the equations expressed in terms of trigonometric, hyperbolic and rational functions. Moreover, some selected solutions are plotted using some specific values for the parameters.展开更多
In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the （2＋1）-dimensional Kaup-Kupershmidt （KK） equation are presented. We successfully obtain more genera...In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the （2＋1）-dimensional Kaup-Kupershmidt （KK） equation are presented. We successfully obtain more general exact travelling wave solutions for （2＋ 1）-dimensional KK equation by the symmetry method and the （G1/G）-expansion method. Consequently, we find some new solutions of （2＋1）-dimensional KK equation, including similarity solutions, solitary wave solutions, and periodic solutions.展开更多
Three (2 + 1)-dimensional equations—Burgers equation, cylindrical Burgers equation and spherical Burgers equation, have been reduced to the classical Burgers equation by different transformation of variables respecti...Three (2 + 1)-dimensional equations—Burgers equation, cylindrical Burgers equation and spherical Burgers equation, have been reduced to the classical Burgers equation by different transformation of variables respectively. The decay mode solutions of the Burgers equation have been obtained by using the extended -expansion method, substituting the solutions obtained into the corresponding transformation of variables, the decay mode solutions of the three (2 + 1)-dimensional equations have been obtained successfully.展开更多
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
Nonlinear Schrödinger-type equations are important models that have emerged from a wide variety of fields,such as fluids,nonlinear optics,the theory of deep-water waves,plasma physics,and so on.In this work,we ob...Nonlinear Schrödinger-type equations are important models that have emerged from a wide variety of fields,such as fluids,nonlinear optics,the theory of deep-water waves,plasma physics,and so on.In this work,we obtain different soliton solutions to coupled nonlinear Schrödinger-type(CNLST)equations by applying three integration tools known as the(G’/G^(2))-expansion function method,the modified direct algebraic method(MDAM),and the generalized Kudryashov method.The soliton and other solutions obtained by these methods can be categorized as single(dark,singular),complex,and combined soliton solutions,as well as hyperbolic,plane wave,and trigonometric solutions with arbitrary parameters.The spectrum of the solitons is enumerated along with their existence criteria.Moreover,2D,3D,and contour profiles of the reported results are also plotted by choosing suitable values of the parameters involved,which makes it easier for researchers to comprehend the physical phenomena of the governing equation.The solutions acquired demonstrate that the proposed techniques are efficient,valuable,and straightforward when constructing new solutions for various types of nonlinear partial differential equation that have important applications in applied sciences and engineering.All the reported solutions are verified by substitution back into the original equation through the software package Mathematica.展开更多
In this paper,two integrating strategies namely exp[-Ф(Х)]and (G'/G^(2))-expansion methods together with the attributes of local-M derivatives have been acknowledged on the electrical microtubule(MT)model to ret...In this paper,two integrating strategies namely exp[-Ф(Х)]and (G'/G^(2))-expansion methods together with the attributes of local-M derivatives have been acknowledged on the electrical microtubule(MT)model to retrieve soliton solutions.The said model performs a significant role in illustrating the waves propagation in nonlinear systems.MTs are also highly productive in signaling,cell motility,and intracellular transport.The proposed algorithms yielded solutions of bright,dark,singular,and combo fractional soliton type.The significance of the fractional parameters of the fetched results is explained and presented vividly.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
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.
文摘In this study, we will introduce the modified (G'/G<sup>2</sup>)-expansion method to explore some of the exact traveling wave solutions of some nonlinear partial differential equations namely, Phi-4 equation, Joseph-Egri (TRLW) equation, and Calogro-Degasperis (CD) equation. As a result, we have obtained solutions for the equations expressed in terms of trigonometric, hyperbolic and rational functions. Moreover, some selected solutions are plotted using some specific values for the parameters.
基金Supported by the Natural Science Foundation of Shandong Province in China under Grant No.Q2005A01
文摘In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the （2＋1）-dimensional Kaup-Kupershmidt （KK） equation are presented. We successfully obtain more general exact travelling wave solutions for （2＋ 1）-dimensional KK equation by the symmetry method and the （G1/G）-expansion method. Consequently, we find some new solutions of （2＋1）-dimensional KK equation, including similarity solutions, solitary wave solutions, and periodic solutions.
文摘Three (2 + 1)-dimensional equations—Burgers equation, cylindrical Burgers equation and spherical Burgers equation, have been reduced to the classical Burgers equation by different transformation of variables respectively. The decay mode solutions of the Burgers equation have been obtained by using the extended -expansion method, substituting the solutions obtained into the corresponding transformation of variables, the decay mode solutions of the three (2 + 1)-dimensional equations have been obtained successfully.
文摘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.
基金the financial support provided for this research via the National Natural Science Foundation of China(11771407-52071298)ZhongYuan Science and Technology Innovation Leadership Program(214200510010)the MOST Innovation Methodproject(2019IM050400)。
文摘Nonlinear Schrödinger-type equations are important models that have emerged from a wide variety of fields,such as fluids,nonlinear optics,the theory of deep-water waves,plasma physics,and so on.In this work,we obtain different soliton solutions to coupled nonlinear Schrödinger-type(CNLST)equations by applying three integration tools known as the(G’/G^(2))-expansion function method,the modified direct algebraic method(MDAM),and the generalized Kudryashov method.The soliton and other solutions obtained by these methods can be categorized as single(dark,singular),complex,and combined soliton solutions,as well as hyperbolic,plane wave,and trigonometric solutions with arbitrary parameters.The spectrum of the solitons is enumerated along with their existence criteria.Moreover,2D,3D,and contour profiles of the reported results are also plotted by choosing suitable values of the parameters involved,which makes it easier for researchers to comprehend the physical phenomena of the governing equation.The solutions acquired demonstrate that the proposed techniques are efficient,valuable,and straightforward when constructing new solutions for various types of nonlinear partial differential equation that have important applications in applied sciences and engineering.All the reported solutions are verified by substitution back into the original equation through the software package Mathematica.
文摘In this paper,two integrating strategies namely exp[-Ф(Х)]and (G'/G^(2))-expansion methods together with the attributes of local-M derivatives have been acknowledged on the electrical microtubule(MT)model to retrieve soliton solutions.The said model performs a significant role in illustrating the waves propagation in nonlinear systems.MTs are also highly productive in signaling,cell motility,and intracellular transport.The proposed algorithms yielded solutions of bright,dark,singular,and combo fractional soliton type.The significance of the fractional parameters of the fetched results is explained and presented vividly.
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.
