In this article, we propose a generalized exp(-Φ(ξ))-expansion method and successfully implement it to find exact traveling wave solutions to the fifth order standard Sawada-Kotera (SK) equation. The exact traveling...In this article, we propose a generalized exp(-Φ(ξ))-expansion method and successfully implement it to find exact traveling wave solutions to the fifth order standard Sawada-Kotera (SK) equation. The exact traveling wave solutions are established in the form of trigonometric, hyperbolic, exponential and rational functions with some free parameters. It is shown that this method is standard, effective and easily applicable mathematical tool for solving nonlinear partial differential equations arises in the field of mathematical physics and engineering.展开更多
As the solution of the two equations for determining the existing fifth order Stokes wave derived by Skjelbreia is complex and tedious, the two equations are simplified into one equation for determining d / L, i. e., ...As the solution of the two equations for determining the existing fifth order Stokes wave derived by Skjelbreia is complex and tedious, the two equations are simplified into one equation for determining d / L, i. e., f(H, T, d / L) = 0. According to this simplified method, three cases of the solution for the Skjelbreia equations have been found: one accurate solution; more than one accurate solution and no accurate solution (but there exists the optimum approximate solution in the area of satisfying Skjelbreia equations). As to the case of more than one accurate solution, the reasonable solution can be judged from the method of variational principle, by means elf which an optimum solution improved from the solution of Skjelbreia equations in the area of satisfying the original mathematical equations of non-vortex and nonlinear wave theory, i. e., the optimum fifth order Stokes wave, is given.展开更多
The aim of the present study is to design a new fifth order system of Emden–Fowler equations and related four types of the model.The standard second order form of the Emden–Fowler has been used to obtain the new mod...The aim of the present study is to design a new fifth order system of Emden–Fowler equations and related four types of the model.The standard second order form of the Emden–Fowler has been used to obtain the new model.The shape factor that appear more than one time discussed in detail for every case of the designed model.The singularity atη=0 at one point or multiple points is also discussed at each type of the model.For validation and correctness of the new designed model,one example of each type based on system of fifth order Emden–Fowler equations are provided and numerical solutions of the designed equations of each type have been obtained by using variational iteration scheme.The comparison of the exact results and present numerical outcomes for solving one problem of each type is presented to check the accuracy of the designed model.展开更多
The nonlocal symmetry of the generalized fifth order KdV equation(FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis...The nonlocal symmetry of the generalized fifth order KdV equation(FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Ba¨cklund transformation is obtained through Lie’s first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method.展开更多
A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) s...A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) scheme and an exponential time integration scheme combined with the best rational approximations based on the Carathéodory-Fejér procedure for time discretization. We check several numerical results of our approach against available analytical solutions. In addition, we computed the conservation laws of the fKdV equation. We find that the DSC approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations.展开更多
New modified Adomian decomposition method is proposed for the solution of the generalized fifth-order Korteweg-de Vries (GFKdV) equation. The numerical solutions are compared with the standard Adomian decomposition me...New modified Adomian decomposition method is proposed for the solution of the generalized fifth-order Korteweg-de Vries (GFKdV) equation. The numerical solutions are compared with the standard Adomian decomposition method and the exact solutions. The results are demonstrated which confirm the efficiency and applicability of the method.展开更多
In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators...In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators.The existence,uniqueness,and stability of the proposed model are discussed.Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator is developed to study the proposed models.Comparative studies with generalized fifth-order Runge-Kutta method are given.Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations are presented.We have showcased the efficiency of the proposed method and garnered robust empirical support for our theoretical findings.展开更多
The solvability of the fifth-order nonlinear dispersive equation δtu+au (δxu)^2+βδx^3u+γδx^5u = 0 is studied. By using the approach of Kenig, Ponce and Vega and some Strichartz estimates for the correspondi...The solvability of the fifth-order nonlinear dispersive equation δtu+au (δxu)^2+βδx^3u+γδx^5u = 0 is studied. By using the approach of Kenig, Ponce and Vega and some Strichartz estimates for the corresponding linear problem,it is proved that if the initial function u0 belongs to H^5(R) and s〉1/4,then the Cauchy problem has a unique solution in C([-T,T],H^5(R)) for some T〉0.展开更多
In this paper,the variable coefficient nonlinear Schrödinger equation with fifth order dispersion in the inhomogeneous optical fiber is investigated to study the impact of fifth order dispersion on attosecond sol...In this paper,the variable coefficient nonlinear Schrödinger equation with fifth order dispersion in the inhomogeneous optical fiber is investigated to study the impact of fifth order dispersion on attosecond soliton propagation.Based on the Darboux transformation method,soliton solution is constructed with modulated coefficients though Lax pair.We show that the coefficients of GVD and fifth order term are the main keys to control the amplitude,width and phase shift in the model.With the suitable choices of coefficients for the obtained soliton solution,some new types of soliton management through fifth order are presented for the first time.Especially,the influence of fifth order dispersion of the higher order NLS equations with and without modulated coefficients is discussed in detail.We will also highlight the impact of the studies on application in ocean engineering.Obtained results may be potentially useful in the soliton shaping and management in inhomogeneous fibers.展开更多
The investigation of closed form solutions for nonlinear evolution equations(NLEEs)is being an attractive subject in the different branches of mathematical and physical sciences.In this article,the enhanced(G'=G)-...The investigation of closed form solutions for nonlinear evolution equations(NLEEs)is being an attractive subject in the different branches of mathematical and physical sciences.In this article,the enhanced(G'=G)-expansion method has been applied to find the closed form solutions for NLEEs,such as the simplified MCH equation and third extended fifth order nonlinear equations which are very important in mathematical physics.Plentiful closed form solutions with arbitrary parameters are successfully obtained by this method which are expressed in terms of hyperbolic and trigonometric functions.It is shown that the obtained solutions are more general and fresh and can be helpful to analyze the NLEES in mathematical physics and engineering problems.展开更多
This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete mu...This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete multi-symplectic conservation law to solve the partial differential equations which are derived from the generalized fifth-order KdV equation numerically. The results of the numerical experiments show that this multi-symplectic algorithm is good in accuracy and its long-time numerical behaviour is also perfect.展开更多
In this paper,aiming to get more insight on the relation between the higher order semidiscrete mKdV equations and higher order mKdV equations,we construct a fifth order semidiscrete mKdV equation from the three known ...In this paper,aiming to get more insight on the relation between the higher order semidiscrete mKdV equations and higher order mKdV equations,we construct a fifth order semidiscrete mKdV equation from the three known semidiscrete mKdV fluxes.We not only give its Lax pairs,Darboux transformation,explicit solutions and infinitely many conservation laws,but also show that their continuous limits yield the corresponding results for the fifth order mKdV equation.We thus conclude that the fifth order discrete mKdV equation is extremely an useful discrete scheme for the fifth order mKdV equation.展开更多
This paper presents a universal fifth-order Stokes solution for steady water waves on the basis of potential theory. It uses a global perturbation parameter, considers a depth uniform current, and thus admits the flex...This paper presents a universal fifth-order Stokes solution for steady water waves on the basis of potential theory. It uses a global perturbation parameter, considers a depth uniform current, and thus admits the flexibilities on the definition of the perturbation parameter and on the determination of the wave celerity. The universal solution can be extended to that of Chappelear (1961), confirming the correctness for the universal theory. Furthermore, a particular fifth-order solution is obtained where the wave steepness is used as the perturbation parameter. The applicable range of this solution in shallow depth is analyzed. Comparisons with the Fourier approximated results and with the experimental measurements show that the solution is fairly suited to waves with the Ursell number not exceeding 46.7.展开更多
By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painleve test for integrability only for three distin...By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painleve test for integrability only for three distinct cases. Moreover, the multi- soliton solutions are presented for this equation under three sets of integrable conditions. Finally, by selecting appropriate parameters, we analyze the evolution of two solitons, which is especially interesting as it may describe the overtaking and the head-on collisions of solitary waves of different shapes and different types.展开更多
Based on the method developed by Nucci,the pseudopotentials,Lax pairs and the singularity manifoldequations of the generalized fifth-order KdV equation are derived.By choosing different coefficient,the correspondingre...Based on the method developed by Nucci,the pseudopotentials,Lax pairs and the singularity manifoldequations of the generalized fifth-order KdV equation are derived.By choosing different coefficient,the correspondingresults and the Backlund transformations can be obtained on three conditioners which include Caudrey-Dodd-GibbonSawada-Kotera equation,the Lax equation and the Kaup-kupershmidt equation.展开更多
This work presents the application of the recently developed “Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5<sup>th</sup>-CASAM-N)” to a simplified Bernoulli ...This work presents the application of the recently developed “Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5<sup>th</sup>-CASAM-N)” to a simplified Bernoulli model. The 5<sup>th</sup>-CASAM-N builds upon and incorporates all of the lower-order (i.e., the first-, second-, third-, and fourth-order) adjoint sensitivities analysis methodologies. The Bernoulli model comprises a nonlinear model response, uncertain model parameters, uncertain model domain boundaries and uncertain model boundary conditions, admitting closed-form explicit expressions for the response sensitivities of all orders. Illustrating the specific mechanisms and advantages of applying the 5<sup>th</sup>-CASAM-N for the computation of the response sensitivities with respect to the uncertain parameters and boundaries reveals that the 5<sup>th</sup>-CASAM-N provides a fundamental step towards overcoming the curse of dimensionality in sensitivity and uncertainty analysis.展开更多
In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used...In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.展开更多
We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem wit...We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem with the well-known from combinatorial topology Sperner’s lemma. We also notice that our geometric approach is strongly based on the associated vector field.展开更多
文摘In this article, we propose a generalized exp(-Φ(ξ))-expansion method and successfully implement it to find exact traveling wave solutions to the fifth order standard Sawada-Kotera (SK) equation. The exact traveling wave solutions are established in the form of trigonometric, hyperbolic, exponential and rational functions with some free parameters. It is shown that this method is standard, effective and easily applicable mathematical tool for solving nonlinear partial differential equations arises in the field of mathematical physics and engineering.
文摘As the solution of the two equations for determining the existing fifth order Stokes wave derived by Skjelbreia is complex and tedious, the two equations are simplified into one equation for determining d / L, i. e., f(H, T, d / L) = 0. According to this simplified method, three cases of the solution for the Skjelbreia equations have been found: one accurate solution; more than one accurate solution and no accurate solution (but there exists the optimum approximate solution in the area of satisfying Skjelbreia equations). As to the case of more than one accurate solution, the reasonable solution can be judged from the method of variational principle, by means elf which an optimum solution improved from the solution of Skjelbreia equations in the area of satisfying the original mathematical equations of non-vortex and nonlinear wave theory, i. e., the optimum fifth order Stokes wave, is given.
文摘The aim of the present study is to design a new fifth order system of Emden–Fowler equations and related four types of the model.The standard second order form of the Emden–Fowler has been used to obtain the new model.The shape factor that appear more than one time discussed in detail for every case of the designed model.The singularity atη=0 at one point or multiple points is also discussed at each type of the model.For validation and correctness of the new designed model,one example of each type based on system of fifth order Emden–Fowler equations are provided and numerical solutions of the designed equations of each type have been obtained by using variational iteration scheme.The comparison of the exact results and present numerical outcomes for solving one problem of each type is presented to check the accuracy of the designed model.
基金supported by the National Natural Science Foundation of China(Grant Nos.11347183,11405110,11275129,and 11305106)the Natural Science Foundation of Zhejiang Province of China(Grant Nos.Y7080455 and LQ13A050001)
文摘The nonlocal symmetry of the generalized fifth order KdV equation(FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Ba¨cklund transformation is obtained through Lie’s first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method.
文摘A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) scheme and an exponential time integration scheme combined with the best rational approximations based on the Carathéodory-Fejér procedure for time discretization. We check several numerical results of our approach against available analytical solutions. In addition, we computed the conservation laws of the fKdV equation. We find that the DSC approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations.
文摘New modified Adomian decomposition method is proposed for the solution of the generalized fifth-order Korteweg-de Vries (GFKdV) equation. The numerical solutions are compared with the standard Adomian decomposition method and the exact solutions. The results are demonstrated which confirm the efficiency and applicability of the method.
文摘In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators.The existence,uniqueness,and stability of the proposed model are discussed.Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator is developed to study the proposed models.Comparative studies with generalized fifth-order Runge-Kutta method are given.Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations are presented.We have showcased the efficiency of the proposed method and garnered robust empirical support for our theoretical findings.
文摘The solvability of the fifth-order nonlinear dispersive equation δtu+au (δxu)^2+βδx^3u+γδx^5u = 0 is studied. By using the approach of Kenig, Ponce and Vega and some Strichartz estimates for the corresponding linear problem,it is proved that if the initial function u0 belongs to H^5(R) and s〉1/4,then the Cauchy problem has a unique solution in C([-T,T],H^5(R)) for some T〉0.
文摘In this paper,the variable coefficient nonlinear Schrödinger equation with fifth order dispersion in the inhomogeneous optical fiber is investigated to study the impact of fifth order dispersion on attosecond soliton propagation.Based on the Darboux transformation method,soliton solution is constructed with modulated coefficients though Lax pair.We show that the coefficients of GVD and fifth order term are the main keys to control the amplitude,width and phase shift in the model.With the suitable choices of coefficients for the obtained soliton solution,some new types of soliton management through fifth order are presented for the first time.Especially,the influence of fifth order dispersion of the higher order NLS equations with and without modulated coefficients is discussed in detail.We will also highlight the impact of the studies on application in ocean engineering.Obtained results may be potentially useful in the soliton shaping and management in inhomogeneous fibers.
文摘The investigation of closed form solutions for nonlinear evolution equations(NLEEs)is being an attractive subject in the different branches of mathematical and physical sciences.In this article,the enhanced(G'=G)-expansion method has been applied to find the closed form solutions for NLEEs,such as the simplified MCH equation and third extended fifth order nonlinear equations which are very important in mathematical physics.Plentiful closed form solutions with arbitrary parameters are successfully obtained by this method which are expressed in terms of hyperbolic and trigonometric functions.It is shown that the obtained solutions are more general and fresh and can be helpful to analyze the NLEES in mathematical physics and engineering problems.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10572119, 10772147 and 10632030)the Doctoral Program Foundation of Education Ministry of China (Grant No 20070699028)+1 种基金the National Natural Science Foundation of Shaanxi Province of China (Grant No 2006A07)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment
文摘This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete multi-symplectic conservation law to solve the partial differential equations which are derived from the generalized fifth-order KdV equation numerically. The results of the numerical experiments show that this multi-symplectic algorithm is good in accuracy and its long-time numerical behaviour is also perfect.
基金supported by National Natural Science Foundation of China (Grant No.10971136)the Ministry of Education and Innovation of Spain (Grant No.MTM2009-12670)
文摘In this paper,aiming to get more insight on the relation between the higher order semidiscrete mKdV equations and higher order mKdV equations,we construct a fifth order semidiscrete mKdV equation from the three known semidiscrete mKdV fluxes.We not only give its Lax pairs,Darboux transformation,explicit solutions and infinitely many conservation laws,but also show that their continuous limits yield the corresponding results for the fifth order mKdV equation.We thus conclude that the fifth order discrete mKdV equation is extremely an useful discrete scheme for the fifth order mKdV equation.
基金supported by the Jiangsu Province Natural Science Foundation for the Young Scholars(Grant No.BK20130827)the National Natural Science Foundation of China(Grant Nos.41076008 and 51479055)
文摘This paper presents a universal fifth-order Stokes solution for steady water waves on the basis of potential theory. It uses a global perturbation parameter, considers a depth uniform current, and thus admits the flexibilities on the definition of the perturbation parameter and on the determination of the wave celerity. The universal solution can be extended to that of Chappelear (1961), confirming the correctness for the universal theory. Furthermore, a particular fifth-order solution is obtained where the wave steepness is used as the perturbation parameter. The applicable range of this solution in shallow depth is analyzed. Comparisons with the Fourier approximated results and with the experimental measurements show that the solution is fairly suited to waves with the Ursell number not exceeding 46.7.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11201290 and 71103118)
文摘By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painleve test for integrability only for three distinct cases. Moreover, the multi- soliton solutions are presented for this equation under three sets of integrable conditions. Finally, by selecting appropriate parameters, we analyze the evolution of two solitons, which is especially interesting as it may describe the overtaking and the head-on collisions of solitary waves of different shapes and different types.
基金Supported by the National Natural Science Foundation of China under Grant No.60772023by the Slpported Project under Grant No.SKLSDE-2010ZX-07 of the State Key Laboratory of Software Development Environment,Beijing University of Aeronautics and As tronautics+2 种基金by the Specialized Research Fund for the Doctoral Program of Higher Educatioi under Grant No.200800130006Chinese Ministry of Education,and by the Innovation Foundation for Ph.D.Graduates under Grant Nos.30-0350 and 30-0366Beijing University of Aeronautics and Astronautics
基金The project supported by the Key Project of the Chinese Ministry of Education under Grant No.106033the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20060006024+2 种基金Chinese Ministry of Education,the National Natural Science Foundation of China under Grant Nos.60772023 and 60372095the Open Fund of the State Key Laboratory of Software Development Environment under Grant No.SKLSDE-07-001Beijing University of Aeronautics and Astronautics,and by the National Basic Research Program of China(973 Program)under Grant No.2005CB321901
基金Supported by the National Natural Science Foundation of China under Grant Nos.10735030,11075055,and 90718041the Shanghai Leading Academic Discipline Project,China under Grant No.B412+1 种基金the Program for Changjiang Scholars,the Innovative Research Team in University of Ministry of Education of China under Grant No.IRT 0734the K.C.Wong Magna Fund in Ningbo University
文摘Based on the method developed by Nucci,the pseudopotentials,Lax pairs and the singularity manifoldequations of the generalized fifth-order KdV equation are derived.By choosing different coefficient,the correspondingresults and the Backlund transformations can be obtained on three conditioners which include Caudrey-Dodd-GibbonSawada-Kotera equation,the Lax equation and the Kaup-kupershmidt equation.
文摘This work presents the application of the recently developed “Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5<sup>th</sup>-CASAM-N)” to a simplified Bernoulli model. The 5<sup>th</sup>-CASAM-N builds upon and incorporates all of the lower-order (i.e., the first-, second-, third-, and fourth-order) adjoint sensitivities analysis methodologies. The Bernoulli model comprises a nonlinear model response, uncertain model parameters, uncertain model domain boundaries and uncertain model boundary conditions, admitting closed-form explicit expressions for the response sensitivities of all orders. Illustrating the specific mechanisms and advantages of applying the 5<sup>th</sup>-CASAM-N for the computation of the response sensitivities with respect to the uncertain parameters and boundaries reveals that the 5<sup>th</sup>-CASAM-N provides a fundamental step towards overcoming the curse of dimensionality in sensitivity and uncertainty analysis.
基金the National Basic Research Program of China(Grant No.2012CB025903)
文摘In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.
文摘We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem with the well-known from combinatorial topology Sperner’s lemma. We also notice that our geometric approach is strongly based on the associated vector field.