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
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
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
The main purpose of this paper is to investigate global asymptotic stability of the zero solution of the fifth-order nonlinear delay differential equation on the following form By constructing a Lyapunov functional, s...The main purpose of this paper is to investigate global asymptotic stability of the zero solution of the fifth-order nonlinear delay differential equation on the following form By constructing a Lyapunov functional, sufficient conditions for the stability of the zero solution of this equation are established.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
Independent component analysis (ICA) is the primary statistical method for solving the problems of blind source separation. The fast ICA is a famous and excellent algorithm and its contrast function is optimized by ...Independent component analysis (ICA) is the primary statistical method for solving the problems of blind source separation. The fast ICA is a famous and excellent algorithm and its contrast function is optimized by the quadratic convergence of Newton iteration method. In order to improve the convergence speed and the separation precision of the fast ICA, an improved fast ICA algorithm is presented. The algorithm introduces an efficient Newton's iterative method with fifth-order convergence for optimizing the contrast function and gives the detail derivation process and the corresponding condition. The experimental results demonstrate that the convergence speed and the separation precision of the improved algorithm are better than that of the fast ICA.展开更多
The (2 + 1)-dimensional fifth-order KdV equation is an important higher-dimensional and higher-order extension of the famous KdV equation in fluid dynamics. In this paper, by constructing new test functions, we invest...The (2 + 1)-dimensional fifth-order KdV equation is an important higher-dimensional and higher-order extension of the famous KdV equation in fluid dynamics. In this paper, by constructing new test functions, we investigate the periodic solitary wave solutions for the (2 + 1)-dimensional fifth-order KdV equation by virtue of the Hirota bilinear form. Several novel analytic solutions for such a model are obtained and verified with the help of symbolic computation.展开更多
In this paper, we develop a method to calculate numerical and approximate solution of some fifth-order Korteweg-de Vries equations with initial condition with the help of Laplace Decomposition Method (LDM). The techni...In this paper, we develop a method to calculate numerical and approximate solution of some fifth-order Korteweg-de Vries equations with initial condition with the help of Laplace Decomposition Method (LDM). The technique is based on the application of Laplace transform to some fifth-order Kdv equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of four examples and results of the present technique have closed agreement with approximate solutions obtained with the help of (LDM).展开更多
A numerical method based on septic B-spline function is presented for the solution of linear and nonlinear fifth-order boundary value problems. The method is fourth order convergent. We use the quesilinearization tech...A numerical method based on septic B-spline function is presented for the solution of linear and nonlinear fifth-order boundary value problems. The method is fourth order convergent. We use the quesilinearization technique to reduce the nonlinear problems to linear problems and use B-spline collocation method, which leads to a seven nonzero bands linear system. Illustrative example is included to demonstrate the validity and applicability of the proposed techniques.展开更多
In this paper,we investigate the fifth-order modified Korteweg-de Vries(mKdV)equation on the half-line via the Fokas unified transformation approach.We show that the solution u(x,t)of the fifth-order mKdV equation can...In this paper,we investigate the fifth-order modified Korteweg-de Vries(mKdV)equation on the half-line via the Fokas unified transformation approach.We show that the solution u(x,t)of the fifth-order mKdV equation can be represented by the solution of the matrix Riemann-Hilbert problem constructed on the plane of complex spectral parameter θ.The jump matrix L(x,t,θ)has an explicit representation dependent on x,t and it can be represented exactly by the two pairs of spectral functions y(θ),z(θ)(obtained from the initial value u0(x))and Y(θ),Z(θ)(obtained from the boundary conditions v0(t),{vk(t)}_(1)^(4)).Furthermore,the two pairs of spectral functions y(θ),z(θ)and Y(θ),Z(θ)are not independent of each other,but are related to the compatibility condition,the so-called global relation.展开更多
In this paper,we discussed the enhanced Kudryashov’s and general projective Riccati equations tech-niques for obtaining exact solutions to the fifth-order nonlinear water wave(FONLWWE)equation.Using the enhanced Kudr...In this paper,we discussed the enhanced Kudryashov’s and general projective Riccati equations tech-niques for obtaining exact solutions to the fifth-order nonlinear water wave(FONLWWE)equation.Using the enhanced Kudryashov’s method,we were able to achieve solitary wave and singular soliton solutions.Solitary-shock hybrid wave,singular soliton,and periodic wave solutions were discovered when we em-ployed the general projective Riccati equations approach.We can say the given methods are effective and powerful for obtaining exact solutions.Our findings in this paper are critical for explaining a wide range of scientific and oceanographic applications involving ocean gravity waves and other related phenomena.展开更多
Based on Biot's consolidation theory, a two-dimensional model for computation of the seabed response to waves is presented with the finite element method. Numerical results for different wave conditions are obtained,...Based on Biot's consolidation theory, a two-dimensional model for computation of the seabed response to waves is presented with the finite element method. Numerical results for different wave conditions are obtained, and the effects of wave non-lineafity on the wave-induced seabed response are examined. Moreover, the wave-induced momentary liquefaction in uniform and inhomogeneous seabeds is investigated. It is shown that the wave non-linearity affects the distribution of the wave-induced pore pressure and effective stresses, while the influence of wave non-linearity on the seabed liquefaction potential is not so significant.展开更多
This paper is concerned with the fifth-order modified Korteweg-de Vries(fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion(CRE) solvable. Three special form of soliton-cnoidal wave i...This paper is concerned with the fifth-order modified Korteweg-de Vries(fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion(CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion(CTE) method, the nonlocal symmetry related to the consistent tanh expansion(CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlev′e method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed.展开更多
We prove the global well-posedness for the Cauchy problem of fifth-order modified Korteweg-de Vries equation in Sobolev spaces H^3(R) for s〉-3/22.The main approach is the "I-method" together with the multilinear ...We prove the global well-posedness for the Cauchy problem of fifth-order modified Korteweg-de Vries equation in Sobolev spaces H^3(R) for s〉-3/22.The main approach is the "I-method" together with the multilinear multiplier analysis.展开更多
Chebyshev collocation method is used to approximate solutions of two-point BVP arising in modelling viscoelastic flow. The scheme is tested on four nonlinear problems. The comparison with other methods is made. The re...Chebyshev collocation method is used to approximate solutions of two-point BVP arising in modelling viscoelastic flow. The scheme is tested on four nonlinear problems. The comparison with other methods is made. The results demonstrate the reliability and efficiency of the algorithm developed.展开更多
Taking the potential fifth-order MKdV equation as an example to introduce a possible way to construct invariance of a nonlinear PDE. Based on an obtained Backlund transformation of the potential fifth-order MKdV equat...Taking the potential fifth-order MKdV equation as an example to introduce a possible way to construct invariance of a nonlinear PDE. Based on an obtained Backlund transformation of the potential fifth-order MKdV equation and by solving the corresponding Ricatti form Lax pairs, an invariance of the potential fifth-order MKdV equation is digged out. Thus, just by differential and quadrature procedure,the solutions of the potential fifth-order MKdV equation can be obtained from a known solution.展开更多
In this paper,we show by Hilbert Uniqueness Method that the boundary value problem of fifth-order KdV equation{y_(t)-y_(5x)=0,(x,t)∈(0,2π)×(0,T),y(t,2π)-y(t,0)=h_(0)(t),y_(x)(t,2π)-y_(x)(t,0)=h_(1)(t),y_(2x)(...In this paper,we show by Hilbert Uniqueness Method that the boundary value problem of fifth-order KdV equation{y_(t)-y_(5x)=0,(x,t)∈(0,2π)×(0,T),y(t,2π)-y(t,0)=h_(0)(t),y_(x)(t,2π)-y_(x)(t,0)=h_(1)(t),y_(2x)(t,2π)-y_(2x)(t,0)=h_(2)(t),y_(3x)(t,2π)-y_(3x)(t,0)=h_(3)(t),y_(4x)(t,2π)-y_(4x)(t,0)=h_(4)(t),(with boundary data as control inputs)is exact controllability.展开更多
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 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.
文摘The main purpose of this paper is to investigate global asymptotic stability of the zero solution of the fifth-order nonlinear delay differential equation on the following form By constructing a Lyapunov functional, sufficient conditions for the stability of the zero solution of this equation are established.
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.
文摘Independent component analysis (ICA) is the primary statistical method for solving the problems of blind source separation. The fast ICA is a famous and excellent algorithm and its contrast function is optimized by the quadratic convergence of Newton iteration method. In order to improve the convergence speed and the separation precision of the fast ICA, an improved fast ICA algorithm is presented. The algorithm introduces an efficient Newton's iterative method with fifth-order convergence for optimizing the contrast function and gives the detail derivation process and the corresponding condition. The experimental results demonstrate that the convergence speed and the separation precision of the improved algorithm are better than that of the fast ICA.
文摘The (2 + 1)-dimensional fifth-order KdV equation is an important higher-dimensional and higher-order extension of the famous KdV equation in fluid dynamics. In this paper, by constructing new test functions, we investigate the periodic solitary wave solutions for the (2 + 1)-dimensional fifth-order KdV equation by virtue of the Hirota bilinear form. Several novel analytic solutions for such a model are obtained and verified with the help of symbolic computation.
文摘In this paper, we develop a method to calculate numerical and approximate solution of some fifth-order Korteweg-de Vries equations with initial condition with the help of Laplace Decomposition Method (LDM). The technique is based on the application of Laplace transform to some fifth-order Kdv equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of four examples and results of the present technique have closed agreement with approximate solutions obtained with the help of (LDM).
文摘A numerical method based on septic B-spline function is presented for the solution of linear and nonlinear fifth-order boundary value problems. The method is fourth order convergent. We use the quesilinearization technique to reduce the nonlinear problems to linear problems and use B-spline collocation method, which leads to a seven nonzero bands linear system. Illustrative example is included to demonstrate the validity and applicability of the proposed techniques.
基金supported by the National Natural Science Foundation of China under Grant Nos.12147115 and 11835011the Natural Science Foundation of Anhui Province under Grant No.2108085QA09+3 种基金the University Natural Science Research Project of Anhui Province under Grant No.KJ2021A1094China Postdoctoral Science Foundation under Grant No.2022M712833the Program for Science and Technology Innovation Talents in Universities of Henan Province under Grant No.22HASTIT019the Natural Science Foundation of Henan Province under Grant No.202300410524
文摘In this paper,we investigate the fifth-order modified Korteweg-de Vries(mKdV)equation on the half-line via the Fokas unified transformation approach.We show that the solution u(x,t)of the fifth-order mKdV equation can be represented by the solution of the matrix Riemann-Hilbert problem constructed on the plane of complex spectral parameter θ.The jump matrix L(x,t,θ)has an explicit representation dependent on x,t and it can be represented exactly by the two pairs of spectral functions y(θ),z(θ)(obtained from the initial value u0(x))and Y(θ),Z(θ)(obtained from the boundary conditions v0(t),{vk(t)}_(1)^(4)).Furthermore,the two pairs of spectral functions y(θ),z(θ)and Y(θ),Z(θ)are not independent of each other,but are related to the compatibility condition,the so-called global relation.
文摘In this paper,we discussed the enhanced Kudryashov’s and general projective Riccati equations tech-niques for obtaining exact solutions to the fifth-order nonlinear water wave(FONLWWE)equation.Using the enhanced Kudryashov’s method,we were able to achieve solitary wave and singular soliton solutions.Solitary-shock hybrid wave,singular soliton,and periodic wave solutions were discovered when we em-ployed the general projective Riccati equations approach.We can say the given methods are effective and powerful for obtaining exact solutions.Our findings in this paper are critical for explaining a wide range of scientific and oceanographic applications involving ocean gravity waves and other related phenomena.
文摘Based on Biot's consolidation theory, a two-dimensional model for computation of the seabed response to waves is presented with the finite element method. Numerical results for different wave conditions are obtained, and the effects of wave non-lineafity on the wave-induced seabed response are examined. Moreover, the wave-induced momentary liquefaction in uniform and inhomogeneous seabeds is investigated. It is shown that the wave non-linearity affects the distribution of the wave-induced pore pressure and effective stresses, while the influence of wave non-linearity on the seabed liquefaction potential is not so significant.
基金Supported by National Natural Science Foundation of China under Grant No.11505090Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No.BS2015SF009
文摘This paper is concerned with the fifth-order modified Korteweg-de Vries(fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion(CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion(CTE) method, the nonlocal symmetry related to the consistent tanh expansion(CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlev′e method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed.
文摘We prove the global well-posedness for the Cauchy problem of fifth-order modified Korteweg-de Vries equation in Sobolev spaces H^3(R) for s〉-3/22.The main approach is the "I-method" together with the multilinear multiplier analysis.
文摘Chebyshev collocation method is used to approximate solutions of two-point BVP arising in modelling viscoelastic flow. The scheme is tested on four nonlinear problems. The comparison with other methods is made. The results demonstrate the reliability and efficiency of the algorithm developed.
文摘Taking the potential fifth-order MKdV equation as an example to introduce a possible way to construct invariance of a nonlinear PDE. Based on an obtained Backlund transformation of the potential fifth-order MKdV equation and by solving the corresponding Ricatti form Lax pairs, an invariance of the potential fifth-order MKdV equation is digged out. Thus, just by differential and quadrature procedure,the solutions of the potential fifth-order MKdV equation can be obtained from a known solution.
基金supported by the Zhejiang Provincial Natural Science Foundation of China(No.LY18A010024)National Natural Science Foundation of China(No.12075208).
文摘In this paper,we show by Hilbert Uniqueness Method that the boundary value problem of fifth-order KdV equation{y_(t)-y_(5x)=0,(x,t)∈(0,2π)×(0,T),y(t,2π)-y(t,0)=h_(0)(t),y_(x)(t,2π)-y_(x)(t,0)=h_(1)(t),y_(2x)(t,2π)-y_(2x)(t,0)=h_(2)(t),y_(3x)(t,2π)-y_(3x)(t,0)=h_(3)(t),y_(4x)(t,2π)-y_(4x)(t,0)=h_(4)(t),(with boundary data as control inputs)is exact controllability.