We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler ...We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler method for temporal discretization. It is verified that the proposed method has corresponding discrete multi-symplectic conservation laws. Numerical simulations indicate that the proposed scheme is characterized by excellent conservation.展开更多
By means of the classical method, we investigate the (3+1)-dimensional Zakharov-Kuznetsov equation. The symmetry group of the (3+1)-dimensional Zakharov-Kuznetsov equation is studied first and the theorem of gro...By means of the classical method, we investigate the (3+1)-dimensional Zakharov-Kuznetsov equation. The symmetry group of the (3+1)-dimensional Zakharov-Kuznetsov equation is studied first and the theorem of group invariant solutions is constructed. Then using the associated vector fields of the obtained symmetry, we give the one-, two-, and three-parameter optimal systems of group-invariant solutions. Based on the optimal system, we derive the reductions and some new solutions of the (3+1)-dimensional Zakharov-Kuznetsov equation.展开更多
his paper studies the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation using the (G'/G)-expand method, we obtain many new explicit solutions of the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equ...his paper studies the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation using the (G'/G)-expand method, we obtain many new explicit solutions of the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation, which include hyperbolic function solutions, trigonometric function solutions and rational function solutions and so on.展开更多
The modified Zakharov-Kuznetsov equation with the initial value problem is studied numerically by means of homotopy perturbation method. The analytical approximate solutions of the modified Zakharov-Kuznetsov equation...The modified Zakharov-Kuznetsov equation with the initial value problem is studied numerically by means of homotopy perturbation method. The analytical approximate solutions of the modified Zakharov-Kuznetsov equation are obtained. Choosing the form of the initial value, the single solitary wave, two solitary waves and rational solutions are presented, some of which are shown by the plots.展开更多
In this paper, we get many new analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov Kuznetsov (mKDV-ZK) equation by means of a new approach namely method of undetermined coefficients based...In this paper, we get many new analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov Kuznetsov (mKDV-ZK) equation by means of a new approach namely method of undetermined coefficients based on a fractional complex transform. These solutions have physics meanings in natural sciences. This method can be used to other nonlinear fractional differential equations.展开更多
The exact solutions of the generalized (2+1)-dimensional nonlinear Zakharov-Kuznetsov (Z-K) equationare explored by the method of the improved generalized auxiliary differential equation.Many explicit analytic solutio...The exact solutions of the generalized (2+1)-dimensional nonlinear Zakharov-Kuznetsov (Z-K) equationare explored by the method of the improved generalized auxiliary differential equation.Many explicit analytic solutionsof the Z-K equation are obtained.The methods used to solve the Z-K equation can be employed in further work toestablish new solutions for other nonlinear partial differential equations.展开更多
The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathem...The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathematical community.Thus,this paper purveys an analytical study of a generalized extended(2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering.The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model.We further reduce the equation using the subalgebras obtained.Besides,more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique.Consequently,we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright,dark,periodic(cnoidal and snoidal),compact-type as well as singular solitons.The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time,are outlined with suitable diagrams.This can be of interest to oceanographers and ocean engineers for future analysis.Furthermore,to visualize the dynamics of the results found,we present the graphic display of each of the solutions.Conclusively,we construct conservation laws of the understudy equation via the application of Noether’s theorem.展开更多
The aim of this paper is to obtain the approximate analytical solution of a fractional Zakharov-Kuznetsov equation by using homotopy perturbation method (HPM). The fractional derivatives are described in the Caputo se...The aim of this paper is to obtain the approximate analytical solution of a fractional Zakharov-Kuznetsov equation by using homotopy perturbation method (HPM). The fractional derivatives are described in the Caputo sense. Several examples are given and the results are compared to exact solutions. The results reveal that the method is very effective and simple.展开更多
The application of the q-homotopy analysis Shehu transform method(q-HAShTM)to discover the esti-mated solution of fractional Zakharov-Kuznetsov equations is investigated in this study.In the presence of a uniform magn...The application of the q-homotopy analysis Shehu transform method(q-HAShTM)to discover the esti-mated solution of fractional Zakharov-Kuznetsov equations is investigated in this study.In the presence of a uniform magnetic field,the Zakharov-Kuznetsov equations regulate the behaviour of nonlinear acoustic waves in a plasma containing cold ions and hot isothermal electrons.The q-HAShTM is a stable analytical method that combines homotopy analysis and the Shehu transform.This q-homotopy investigation Shehu transform is a constructive method that leads to the Zakharov-Kuznetsov equations,which regulate the propagation of nonlinear ion-acoustic waves in a plasma.It is a more semi-analytical method for adjust-ing and controlling the convergence region of the series solution and overcoming some of the homotopy analysis method’s limitations.展开更多
Three modified sine-Hilbert(sH)-type equations, i.e., the modified sH equation, the modified damped sH equation, and the modified nonlinear dissipative system, are proposed, and their bilinear forms are provided.Based...Three modified sine-Hilbert(sH)-type equations, i.e., the modified sH equation, the modified damped sH equation, and the modified nonlinear dissipative system, are proposed, and their bilinear forms are provided.Based on these bilinear equations, some exact solutions to the three modified equations are derived.展开更多
By the modifying loss function MSE and training area of physics-informed neural networks(PINNs),we propose a neural networks model,namely prior-information PINNs(PIPINNs).We demonstrate the advantages of PIPINNs by si...By the modifying loss function MSE and training area of physics-informed neural networks(PINNs),we propose a neural networks model,namely prior-information PINNs(PIPINNs).We demonstrate the advantages of PIPINNs by simulating Ai-and Bi-soliton solutions of the cylindrical Korteweg-de Vries(cKdV)equation.展开更多
In this paper, the matrix Riccati equation is considered. There is no general way for solving the matrix Riccati equation despite the many fields to which it applies. While scalar Riccati equation has been studied tho...In this paper, the matrix Riccati equation is considered. There is no general way for solving the matrix Riccati equation despite the many fields to which it applies. While scalar Riccati equation has been studied thoroughly, matrix Riccati equation of which scalar Riccati equations is a particular case, is much less investigated. This article proposes a change of variable that allows to find explicit solution of the Matrix Riccati equation. We then apply this solution to Optimal Control.展开更多
In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation e...In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary L^(2) weighted estimates.展开更多
On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear ...On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].展开更多
This paper is concerned with a third order in time linear Moore-Gibson-Thompson equation which describes the acoustic velocity potential in ultrasound wave program.Influenced by the work of Kaltenbacher,Lasiecka and M...This paper is concerned with a third order in time linear Moore-Gibson-Thompson equation which describes the acoustic velocity potential in ultrasound wave program.Influenced by the work of Kaltenbacher,Lasiecka and Marchand(Control Cybernet.2011,40:971-988),we establish an observability inequality of the conservative problem,and then discuss the equivalence between the exponential stabilization of a dissipative system and the internal observational inequality of the corresponding conservative system.展开更多
Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained thro...Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.展开更多
The conversion of the cartesian coordinates of a point to its geodetic equivalent coordinates in reference to the geodetic ellipsoid is one of the main challenges in geodesy.The ellipse equation in the meridian plane ...The conversion of the cartesian coordinates of a point to its geodetic equivalent coordinates in reference to the geodetic ellipsoid is one of the main challenges in geodesy.The ellipse equation in the meridian plane significantly influences the value of the geodetic coordinates.This research analyzes this influence and how it can contribute to their solutions.The study investigates the mathematical relation between them and presents an exact formula relating to the geodetic height and the ellipse equation.In addition,a heuristic formula for the relation between the geodetic height and the ellipse equation is proposed,which is independent of the geodetic latitude and has a relative accuracy better than 99.9 %.The calculation is stable,and the cost is low.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10971226, 91130013, and 11001270)the National Basic Research Program of China (Grant No. 2009CB723802)
文摘We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler method for temporal discretization. It is verified that the proposed method has corresponding discrete multi-symplectic conservation laws. Numerical simulations indicate that the proposed scheme is characterized by excellent conservation.
基金supported by the National Natural Science Foundation of China (Grant Nos.10735030 and 90718041)Shanghai Leading Academic Discipline Project,China (Grant No.B412)Program for Changjiang Scholars and Innovative Research Team in University,China (Grant No.IRT0734)
文摘By means of the classical method, we investigate the (3+1)-dimensional Zakharov-Kuznetsov equation. The symmetry group of the (3+1)-dimensional Zakharov-Kuznetsov equation is studied first and the theorem of group invariant solutions is constructed. Then using the associated vector fields of the obtained symmetry, we give the one-, two-, and three-parameter optimal systems of group-invariant solutions. Based on the optimal system, we derive the reductions and some new solutions of the (3+1)-dimensional Zakharov-Kuznetsov equation.
基金The project supported by Natural Science Foundation of Shandong Province of China under Grant 2004 zx 16The authors would like to thank professor Bai Cheng-Lin and the referees for their valuable advices.
文摘his paper studies the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation using the (G'/G)-expand method, we obtain many new explicit solutions of the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation, which include hyperbolic function solutions, trigonometric function solutions and rational function solutions and so on.
文摘The modified Zakharov-Kuznetsov equation with the initial value problem is studied numerically by means of homotopy perturbation method. The analytical approximate solutions of the modified Zakharov-Kuznetsov equation are obtained. Choosing the form of the initial value, the single solitary wave, two solitary waves and rational solutions are presented, some of which are shown by the plots.
文摘In this paper, we get many new analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov Kuznetsov (mKDV-ZK) equation by means of a new approach namely method of undetermined coefficients based on a fractional complex transform. These solutions have physics meanings in natural sciences. This method can be used to other nonlinear fractional differential equations.
基金Supported by the National Natural Science Foundation of China under Grant No.10974160
文摘The exact solutions of the generalized (2+1)-dimensional nonlinear Zakharov-Kuznetsov (Z-K) equationare explored by the method of the improved generalized auxiliary differential equation.Many explicit analytic solutionsof the Z-K equation are obtained.The methods used to solve the Z-K equation can be employed in further work toestablish new solutions for other nonlinear partial differential equations.
文摘The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathematical community.Thus,this paper purveys an analytical study of a generalized extended(2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering.The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model.We further reduce the equation using the subalgebras obtained.Besides,more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique.Consequently,we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright,dark,periodic(cnoidal and snoidal),compact-type as well as singular solitons.The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time,are outlined with suitable diagrams.This can be of interest to oceanographers and ocean engineers for future analysis.Furthermore,to visualize the dynamics of the results found,we present the graphic display of each of the solutions.Conclusively,we construct conservation laws of the understudy equation via the application of Noether’s theorem.
文摘The aim of this paper is to obtain the approximate analytical solution of a fractional Zakharov-Kuznetsov equation by using homotopy perturbation method (HPM). The fractional derivatives are described in the Caputo sense. Several examples are given and the results are compared to exact solutions. The results reveal that the method is very effective and simple.
文摘The application of the q-homotopy analysis Shehu transform method(q-HAShTM)to discover the esti-mated solution of fractional Zakharov-Kuznetsov equations is investigated in this study.In the presence of a uniform magnetic field,the Zakharov-Kuznetsov equations regulate the behaviour of nonlinear acoustic waves in a plasma containing cold ions and hot isothermal electrons.The q-HAShTM is a stable analytical method that combines homotopy analysis and the Shehu transform.This q-homotopy investigation Shehu transform is a constructive method that leads to the Zakharov-Kuznetsov equations,which regulate the propagation of nonlinear ion-acoustic waves in a plasma.It is a more semi-analytical method for adjust-ing and controlling the convergence region of the series solution and overcoming some of the homotopy analysis method’s limitations.
基金supported by the National Natural Science Foundation of China (Grant Nos. 11931017 and 12071447)。
文摘Three modified sine-Hilbert(sH)-type equations, i.e., the modified sH equation, the modified damped sH equation, and the modified nonlinear dissipative system, are proposed, and their bilinear forms are provided.Based on these bilinear equations, some exact solutions to the three modified equations are derived.
基金supported by the National Natural Science Foundation of China(Grant Nos.12175111 and 12235007)the K.C.Wong Magna Fund in Ningbo University。
文摘By the modifying loss function MSE and training area of physics-informed neural networks(PINNs),we propose a neural networks model,namely prior-information PINNs(PIPINNs).We demonstrate the advantages of PIPINNs by simulating Ai-and Bi-soliton solutions of the cylindrical Korteweg-de Vries(cKdV)equation.
文摘In this paper, the matrix Riccati equation is considered. There is no general way for solving the matrix Riccati equation despite the many fields to which it applies. While scalar Riccati equation has been studied thoroughly, matrix Riccati equation of which scalar Riccati equations is a particular case, is much less investigated. This article proposes a change of variable that allows to find explicit solution of the Matrix Riccati equation. We then apply this solution to Optimal Control.
基金supported by the NSFC(12101012)the PhD Scientific Research Start-up Foundation of Anhui Normal University.Zeng’s research was supported by the NSFC(11961160716,11871054,12131017).
文摘In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary L^(2) weighted estimates.
基金Supported by the National Natural Science Foundation of China(12261023,11861023)the Foundation of Science and Technology project of Guizhou Province of China([2018]5769-05)。
文摘On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].
基金Supported by the National Natural Science Foundation of China(11771216)the Key Research and Development Program of Jiangsu Province(Social Development)(BE2019725)the Qing Lan Project of Jiangsu Province。
文摘This paper is concerned with a third order in time linear Moore-Gibson-Thompson equation which describes the acoustic velocity potential in ultrasound wave program.Influenced by the work of Kaltenbacher,Lasiecka and Marchand(Control Cybernet.2011,40:971-988),we establish an observability inequality of the conservative problem,and then discuss the equivalence between the exponential stabilization of a dissipative system and the internal observational inequality of the corresponding conservative system.
基金Project supported by the National Natural Science Foundation of China (Grant No. 12161061)the Fundamental Research Funds for the Inner Mongolia University of Finance and Economics (Grant No. NCYWT23036)+2 种基金the Young Innovative and Entrepreneurial Talents of the Inner Mongolia Grassland Talents Project in 2022,Autonomous Region “Five Major Tasks” Research Special Project for the Inner Mongolia University of Finance and Economics in 2024 (Grant No. NCXWD2422)High Quality Research Achievement Cultivation Fund for the Inner Mongolia University of Finance and Economics in 2024 (Grant No. GZCG2426)the Talent Development Fund of Inner Mongolia Autonomous Region, China。
文摘Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.
文摘The conversion of the cartesian coordinates of a point to its geodetic equivalent coordinates in reference to the geodetic ellipsoid is one of the main challenges in geodesy.The ellipse equation in the meridian plane significantly influences the value of the geodetic coordinates.This research analyzes this influence and how it can contribute to their solutions.The study investigates the mathematical relation between them and presents an exact formula relating to the geodetic height and the ellipse equation.In addition,a heuristic formula for the relation between the geodetic height and the ellipse equation is proposed,which is independent of the geodetic latitude and has a relative accuracy better than 99.9 %.The calculation is stable,and the cost is low.