二元一阶常系数线性微分方程组的新解法

对于二元一阶常系数线性微分方程组:x′=Ax+f(t),引入特征根方程|A-λE|=0的特征行向量K=(k_1,k_2)(其中K满足:K^T(A-λE)=0)概念,将二元一阶常系数线性微分方程组,化为二元一次代数线性方程:k_1x_1+k_2x_2=C_1e^(λt)+e^(λt)∫(k_1f_1+k_2f_2)e^(-λt)dt,并结合代数线性方程和一阶线性微分方程的理论,给出原微分方程组的解.

常系数线性微分方程解法研究的新认识

对于常系数线性微分方程L(x)=f(t),通过变换将其化为常系数线性微分方程组x'=Ax+f(t).分别就常系数线性微分方程的特征根为单根和重根情况,探求函数f(t)构成的可积条件,并对于可解的常系数线性微分方程给出其解,对于不可解的常系数线性微分方程进行讨论.

常系数线性微分方程组解结构的再认识

对于常系数线性微分方程组:dx/dt=Ax+f (t)(A是n阶实常数矩阵),引入特征根方程A-||λE=0的特征行向量K=(k_1,k_2,?,k_n)(其中K满足:K(A-λE)=0)概念,将n元一阶常系数线性微分方程组化为一阶线性微分方程形式.

三元一阶常系数线性微分方程组的解构造

对三元一阶线性非齐次微分方程组x'=Ax+f(t)的解法进行深入讨论,提出"行向量"概念,并且利用该概念给出其方程组解的本质结构.

二元一阶常系数线性微分方程组的本质解法

对于二元一阶常系数线性微分方程组:x′=Ax+f(t),引入特征根方程|A-λE|=0的特征行向量K=(k_1,k_2)(其中K满足:K(A-λE)=0)概念,将二元一阶常系数线性微分方程组,化为二元一次代数线性方程形式:(K_2x_2)′=λ(K_2x_2)+(K_2f),(K_1x_1)′=λ(K_1x_1)+K_1x_2+K_1f,从中给出原微分方程组的解.

Applications of Computer Algebra in Solving Nonlinear Evolution Equations

With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a result, their abundant new soliton-like solutions and period form solutions are found.

Computer Algebra and Solutions to the Karamoto-Sivashinsky Equation

The method of Riccati equation is extended for constructing travelling wave solutions of nonlinear partial differential equations. It is applied to solve the Karamoto-Sivashinsky equation and then its more new explicit solutions have been obtained. From the results given in this paper, one can see the computer algebra plays an important role in this procedure.

Lie Algebras for Constructing Nonlinear Integrable Couplings

Two new explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Giachetti- Johnson （G J） hierarchy and the Yang hierarchy are obtained, respectively. By employing the variational identity their ttamiltonian structures are also generated. The approach presented in the paper can also provide nonlinear integrable couplings of other soliton hierarchies of evolution equations.

A Study for Obtaining New and More General Solutions of Special-Type Nonlinear Equation

The generalized algebraic method with symbolic computation is extended to some special-type nonlinear equations for constructing a series of new and more general travelling wave solutions in terms of special functions. Such equations cannot be directly dealt with by the method and require some kinds of pre-processing techniques. It is shown that soliton solutions and triangular periodic solutions can be established as the limits of the Jacobi doubly periodic wave solutions.

A New General Algebraic Method and Its Application to Shallow Long Wave Approximate Equations

A new general algebraic method is presented to uniformly construct a series of exact solutions for nonlinear evolution equations （NLEEs）. For illustration, we apply the new method to shallow long wave approximate equations and successfully obtain abundant new exact solutions, which include rational solitary wave solutions and rational triangular periodic wave solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.

Inverse problem in linear algebra

With the increasingly widespread application of linear algebra theory, and in its opposite direction is not enough emphasis, linear algebra, several important points： matrix, determinant, linear equations, linear transformations, matrix keratosis and other anti-deepening understanding of the basics and improve the comprehensive ability to solve problems.

Aerodynamic Performances of Wind Turbine Airfoils Using a Panel Method

One of the key features of Laplace＇s Equation is the property that allows the equation governing the flow field to be converted from a 3D problem throughout the field to a 2D problem for finding the potential on the surface. The solution is then found using this property by distributing ＂singularities＂ of unknown strength over discretized portions of the surface： panels. Hence the flow field solution is found by representing the surface by a number of panels, and solving a linear set of algebraic equations to determine the unknown strengths of the singularities. In this paper a Hess-Smith Panel Method is then used to examine the aerodynamics of NACA 4412 and NACA 23015 wind turbine airfoils. The lift coefficient and the pressure distribution are predicted and compared with experimental result for low Reynolds number. Results show a good agreement with experimental data.

Studies of Rigid Rotor-Rigid Surface Scattering in Dynamical Lie Algebraic Method

The dynamical Lie algebraic method is used for the description of statistical mechanics of rotationally inelastic molecule-surface scattering. It can give the time-evolution operators about the low power of and by solving a set of coupled nonlinear differential equations. For considering the contribution of the high power of and , we use the Magnus formula. Thus, with the time-evolution operators we can get the statistical average values of the measurable quantities in terms of the density operator formalism in statistical mechanics. The method is applied to the scattering of (rigid rotor) by a flat, rigid surface to illustrate its general procedure. The results demonstrate that the method is useful for describing the statistical dynamics of gas-surface scattering.

Variational algorithms for linear algebra

Quantum algorithms have been developed for efficiently solving linear algebra tasks.However,they generally require deep circuits and hence universal fault-tolerant quantum computers.In this work,we propose variational algorithms for linear algebra tasks that are compatible with noisy intermediate-scale quantum devices.We show that the solutions of linear systems of equations and matrix–vector multiplications can be translated as the ground states of the constructed Hamiltonians.Based on the variational quantum algorithms,we introduce Hamiltonian morphing together with an adaptive ans?tz for efficiently finding the ground state,and show the solution verification.Our algorithms are especially suitable for linear algebra problems with sparse matrices,and have wide applications in machine learning and optimisation problems.The algorithm for matrix multiplications can be also used for Hamiltonian simulation and open system simulation.We evaluate the cost and effectiveness of our algorithm through numerical simulations for solving linear systems of equations.We implement the algorithm on the IBM quantum cloud device with a high solution fidelity of 99.95%.

Optimization of formation for multi-agent systems based on LQR

In this paper,three optimal linear formation control algorithms are proposed for first-order linear multiagent systems from a linear quadratic regulator(LQR) perspective with cost functions consisting of both interaction energy cost and individual energy cost,because both the collective ob ject(such as formation or consensus) and the individual goal of each agent are very important for the overall system.First,we propose the optimal formation algorithm for first-order multi-agent systems without initial physical couplings.The optimal control parameter matrix of the algorithm is the solution to an algebraic Riccati equation(ARE).It is shown that the matrix is the sum of a Laplacian matrix and a positive definite diagonal matrix.Next,for physically interconnected multi-agent systems,the optimal formation algorithm is presented,and the corresponding parameter matrix is given from the solution to a group of quadratic equations with one unknown.Finally,if the communication topology between agents is fixed,the local feedback gain is obtained from the solution to a quadratic equation with one unknown.The equation is derived from the derivative of the cost function with respect to the local feedback gain.Numerical examples are provided to validate the effectiveness of the proposed approaches and to illustrate the geometrical performances of multi-agent systems.

A Corresponding Lie Algebra of a Reductive homogeneous Group and Its Applications

With the help of a Lie algebra of a reductive homogeneous space G/K, where G is a Lie group and K is a resulting isotropy group, we introduce a Lax pair for which an expanding （2＋1）-dimensional integrable hierarchy is obtained by applying the binormial-residue representation （BRR） method, whose ttamiltonian structure is derived from the trace identity for deducing （2＋1）-dimensional integrable hierarchies, which was proposed by Tu, et al. We further consider some reductions of the expanding integrable hierarchy obtained in the paper. The first reduction is just right the （2＋1 ）-dimensionai AKNS hierarchy, the second-type reduction reveals an integrable coupling of the （2＋1）-dimensional AKNS equation （also called the Davey-Stewartson hierarchy）, a kind of （2＋1）-dimensionai Sehr6dinger equation, which was once reobtained by Tu, Feng and Zhang. It is interesting that a new （2＋1）-dimensional integrable nonlinear coupled equation is generated from the reduction of the part of the （2＋1）-dimensional integrable coupling, which is further reduced to the standard （2＋1）-dimensionaJ diffusion equation along with a parameter. In addition, the well-known （1＋1）-dimensional AKNS hierarchy, the （1＋1）-dimensional nonlinear Schr6dinger equation are all special cases of the （2＋1）-dimensional expanding integrable hierarchy. Finally, we discuss a few discrete difference equations of the diffusion equation whose stabilities are analyzed by making use of the yon Neumann condition and the Fourier method. Some numerical solutions of a special stationary initial value problem of the （2＋1）-dimensional diffusion equation are obtained and the resulting convergence and estimation formula are investigated.

On the universal third order Stokes wave solution

This paper presents a universal third-order Stokes solution with uniform current. This solution is derived on the basis of potential theory by expanding the free surface and potential function in Fourier series and determining the Fourier coefficients by solving a set of nonlinear algebraic equations through the Taylor expansion and perturbation method. The universal solution is expressed upon the still water depth with the still water level as datum and retains a global perturbation parameter. The wave set-up term generated by the self-interaction of oscillatory waves is explicitly included in the free surface function. With the use of different definitions for the wave celerity, different water levels as the datum, different non-dimensional variables as the perturbation parameter, and different treatments for the total head, the universal solution can be reduced to the existing various Stokes solutions, thus explaining the reasons and the physical significance of different non-periodic terms in them, such as the positive or negative constant term in the free surface expression and the time-or space-proportional term in the potential function.

BIFURCATION OF PERIODIC ORBITS OF A THREE-DIMENSIONAL SYSTEM

Consider a three-dimensional system having an invariant surface. By using bifurca- tion techniques and analyzing the solutions of bifurcation equations, the authors study the spacial bifurcation phenomena of a k multiple closed orbit in the invariant surface. The su?cient conditions of the existence of many closed orbits bifurcate from the k multiple closed orbit are obtained.

Probability density function solution to nonlinear ship roll motion excited by external Poisson white noise

The stationary probability density function (PDF) solution to nonlinear ship roll motion excited by Poisson white noise is analyzed. Subjected to such random excitation, the joint PDF solution to the roll angle and angular velocity is governed by the generalized Fokker-Planck-Kolmogorov (FPK) equation. To solve this equation, the exponential-polynomial closure (EPC) method is adopted. With the EPC method, the PDF solution is assumed to be an exponential-polynomial function of state variables. Special measure is taken such that the generalized FPK equation is satisfied in the average sense of integration with the assumed PDF. The problem of determining the unknown parameters in the approximate PDF finally results in solving simultaneous nonlinear algebraic equations. Both slight and high nonlinearities are considered in the illustrative examples. The analysis shows that when a second-order polynomial is taken, the result of the EPC method is the same as the one given by the equivalent linearization (EQL) method. The EQL results differ significantly from the simulated results in the case of high nonlinearity. When a fourth-order or sixth-order polynomial is taken, the results of the EPC method agree well with the simulated ones, especially in the tail regions of the PDF. This agreement is observed in the cases of both slight and high nonlinearities.

Compatible Lie Bialgebras

A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket. We construct a bialgebra theory of compatible Lie Mgebras as an analogue of a piiLie bialgebra. They can also be regarded as a ＂compatible version＂ of Lie bialgebras, that is, a pair of Lie biaJgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra. Many properties of compatible Lie bialgebras as the ＂compatible version＂ of the corresponding properties of Lie biaJgebras are presented. In particular, there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang-Baxter equation in compatible Lie algebras as a combination of two classical Yang-Baxter equations in lAe algebras. FUrthermore, a notion of compatible pre-Lie Mgebra is introduced with an interpretation of its close relation with the classical Yang-Baxter equation in compatible Lie a/gebras which leads to a construction of the solutions of the latter. As a byproduct, the compatible Lie bialgebras fit into the framework to construct non-constant solutions of the classical Yang-Baxter equation given by Golubchik and Sokolov.