Lie symmetry and conserved quantity of a system of first-order differential equations

This paper focuses on studying the Lie symmetry and a conserved quantity of a system of first-order differential equations. The determining equations of the Lie symmetry for a system of first-order differential equations, from which a kind of conserved quantity is deduced, are presented. And their general conclusion is applied to a Hamilton system, a Birkhoff system and a generalized Hamilton system. Two examples are given to illustrate the application of the results.

First-Order Symmetry Energy Induced by Neutron-Proton Mass Difference

The 1st-order symmetry energy coefficient of nuclear matter induced merely by the neutron-proton （n p） mass difference is derived analytically, which turns out to be completely model-independent. Based on this result, （npDM） the 1st-order symmetry energy Esym,1 （A） of heavy nuclei such as 2~spb induced by the np mass difference is investigated with the help of a local density approximation combined with the Skyrme energy density functionals. Although /U（npDM） Esym,1 （A） is small compared with the second-order symmetry energy, it cannot be dropped simply for an accurate estimation of nuclear masses as it is still larger than the rms deviation given by some accurate mass formulas. It is therefore suggested that one perhaps needs to distinguish the neutron mass from the proton one in the construction of nuclear density funetionals.

Differential characteristic set algorithm for the complete symmetry classification of partial differential equations

In this paper, we present a differential polynomial characteristic set algorithm for the complete symmetry classification of partial differential equations （PDEs） with some parameters. It can make the solution to the complete symmetry classification problem for PDEs become direct and systematic. As an illustrative example, the complete potential symmetry classifications of nonlinear and linear wave equations with an arbitrary function parameter are presented. This is a new application of the differential form characteristic set algorithm, i.e., Wu＇s method, in differential equations.

Symmetry Classification of Partial Differential Equations Based on Wu's Method

Lie algorithm combined with differential form Wu's method is used to complete the symmetry classification of partial differential equations(PDEs)containing arbitrary parameter.This process can be reduced to solve a large system of determining equations,which seems rather difficult to solve,then the differential form Wu's method is used to decompose the determining equations into a series of equations,which are easy to solve.To illustrate the usefulness of this method,we apply it to some test problems,and the results show the performance of the present work.

Nonlinear amplitude versus angle inversion for transversely isotropic media with vertical symmetry axis using new weak anisotropy approximation equations

In VTI media,the conventional inversion methods based on the existing approximation formulas are difficult to accurately estimate the anisotropic parameters of reservoirs,even more so for unconventional reservoirs with strong seismic anisotropy.Theoretically,the above problems can be solved by utilizing the exact reflection coefficients equations.However,their complicated expression increases the difficulty in calculating the Jacobian matrix when applying them to the Bayesian deterministic inversion.Therefore,the new reduced approximation equations starting from the exact equations are derived here by linearizing the slowness expressions.The relatively simple form and satisfactory calculation accuracy make the reduced equations easy to apply for inversion while ensuring the accuracy of the inversion results.In addition,the blockiness constraint,which follows the differentiable Laplace distribution,is added to the prior model to improve contrasts between layers.Then,the concept of GLI and an iterative reweighted least-squares algorithm is combined to solve the objective function.Lastly,we obtain the iterative solution expression of the elastic parameters and anisotropy parameters and achieve nonlinear AVA inversion based on the reduced equations.The test results of synthetic data and field data show that the proposed method can accurately obtain the VTI parameters from prestack AVA seismic data.

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

Symmetry solutions of a nonlinear elastic wave equation with third-order anharmonic corrections

Lie symmetry method is applied to analyze a nonlinear elastic wave equation for longitudinal deformations with third-order anharmonic corrections to the elastic energy. Symmetry algebra is found and reductions to second-order ordinary differential equations （ODEs） are obtained through invariance under different symmetries. The reduced ODEs are further analyzed to obtain several exact solutions in an explicit form. It was observed in the literature that anharmonic corrections generally lead to solutions with time-dependent singularities in finite times singularities, we also obtain solutions which Along with solutions with time-dependent do not exhibit time-dependent singularities.

IMF production and symmetry energy in heavy ion collisions near Fermi energy

The symmetry energy at the time of the production of intermediate mass fragments(IMFs) is studied using experimentally observed IMF multiplicities combined with quantum statistical model calculations(QSM of Hahn and St cker).The ratios of difference in chemical potentials between neutrons and protons relative to the temperature,(μn-μp)/T,and the double ratio temperature,T,were extracted experimentally in the reactions of64,70Zn,64Ni+58,64Ni,112,124Sn,197Au,232Th at 40A MeV.The extracted(μn-μp)/TTscales linearly with δNN,where δNN is the asymmetry parameter,(N-Z)/A,of the emitting source and(μn-μp)/T=(11.1 1.4)δNN0.21 was derived.The experimentally extracted(μn-μp)/T and the double ratio temperatures are compared with those from the QSM calculations.The temperatures-,T,and densities,ρ,extracted from the(μn-μp)/T values agreed with those from the double ratio thermometer which used the yield ratios of d,t,h and α particles.However the two analyses of the differential chemical potential analysis and the initial temperature analysis end up almost identical relation between T and ρ.T=5.25±0.75 MeV is evaluated from the(μn-μp)/Tanalysis,but no density determination was possible.From the extracted T value,the symmetry energy coefficient Esym =14.6±3.5 MeV is determined for the emitting source of T=5.25±0.75 MeV.

Finite Element Approach for the Solution of First-Order Differential Equations

The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differential equation into a system of algebraic equations by application of the method of weighted residuals in conjunction with a finite element ansatz. However, this procedure is restricted to even-ordered differential equations and leads to symmetric system matrices as a key property of the finite element method. This paper aims in a generalization of the finite element method towards the solution of first-order differential equations. This is achieved by an approach which replaces the first-order derivative by fractional powers of operators making use of the square root of a Sturm-Liouville operator. The resulting procedure incorporates a finite element formulation and leads to a symmetric but dense system matrix. Finally, the scheme is applied to the barometric equation where the results are compared with the analytical solution and other numerical approaches. It turns out that the resulting numerical scheme shows excellent convergence properties.

Quasiperiodic Solutions for Nonlinear Differential Equations of Second Order with Symmetry

In this paper,we study the existence of quasi-periodic solutions and the bound- edness of solutions for a wide class nonlinear differential equations of second order.Using the KAM theorem of reversible systems and the theory of transformations,we obtain the existence of quasi-periodic solutions and the boundedness of solutions under some reasonable conditions.

Symmetries, Symmetry Reductions and Exact Solutions to the Generalized Nonlinear Fractional Wave Equations

In this paper, the Lie group classification method is performed on the fractional partial differential equation(FPDE), all of the point symmetries of the FPDEs are obtained. Then, the symmetry reductions and exact solutions to the fractional equations are presented, the compatibility of the symmetry analysis for the fractional and integer-order cases is verified. Especially, we reduce the FPDEs to the fractional ordinary differential equations(FODEs) in terms of the Erd′elyi-Kober(E-K) fractional operator method, and extend the power series method for investigating exact solutions to the FPDEs.

Comparative Studies between Picard’s and Taylor’s Methods of Numerical Solutions of First Ordinary Order Differential Equations Arising from Real-Life Problems

To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.

Existence of Many and Infinitely Many Periodic Solutions for Some Types of Differential Delay Equations

J.Kaplan and J.Yorke's method is extended to establish the exis- tence of many and infinitely many periodic solutions for the DDEs (t) =±f(x(t-1))±f(x(t-2))and (t)=±f(x(t-1).

LIE SYMMETRIES AND CONSERVED QUANTITIES OF ROTATIONAL RELATIVISTIC SYSTEMS

The Lie symmetries and conserved quantities of the rotational relativistic holonomic and nonholonomic systems were studied. By defining the infinitesimal transformations' generators and by using the invariance of the differential equations under the infinitesimal transformations, the determining equations of Lie symmetries for the rotational relativistic mechanical systems are established. The structure equations and the forms of conserved quantities are obtained. An example to illustrate the application of the results is given.

Hojman's theorem of the third-order ordinary differential equation

This paper extends Hojman＇s conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results.

Application of Differential Invariant Method for Solving the Electromagnetic Fields

We study the first integral and the solution of electromagnetic field by Lie symmetry technique and the differential invariant method.The definition and properties of differential invariants are introduced and the infinitesimal generators of Lie symmetries and the differential invariants of electromagnetic field are obtained.The first integral and the solution of electromagnetic field are given by the Lie symmetry technique and the differential invariants method.A typical example is presented to illustrate the application of our theoretical results.

An Effective Method for Seeking Conservation Laws of Partial Differential Equations

This paper introduces an effective method for seeking local conservation laws of general partial differential equations （PDEs）. The well-known variational principle does not involve in this method. Alternatively, the conservation laws can be derived from symmetries, which include t/he symmetries of t/he associated linearized equation of t/he PDEs, and the adjoint symmetries of the adjoint eqUation of the PDEs.

Lie symmetry analysis,explicit solutions,and conservation laws of the time-fractional Fisher equation in two-dimensional space

In these analyses,we consider the time-fractional Fisher equation in two-dimensional space.Through the use of the Riemann-Liouville derivative approach,the well-known Lie point symmetries of the utilized equation are derived.Herein,we overturn the fractional fisher model to a fractional differential equation of nonlinear type by considering its Lie point symmetries.The diminutive equation’s derivative is in the Erdélyi-Kober sense,whilst we use the technique of the power series to conclude explicit solutions for the diminutive equations for the first time.The conservation laws for the dominant equation are built using a novel conservation theorem.Several graphical countenances were utilized to award a visual performance of the obtained solutions.Finally,some concluding remarks and future recommendations are utilized.

Lie symmetry analysis and invariant solutions for(2+1) dimensional Bogoyavlensky-Konopelchenko equation with variable-coefficient in wave propagation

This work aims to present nonlinear models that arise in ocean engineering.There are many models of ocean waves that are present in nature.In shallow water,the linearization of the equations requires critical conditions on wave capacity than it make in deep water,and the strong nonlinear belongings are spotted.We use Lie symmetry analysis to obtain different types of soliton solutions like one,two,and three-soliton solutions in a(2+1)dimensional variable-coefficient Bogoyavlensky Konopelchenko(VCBK)equation that describes the interaction of a Riemann wave reproducing along the y-axis and a long wave reproducing along the x-axis in engineering and science.We use the Lie symmetry analysis then the integrating factor method to obtain new solutions of the VCBK equation.To demonstrate the physical meaning of the solutions obtained by the presented techniques,the graphical performance has been demonstrated with some values.The presented equation has fewer dimensions and is reduced to ordinary differential equations using the Lie symmetry technique.

Linear Quadratic Optimal Control for Systems Governed by First-Order Hyperbolic Partial Differential Equations

This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation.