Characteristic solutions of scalar newton equations in one space dimension

Newton equations are dynamical systems on the space of fields. The solutions of a given equation which are curves of characteristic fields for its force are planar and have constant angular momentum. Separable solutions are characteristic with angular momentum equal to zero. A Newton equation is separable if and only if its characteristic equation is homogeneous. Separable equations correspond to invariants of homogeneous ordinary differential equations, and those associated with a given homogenous equation correspond to its generalized dilation symmetries. A Newton equation is compatible with the characteristic condition if and only if its characteristic equation is linear. Such equations correspond to invariants of linear ordinary differential equations. Those associated with a given linear equation correspond to the central force problems on its solution space. Regardless of compatibility, any Newton equation with a plane of characteristic fields has non-separable characteristic solutions.     

Integration of third order ordinary differential equations possessing two-parameter symmetry group by Lie’s method

The solution of a class of third order ordinary differential equations possessing two parameter Lie symmetry group is obtained by group theoretic means. It is shown that reduction to quadratures is possible according to two scenarios: (1) if upon first reduction of order the obtained second order ordinary differential equation besides the inherited point symmetry acquires at least one more new point symmetry (possibly a hidden symmetry of Type II). (2) First, reduction paths of the fourth order differential equations with four parameter symmetry group leading to the first order equation possessing one known (inherited) symmetry are constructed. Then, reduction paths along which a third order equation possessing two-parameter symmetry group appears are singled out and followed until a first order equation possessing one known (inherited) symmetry are obtained. The method uses conditions for preservation, disappearance and reappearance of point symmetries.     

A unified compatibility method for exact solutions of non-linear flow models of Newtonian and non-Newtonian fluids

Criteria are established for higher order ordinary differential equations to be compatible with lower order ordinary differential equations. Necessary and sufficient compatibility conditions are derived which can be used to construct exact solutions of higher order ordinary differential equations subject to lower order equations. We provide the connection to generalized groups through conditional symmetries. Using this approach of compatibility and generalized groups, new exact solutions of non-linear flow problems arising in the study of Newtonian and non-Newtonian fluids are derived. The ansatz approach for obtaining exact solutions for non-linear flow models of Newtonian and non-Newtonian fluids is unified with the application of the compatibility and generalized group criteria.     

Painlevé analysis and invariant solutions of generalized fifth-order nonlinear integrable equation

In present work, new form of generalized fifth-order nonlinear integrable equation has been investigated by locating movable critical points with aid of Painlevé analysis and it has been found that this equation passes Painlevé test for \(\alpha =\beta \) which implies affirmation toward the complete integrability. Lie symmetry analysis is implemented to obtain the infinitesimals of the group of transformations of underlying equation, which has been further pre-owned to furnish reduced ordinary differential equations. These are then used to establish new abundant exact group-invariant solutions involving various arbitrary constants in a uniform manner.     

Isospectral Euler-Bernoulli beams via factorization and the Lie method

We obtain isospectral Euler-Bernoulli beams by using factorization and Lie symmetry techniques. The canonical Euler-Bernoulli beam operator is factorized as the product of a second-order linear differential operator and its adjoint. The factors are then reversed to obtain isospectral beams. The factorization is possible provided the coefficients of the factors satisfy a system of non-linear ordinary differential equations. The uncoupling of this system yields a single non-linear third-order ordinary differential equation. This ordinary differential equation, called the principal equation, is analyzed, reduced or solved using Lie group methods. We show that the principal equation may admit a one-dimensional or three-dimensional symmetry Lie algebra. When the principal system admits a unique symmetry, the best we can do is to depress its order by one. We obtain a one-parameter family of invariant solutions in this case. The maximally symmetric case is shown to be isomorphic to a Chazy equation which is solved in closed form to derive the general solution of the principal equation. The transformations connecting isospectral pairs are obtained by numerically solving systems of ordinary differential equations using the fourth-order Runge-Kutta method.     

Group classification and exact solutions of a radially symmetric porous-medium equation

In this paper we perform a group classification for the generalized radial porous-medium equation. We also classify symmetry reductions of the equation to first- or second-order ordinary differential equations (ODEs) and hence construct invariant solutions in a systematic manner. We show that the reduced second-order equations are invariant under either a two-parameter or one-parameter Lie groups. In the first case, they are completely integrated by a pair of quadratures. In the latter, they are often reduced to first-order ODEs of Abel type.     

具有可积微分约束的力学系统的Lie对称性

研究具有可积微分约束的力学系统的Ｌｉｅ对称性与守恒量。采用两种方法：一是用不可积微分约束系统的方法;另一是用积分后降阶系统的方法,研究两种方法之间的关系。     

Theoretical and Numerical Investigation of Gas Lubrication Processes on the Basis of Aerodynamic Equations

The dimensionless parameters of the complete system of Navier-Stokes equations of a compressible gas are estimated with reference to a typical gas bearing. It is found that the three-dimensional compressible boundary layer equations should be used as the determining equations for describing gas lubrication processes. After introducing certain assumptions with respect to the dimensionless parameters in the determining equations, an equation for the pressure, the generalized Reynolds equation, is obtained.Use of the spectral method of analysis makes it possible to transform the generalized Reynolds equation into a system of ordinary differential equations. An analytic solution of the entire boundary value problem is obtained for a journal bearing with fairly small eccentricity. By comparing the numerical results obtained using both the solution of the generalized Reynolds equation and the traditional theory it is possible to estimate the effect of the inertia forces, dissipation processes, and heat transfer.     

Existence,Uniqueness, and Stability of Generalized Traveling Waves in Time Dependent Monostable Equations

The current paper is devoted to the study of traveling wave solutions of spatially homogeneous monostable reaction diffusion equations with ergodic or recurrent time dependence, which includes periodic and almost periodic time dependence as special cases. Such an equation has two spatially homogeneous and time recurrent solutions with one of them being stable and the other being unstable. Traveling wave solutions are a type of entire solutions connecting the two spatially homogeneous and time recurrent solutions. Recently, the author of the current paper proved that a spatially homogeneous time almost periodic monostable equation has a spreading speed in any given direction. This result can be easily extended to monostable equations with recurrent time dependence. In this paper, we introduce generalized traveling wave solutions for time recurrent monostable equations and show the existence of such solutions in any given direction with average propagating speed greater than or equal to the spreading speed in that direction and non-existence of such solutions of slower average propagating speed. We also show the uniqueness and stability of generalized traveling wave solutions in any given direction with average propagating speed greater than the spreading speed in that direction. Moreover, we show that a generalized traveling wave solution in a given direction with average propagating speed greater than the spreading speed in that direction is unique ergodic in the sense that its wave profile and wave speed are unique ergodic, and if the time dependence of the monostable equation is almost periodic, it is almost periodic in the sense that its wave profile and wave speed are almost periodic.     

Construction of approximate analytical solutions to strongly nonlinear damped oscillators

An analytical approximate method for strongly nonlinear damped oscillators is proposed. By introducing phase and amplitude of oscillation as well as a bookkeeping parameter, we rewrite the governing equation into a partial differential equation with solution being a periodic function of the phase. Based on combination of the Newton’s method with the harmonic balance method, the partial differential equation is transformed into a set of linear ordinary differential equations in terms of harmonic coefficients, which can further be converted into systems of linear algebraic equations by using the bookkeeping parameter expansion. Only a few iterations can provide very accurate approximate analytical solutions even if the nonlinearity and damping are significant. The method can be applied to general oscillators with odd nonlinearities as well as even ones even without linear restoring force. Three examples are presented to illustrate the usefulness and effectiveness of the proposed method.     

Adapting simulation codes to compute unstable solutions and bifurcation behaviour

Simulation codes for solving large systems of ordinary differential equations suffer from the disadvantage that bifurcation‐theoretic results about the underlying dynamical system cannot be obtained from them easily, if at all. Bifurcation behaviour typically can be inferred only after significant computational effort, and even then the exact location and nature of the bifurcation cannot always be determined definitively. By incorporating relatively minor changes to an existing simulation code for the Taylor–Couette problem, specifically, by implementing the Newton–Picard method, we have developed a computational structure that enables us to overcome some of the inherent limitations of the simulation code and begin to perform bifurcation‐theoretic tasks. While a complete bifurcation picture was not developed, three distinct solution branches of the Taylor–Couette problem were analysed. These branches exhibit a wide variety of behaviours, including Hopf bifurcation points, symmetry‐breaking bifurcation points, turning points and bifurcation to motion on a torus. Unstable equilibrium and time‐periodic solutions were also computed. Copyright © 2007 John Wiley & Sons, Ltd.     

Analytical solutions for some nonlinear evolution equations

IntroductionItiswell_knownthatmanyimportantdynamicsprocessescanbedescribedbyspecificnonlinearpartialdifferentialequations .Whenanonlinearpartialdifferentialequationisusedtodescribeaphysicalparameterthatshowssomekindsofpropagationoraggregationproperties,oneofthemostimportantphysicalmotivationsistosolvethepartialdifferentialequationwithacertaintypeoftravellingwavesolution .Inthepastseveraldecades,therehavebeenmanyattemptsinthisfieldbothbymathematiciansandphysicists[1]- [16 ],however,duetothecomp…     

A symmetry analysis of some classes of evolutionary nonlinear (2+1)-diffusion equations with variable diffusivity

In this paper the (2+1)-nonlinear diffusion equation u _t ?div(f(u)grad u)=0 with variable diffusivity is considered. Using the Lie method, a complete symmetry classification of the equation is presented. Reductions, via two-dimensional Lie subalgebras of the equation, to first- or second-order ordinary differential equations are given. In a few interesting cases exact solutions are presented.     

FD-PINN:频域物理信息神经网络

物理信息神经网络(physics-informed neural network, PINN)是将模型方程编码到神经网络中,使网络在逼近定解条件或观测数据的同时最小化方程残差,实现偏微分方程求解.该方法虽然具有无需网格划分、易于融合观测数据等优势,但目前仍存在训练成本高、求解精度低等局限性.文章提出频域物理信息神经网络(frequency domain physics-informed neural network, FD-PINN),通过从周期性空间维度对偏微分方程进行离散傅里叶变换,偏微分方程被退化为用于约束FD-PINN的频域中维度更低的微分方程组,该方程组内各方程不仅具有更少的自变量,并且求解难度更低.因此,与使用原始偏微分方程作为约束的经典PINN相比, FD-PINN实现了输入样本数目和优化难度的降低,能够在降低训练成本的同时提升求解精度.热传导方程、速度势方程和Burgers方程的求解结果表明, FD-PINN普遍将求解误差降低1～2个数量级,同时也将训练效率提升6～20倍.     

Integral transform analysis of natural convection in porous enclosures

A hybrid numerical-analytical solution for steady-state natural convection in a porous cavity is proposed, based on application of the ideas in the generalized integral transform technique. The integral transformation process reduces the original coupled partial differential equations, for temperature and stream function, into an infinite system of non-linear ordinary differential equations for the transformed potentials, which is adaptively truncated and numerically solved through well-established algorithms. The approach is applied to a vertical rectangular enclosure subjected to uniform internal heat generation. The convergence characteristics of the explicit inversion formulae are illustrated and critical comparisons with previously reported purely numerical solutions are performed.     

The displacement solution of conical shell and its application

In this paper,the displacement solution method of the conical shell is presented.Fromthe differential equations in displacement form of conical shell and by introducing adisplacement function,U(s,θ),the differential equations are changed into an eight-ordersoluble partial differential equation about the displacement function U(s,θ)in which thecoefficients are variable.At the same time,the expressions of the displacement and internalforce components of the shell are also given by the displacement function.As special casesof this paper,the displacement function introduced by V.Z.Vlasov in circular cylindricalshell,the basic equation of the cylindrical shell and that of the circular plate are directlyderived.Under the arbitrary loads and boundary conditions,the general bending problem of theconical shell is reduced to finding the displacement function U(s,θ),and the generalsolution of the governing equation is obtained in generalized hypergeometric function,Forthe axisymmetric bending deformation of the     

Stability analysis of a fractional viscoelastic plate strip in supersonic flow under axial loading

The stability of a viscoelastic plate strip, subjected to an axial load with the Kelvin–Voigt fractional order constitutive relationship is studied. Based on the classical plate theory, the structural formulation of the plate is obtained by using the Newton’s second law and the aerodynamic force due to the fluid flow is evaluated by piston theory. The Galerkin method is employed to discretize the equation of motion into a set of ordinary differential equations. To determine the stability margin of plate the obtained set of ordinary differential equations are solved using the Laplace transform method. The effects of variation of the governing parameters such as axial force, retardation time, fractional order and boundary conditions on the stability margin of fractional viscoelastic panel are investigated and finally some conclusions are outlined.     

Sharp-Interface Nematic–Isotropic Phase Transitions without Flow

We derive a supplemental evolution equation for an interface between the nematic and isotropic phases of a liquid crystal when flow is neglected. Our approach is based on the notion of configurational force. As an application, we study the behavior of a spherical isotropic drop surrounded by a radially oriented nematic phase: our supplemental evolution equation then reduces to a simple ordinary differential equation admitting a closed-form solution. In addition to describing many features of isotropic-to-nematic phase transitions, this simplified model yields insight concerning the occurrence and stability of isotropic cores for hedgehog defects in liquid crystals.     

Quasi-similar solutions for blast waves with internal heat transfer effects

The point explosion problem with internal heat transfer effects taken into account is analysed. The classical inviscid solutions to this problem yield a non-physical phenomenon of infinite temperature and zero density at the center of explosion for all times. With heat transfer fluxes considered, the solution near the center of symmetry is improved and finite values for temperature are obtained. The non-self-similar solution of the problem is based on the quasi-similar approximate technique which reduces the non-linear partial differential equations governing the problem to ordinary differential ones. However, this formulation yields a two-point boundary-value problem. To facilitate the integration, the flow field is first divided into two regions: an outer inviscid region and an inner region where dissipation effects are manifested. This results in two sets of ordinary differential equations expressing the conservation equations in the inner and outer regions which are then solved and matched together to yield the composite solution. Secondly, the problem is then transformed into an initial-value one. Using the results of the composite solution, the governing equations can be integrated directly from the center until the shock front. The structure of the non-self-similar flow fields with internal heat transfer effects is then fully determined for specified values of the heat transfer parameters.     

Propagation of discontinuities against a static background

The solution of the ideal gasdynamic equations describing propagation of a shock wave initiated, for example, by the motion of a piston against an inhomogeneous static background is considered. The solution is constructed in the form of Taylor series in a special time variable which is equal to zero on the shock wave. In the case of weak shock waves divergence of the series serves as the constraint for such an approach. Then the solution is constructed by linearizing the equations about the solution with a weak discontinuity. In the case of a given background the last solution can be always found exactly by solving successively a set of transport equations, all these equations are reduced to linear ordinary differential equations. The presentation begins from the one-dimensional solutions with plane waves and ends by discussion of spatial problems.