Mikusinski’s operators solution of three-order linear difference equation with variable coefficients

This paper proceeds the papers [3] [4], we make use of the idea of the variable number operators and some concepts and conclusions of the shifting operators series with variable coefficients in the operational field of Mikusinski, it is devoted to the solution of the general three-order linear difference equation with variable coefficients, and it is also devoted to the better solution formula for the some special three-order linear difference equations with variable coefficients: in addition, we try to provide the idea and method for realizing solution of the more than three-order linear difference equation with variable coefficients. Project Supported by the Science Foundation of Anhui Province     

Differentiator series solution of linear differential ordinary equation

１Ｄｉｆｅｒｅｎｔｉａｔｏｒ,ＩｎｖｅｒｓＯｐｅｒａｔｏｒａｎｄＴｈｅｉｒＰｒｏｐｅｒｔｉｅｓ１．１ＤｉｆｅｒｅｎｔｉａｔｏｒａｎｄｉｎｖｅｒｓｏｐｅｒａｔｏｒＳｕｐｐｏｓｅｔｈｅｌｉｎｅａｒｏｒｄｉｎａｒｙｄｉｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｏｆ...     

A semi-analysis method of differential equations with variable coefficients under complicated boundary conditions

IntroductionInthemostcases,asfarasthesolutiontothedifferentialequationwithvariablecoefficientsisconcerned ,therehasbeennoanysatisfactoryanswer.Forthisreason ,thescholarsbothathomeandabroadhavedonesomeresearchjustasindicatedinRefs.[1 ] ,[2 ]and [3 ] ,etc.Theauthorsinthispaperhaveusedthefiniteelementmodelandthemetricfunctiontheoryincombinationwithadjustableparametermodelwithvariablecoefficientstosimplifythesolutiontothedifferentialequationwithvariablecoefficientsintothefeaturevalueaswellasthefea…     

Instability of solution for the fourth order linear differential equation with varied coefficient

In this paper,we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient,at least one of the characteristic roots of which has positive real part,by means of Liapunov’s second method.     

Exact solutions of a KdV equation with variable coefficients via Exp-function method

In this paper, the Exp-function method is used to obtain generalized solitonary solutions and periodic solutions of a KdV equation with five arbitrary functions. The results show that the Exp-function method with the help of symbolic computation provides a very effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.     

A coupling method of difference with high order accuracy and boundary integral equation for evolutionary equation and its error estimates

In the present paper,a new numerical method for solving initial-boundary valueproblems of evolutionary equations is proposed and studied,combining difference methodwith high accuracy with boundary integral equation method.The numerical approximateschemes for both problems on a bounded or unbounded domain in R~3 are proposed and theirprior error estimates are obtained.     

Approximated-asymptotic solution of the complex variable equation of toroidal shells under axial symmetric loads

The series-approach and the asymptotic-approach are usually used to solve the complex variable equation of the toroidal shells under axial symmetric loads. As is known, the convergence of the series-solution is good only for small values of . On the other hand, the convergence of the asymptotic solution is good only for large values of μ. In this paper, based on an earlier work^[1], a new approach which may be called the approximated-asymptotic solution has been developed and it is valid for both small and large values of μ. It is shown that the results of the approximated-asymptotic solution for toroidal shell with μ=0.5 coincide very well with those of the series-solution, while the results of the asymptotic solution for this value of μ are not as good, and the results of the approximated-asymptotic solution for μ=15 agree with those of the asymptotic solution. This work has been carried out under the direction of Professor Zhang Wei.     

Explicit solutions to the coupled KdV equations with variable coefficients

1　IntroductionandtheProblemPresentedSeekingtheexplicitsolutionofthenonlinearpartialdifferentialequation (NPDEs)isanimportantsubjectinsolitontheoryanditsapplication .Formanyyears,themainattentionwaspaidtotheconstantcoefficientNPDEs[1～ 7],manypowerfulmethodshavebeenproposedanddeveloped.Inrecentyears,moreandmoreattentionshavebeenpaidtovariablecoefficientNPDEs[8～ 13].Manymethodssuchassimilarityreductionmethod ,truncatedexpansionmethodandhomogeneousbalancemethodhavebeenextendedtosolvevaria…     

A UNIFORMLY CONVERGENT DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF A HIGH ORDER ELLIPTIC DIFFERENTIAL EQUATION

ＡＵＮＩＦＯＲＭＬＹＣＯＮＶＥＲＧＥＮＴＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＦＯＲＴＨＥＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＰＲＯＢＬＥＭＯＦＡＨＩＧＨＯＲＤＥＲＥＬＬＩＰＴＩＣＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮ（刘国庆）（苏煜城）ＡＵＮＩＦＯ...     

Constructions of the general solution for a system of partial differential equations with variable coefficients

Solving partial differential equations has not only theoretical significance,but alsopractical value.In this paper.by the property of conjugate operator.we give a method toconstruct the general solutions of a system of partial differential equations.     

A CLASS OF ALTERNATING GROUP METHODOF BURGERS＇ EQUATION

Some new Saul‘ yev type asymmetric difference schemes for Burgers’ equation isgiven, by the use of the schemes, a kind of alternating group four points method for solvingnonlinear Burgers‘ equation is constructed here. The basic idea of the method is that thegrid points on the same time level is divided into a number of groups, the differenceequations of each group can be solved independently, hence the method with intrinsicparallelism can be used directly on parallel computer. The method is unconditionally stableby analysis of linearization procedure. The numerical experiments show that the method has good stability and accuracy.     

On the uniform boundedness of solutions of some non-autonomous differential equations of the fourth order

In this paper, sufficient conditions are established under which all solutions of some non-autonomous differential equations of the fourth order are bounded and uniformly bounded.     

Nonaxisymmetric solutions of Laplace's tidal equation and Rossby waves

The Laplace tidal equation which plays a basic role in tide theory is investigated. A method of numerically-analytic integration of the Laplace tidal equation over the entire sphere without using the β-plane approximation is developed and its nonaxisymmetric solutions are investigated. Harmonics corresponding to long-period oscillations known as planetary or Rossby waves are obtained.     

A－HIGH－ORDER ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING THE EQUATION OF TWO－DIMENSIONAL PARABOLIC TYPE

Ａ－ＨＩＧＨ－ＯＲＤＥＲＡＣＣＵＲＡＣＹＥＸＰＬＩＣＩＴＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＦＯＲＳＯＬＶＩＮＧＴＨＥＥＱＵＡＴＩＯＮＯＦＴＷＯ－ＤＩＭＥＮＳＩＯＮＡＬＰＡＲＡＢＯＬＩＣＴＹＰＥＭａＭｉｎｇｓｈｕ（马明书）（ＲｅｃｅｉｖｅｄＪｕｎｅ２,１...     

Travel time prediction with viscoelastic traffic model

Travel time through a ring road with a total length of 80 km has been predicted by a viscoelastic traffic model(VEM), which is developed in analogous to the non-Newtonian fluid flow. The VEM expresses a traffic pressure for the unfree flow case by space headway, ensuring that the pressure can be determined by the assumption that the relevant second critical sound speed is exactly equal to the disturbance propagation speed determined by the free flow speed and the braking distance measured by the average vehicular length. The VEM assumes that the sound speed for the free flow case depends on the traffic density in some specific aspects, which ensures that it is exactly identical to the free flow speed on an empty road. To make a comparison, the open Navier-Stokes type model developed by Zhang(ZHANG, H. M. Driver memory, traffic viscosity and a viscous vehicular traffic flow model. Transp. Res. Part B, 37, 27–41(2003)) is adopted to predict the travel time through the ring road for providing the counterpart results.When the traffic free flow speed is 80 km/h, the braking distance is supposed to be 45 m,with the jam density uniquely determined by the average length of vehicles l ≈ 5.8 m. To avoid possible singular points in travel time prediction, a distinguishing period for time averaging is pre-assigned to be 7.5 minutes. It is found that the travel time increases monotonically with the initial traffic density on the ring road. Without ramp effects, for the ring road with the initial density less than the second critical density, the travel time can be simply predicted by using the equilibrium speed. However, this simpler approach is unavailable for scenarios over the second critical.     

Stokes＇ first problem for the fourth order fluid in a porous half space

In this study, the flow of a fourth order fluid in a porous half space is modeled. By using the modified Darcy’s law, the flow over a suddenly moving flat plate is studied numerically. The influence of various parameters of interest on the velocity profile is revealed. The English text was polished by Keren Wang.     

DISCUSSION ON THE PERIODIC SOLUTIONS FOR HIGHER-ORDER LINEAR EQUATION OF NEU

ＳｉＪｉａｎｇｕｏ（司建国）（ＲｅｃｅｉｖｅｄＭａｙ３０，１９９４，ＣｏｍｍｕｎｉｃａｔｅｄｂｙＬｉｎＺｏｎｇｃｈｉ）ＤＩＳＣＵＳＳＩＯＮＯＮＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＳＦＯＲＨＩＧＨＥＲ－ＯＲＤＥＲＬＩＮＥＡＲＥＱＵＡＴＩＯＮＯＦＮＥＵ...     

Positive solutions to （n-1,1） m-point boundary value problems with dependence on the first order derivative

Using the extension of Krasnoselskii＇s fixed point theorem in a cone, we prove the existence of at least one positive solution to the nonlinear nth order m-point boundary value problem with dependence on the first order derivative. The associated Green＇s function for the nth order m-point boundary value problem is given, and growth conditions are imposed on the nonlinear term f which ensures the existence of at least one positive solution. A simple example is presented to illustrate applications of the obtained results.     

One-dimensional unsteady inertial flow in phreatic aquifers induced by a sudden change of the boundary head

We are examining the classical problem of unsteady flow in a phreatic semi-infinite aquifer, induced by sudden rise or drawdown of the boundary head, by taking into account the influence of the inertial effects. We demonstrate that for short times the inertial effects are dominant and the equation system describing the flow behavior can be reduced to a single ordinary differential equation. This equation is solved both numerically by the Runge-Kutta method and analytically by the Adomian’s decomposition approach and an adequate polynomial-exponential approximation as well. The influence of the viscous term, occurring for longer times, is also taken into account by solving the full Forchheimer equation by a finite difference approach. It is also demonstrated that as for the Darcian flow, for the case of small fluctuations of the water table, the computation procedure can be simplified by using a linearized form of the mass balance equation. Compact analytical expressions for the computation of the water stored or extracted from an aquifer, including viscous corrections are also developed.     

THE EQUATION OF MOTION FOR THE SYSTEM OF THE VARIABLE MASS IN THE NON－LINEAR NON－HOLONOMIC SPACE

ＴＨＥＥＱＵＡＴＩＯＮＯＦＭＯＴＩＯＮＦＯＲＴＨＥＳＹＳＴＥＭＯＦＴＨＥＶＡＲＩＡＢＬＥ　ＡＳＳＩＮＴＨＥＮＯＮ－ＬＩＮＥＡＲＮＯＮ－ＨＯＬＯＮＯＭＩＣＳＰＡＣＥＴＨＥＥＱＵＡＴＩＯＮＯＦＭＯＴＩＯＮＦＯＲＴＨＥＳＹＳＴＥＭＯＦＴＨＥＶＡＲＩＡＢＬＥ...