An Approximate Method for the Surge Response of the Tension Leg Platform

The solution for the Duffing equation in a nonlinear vibration problem is studied in this paper. Clearly, in the case of the perturb parameter being a larger value, the traditional perturbation method is no longer valid but the Homotopy Perturbation Method(HPM) is applicable usually.HPM is used to solve the weak and strong nonlinear differential equations for finding the perturbed frequency of the response. The obtained frequencies via HPM and the approximate method have good accordance for weak and strong nonlinear differential equations. Additionally, the calculated responses by use of the approximate method are compared with the responses obtained from the Numerical method in the time history of the response and phase plane.The results represent good accordance between them.

MODIFIED CLOSED-FORM NUMERICAL ALGORITHM FOR PERIODIC VIBRATION RESPONSE OF MECHANICAL TRANSMISSIONS

A closed-form numerical algorithm (CFNA) is analyzed in detail. CFNA iswidely used in mechanical dynamics for periodic solution of second-order original differentialequations (SODE) with periodic time-variant coefficients. The principle of the algorithm is todiscretize the motion period into many short time intervals, so the coefficient matrices of theequation set are regarded as constant in a time interval. Defects are found in the originalalgorithm in treating the modal coordinates at the two end-nodes and important modifications to thedefects is made for the algorithm. The modified algorithm is finally used to solve the dynamicproblem of a three-ring planetary gear transmission.

Nonlinear boundary value problems for discontinuous delayed differential equations

In this paper nonlinear boundary value problems for discontinuous delayed differen- tial equations are considered. Some existence and boundedness results of solutions are obtained via the method of upper and lower solutions, which may be discontinuous. Our analysis can be applied to those phenomena from physics and control theory which have been successfully described by delayed differential equations with discontinuous functions, such as electric, pneu- matic, and hydraulic networks. It is also an important step to precede the design of control signals when finite-time or practical stability control are concerned.

Explicit Solutions for Generalized (2+1)-Dimensional Nonlinear Zakharov-Kuznetsov Equation

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.

UNIQUENESS OF POSITIVE SOLUTIONS OF A CLASS OF QUASILINEAR ORDINARY DIFFERENTIAL EQUATIONS

Uniqueness results are obtained for positive solutions of a class of quasilinear ordinary differential equations.The methods rely on the energy analysis and a scale argument.

DISTRIBUTION OF ZEROS OF SOLUTIONS OF FIRST ORDER DELAY DIFFERENTIAL EQUATIONS

The paper gives two estimates of the distance between adjacent zeros of solutions of the first\|order delay differential equation x′(t)+p(t)x(t-τ) =0 in the case when p(t)≥0 and ∫ t t-τ p(s)d s-1e oscillates or p(t) itself oscillates.

IMPROVE EFFICIENCY OF BIOGAS FEEDBACK SUPPLY CHAIN IN RURAL CHINA

Feedback supply chain is a key structure in the supply chain system, and the development of feedback supply chain for biogas biomass energy is one of the important ways of the rural ecological civilization construction. Presently, the efficiency problem of biogas supply chain in rural China has been restricting the development of biogas biomass energy business. This article, on the basis of combination of regulation parameters, describes the dynamic changes in the system, using differential equations integrated with simulation to reveal the rules of regulation parameters to investigate the efficiency problem in the biogas supply chain. First of all, on the basis of the actual situation, the flow level and flow rate system structure model and simulation equation set are established for the biogas energy feedback supply chain from a scale livestock farm to peasant households; On the basis of the differentiability of the simulation equation a third order inhomogeneous differential equation with constant coefficients containing regulative parameters is established for the quantity of biogas stored in the feedback supply chain. A theorem and its corollaries are established for the operating efficiency of supply chain to reveal the change law of the quantity of biogas, the quantity of biogas consumed daily by peasant households and its standard-reaching rate as well as other variables.

Local differential quadrature method using irregularly distributed nodes for solving partial differential equations

In the conventional differential quadrature （DQ） method the functional values along a mesh line are used to approximate derivatives and its application is limited to regular regions. In this paper, a local differential quadrature （LDQ） method was developed by using irregular distributed nodes, where any spatial derivative at a nodal point is approximated by a linear weighted sum of the functional values of nodes in the local physical domain. The weighting coefficients in the new approach are determined by the quadrature rule with the aid of nodal interpolation. Since the proposed method directly approximates the derivative, it can be consistently well applied to linear and nonlinear problems and the mesh-free feature is still kept. Numerical examples are provided to validate the LDQ method.

A New Kind of Iteration Method for Finding Approximate Periodic Solutions to Ordinary Diferential Equations

In this paper, a new kind of iteration technique for solving nonlinear ordinary differential equations is described and used to give approximate periodic solutions for some well-known nonlinear problems. The most interesting features of the proposed methods are its extreme simplicity and concise forms of iteration formula for a wide range of nonlinear problems.

BSDEs with Jumps and Path-Dependent Parabolic Integro-differential Equations

This paper deals with backward stochastic differential equations with jumps,whose data(the terminal condition and coefficient) are given functions of jump-diffusion process paths. The author introduces a type of nonlinear path-dependent parabolic integrodifferential equations, and then obtains a new type of nonlinear Feynman-Kac formula related to such BSDEs with jumps under some regularity conditions.

Convergence error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs

In this work, we theoretically analyze the convergence error estimates of the Crank-Nicolson （C-N） scheme for solving decoupled FBSDEs. Based on the Taylor and ItS-Taylor expansions, the Malliavin calculus theory （e.g., the multiple Malliavin integration-by-parts formula）, and our new truncation error cancelation techniques, we rigorously prove that the strong convergence rate of the C-N scheme is of second order for solving decoupled FBSDEs, which fills the gap between the second-order numerical and theoretical analysis of the C-N scheme.

THE ACCURATE SOLUTION OF THE INITIALVALUE PROBLEM OF THE EQUATION x'(t) + ax(t) + bx(t-т)=0

Consider the RDDE's initial value problemwhere a, b and τ are arbitrary real constants, and τ＞ 0, φ(θ) is a given initial function.In this paper, we find series expansions of the accurate solution of the initial valueproblem (EI).

ON SUFFICIENT AND NECESSARY OF EXISTENCE FOR A CLASS OF SINGULAR OPTIMAL STOCHASTIC CONTROL

We study a class of discounted models of singular stochastic control. In thiskind of models, not only the structure of cost function has been extended to some general type, butalso the state can be represented as the solution of a class of stochastic differential equationswith nonlinear drift and diffusion term. By the various methods of stochastic analysis, we derivethe sufficient and necessary conditions of the existence of optimal control.

On the Existence of Periodic Solutions for a Kind of Second Order Neutral Functional Differential Equation

By means of the continuation theorem of coincidence degree theory, some newresults on the non-existence, existence and unique existence of periodic solutions for a kind ofsecond order neutral functional differential equation are obtained.

PDE Surface Generation with Combined Closedand Non-Closed Form Solutions

Partial differential equations (PDEs) combined with suitably chosen boundaryconditions are effective in creating free form surfaces. In this paper, a fourth order partialdifferential equation and boundary conditions up to tangential continuity are introduced. Thegeneral solution is divided into a closed form solution and a non-closed form one leading to a mixedsolution to the PDE. The obtained solution is applied to a number of surface modelling examplesincluding glass shape design, vase surface creation and arbitrary surface representation.

Shift Harnack inequality and integration by parts formula for semilinear stochastic partial differential equations

Shift Harnack inequality and integration by parts formula are established for semilinear stochastic partial differential equations and stochastic functional partial differential equations by modifying the coupling used by F. -Y. Wang [Ann. Probab., 2012, 42（3）： 994-1019]. Log-Harnack inequality is established for a class of stochastic evolution equations with non- Lipschitz coefficients which includes hyperdissipative Navier-Stokes/Burgers equations as examples. The integration by parts formula is extended to the path space of stochastic functional partial differential equations, then a Dirichlet form is defined and the log-Sobolev inequality is established.