The H-stability of the Solution for Infinite Delay Functional Differential Equation

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＮｏｗａｄａｙｓ，ｔｈｅｒｅｓｅａｒｃｈｏｆｔｈｅｉｎｆｉｎｔｅｄｅｌａｙｆｕｎｃｔｉｏｎａｌｄｉｆｆｅｒｅｎｔｉａｌｑｕａｔｉｏｎｈａｓｄｅｖｅｌｏｐｅｄｄｅｅｐｌｙ．Ｍａｎｙｌｉｔｅｒａｔｕｒｅｓｈａｖｅｂｅｅｎｃｏｎｃｅｒｎｅｄｆｏｒｒｅｓｅａｒｃｈｔｈｅｉｎｔｅｇｒａｌｄｉｆｆｅｒｅｎｔｉａｌｑｕａｔｉｏｎａｓｆｏｌｌｏｗｓｘ（ｔ）＝－ａ（ｔ）ｘ（ｔ）＋∫ｔ－∞ｋ（ｔ，ｓ－ｔ，ｘ（ｓ））ｄｓ．（１）　　Ｉｎｏｒｄｅｒｔｏｓｔｕｄｙｔｈｅｉｎｆｉｎｉｔｅｄ…     

Oscillation of Second-order Delay Differential Equation

By using some differential inequality, a second-order delay differential equation(r(t)x′(t))′ p(t)x(q(t)) = 0has been investigated and some necessary condition for this equation has a nonoscillatorysolution and some sufficient condition which ensures that all of the solutions of the aboveequation are oscillatory are obtained.     

Stability of Solution of a Scalar Differential Equation

Lyapunov's second method in fact is that which may be described as follows:applying the comparison principle to the V-function, we may render the stability ofthe solution of a vector differential.equation to that of the scalar differential equation du/dt=ω(t,u),ω(t,0)≡0,0≤u≤ρ(t),t∈ [cf. C. Corduneanu, 1960]. Unfortunately, the stability of the solutions of scalardifferential equations had not been well-discussed. In literatures, ρ is assumed to be     

On the Higher-Order Method for the Solution of a Nonlinear Scalar Equation

In this paper, a higher-order method for the solution of a nonlinear scalar equation is presented. It is proved that the new method is locally convergent with an order of (m+2), where m is the highest order derivative used in the iterative formula. Some numerical examples are used to demonstrate the new method.     

Variational Iteration Method for Delay Differential Equations

Since1930'sand40's,theexamplesofdelaydifferentialequationsarisinginpracticalapplicationshavebeenescalatedrapidly,andhavebeenstudiedextensively(fordetails,see[1]).Inthispaperwewillproposeanovelmethodcalledvariationaliterationmethod[2]tosolvesuchproblems.Considerfollowingpopulationgrowthmodel[1]x′(t)+cθ(t-1)x(t)+cθ(t-1)=0(1a)x(0)=θ(0),-1≤t≤0(1b)　　Accordingtovariationaliterationmethod[2],thecorrectionfunctionalcanbeconstructedasfollowsxn+1(t)=xn(t)+∫t0λ[x′nτ+cθ(τ-1)xn(τ)+cθ(τ-1…     

Periodic Solution of Nonlinear Parabolic Differential Equation With Time Lag

ＰｅｒｉｏｄｉｃＳｏｌｕｔｉｏｎｏｆＮｏｎｌｉｎｅａｒＰａｒａｂｏｌｉｃＤｉｆｆｅｒｅｎｔｉａｌＥｑｕａｔｉｏｎＷｉｔｈＴｉｍｅＬａｇ￥（王克）ＷａｎｇＫｅ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＮｏｒｔｈｅａｓｔＮｏｒｍａｌＵｎｉｖｅｒｓ...     

On Existence of Solution, Oscillatory and Non-Oscillatory Properties of Delay Impulsive Differential Equations Containing ‘Maximum’

In this paper, we determined the criteria for the existence of solutions for delay impulsive differential equations containing ‘maximum’ (DIDEM) using the Morales’ fixed point theorem. An impulsive analogue of the Gronwall-Bellman’s inequality with deviating argument is also considered. Some results on the oscillation and non-oscillation of solutions of the (DIDEM) are also obtained. Applications of the oscillatory results obtained are made to four non-linear examples one of which is a military model and upper bounds found for the solutions of the impulsive Fitzhugh-Naguno model and an impulsive nonlinear control system.     

Homotopy Method for a General Multiobjective Programming Problem

In this paper, we study a class of vector minimization problems on a complete metric space X which is identified with the corresponding complete metric space of objective functions . We do not impose any compactness assumption on X. We show that, for most (in the sense of Baire category) functions , the corresponding vector optimization problem has a solution.     

Razumikhin-type Theorems on Exponential Stability of Stochastic Functional Differential Equations with Infinite Delay

The main aim of this paper is to study the stability of the stochastic functional differential equations with infinite delay. We establish several Razumikhin-type theorems on the exponential stability for stochastic functional differential equations with infinite delay. By applying these results to stochastic differential equations with distributed delay, we obtain some sufficient conditions for both pth moment and almost surely exponentially stable. Finally, some examples are presented to illustrate our theory.     

The Stability Analysis of the $θ$-Methods for Delay Differential Equations

This paper deals with the stability analysis of $\theta -$methods for the numerical solution of delay differential equations (DDEs). We focus on the behaviour of such methods in the solution of the linear test equation $y^{\prime}(t)=a(t)y(t)+b(t)y(t-\tau )$, where $\tau >0$, $a(t)$ and $b(t)$ are functions from $R$ to $C$. It is proved that the linear $\theta -$method and the one-leg $\theta -$method are TGP-stable if and only if $\theta =1.$     

Existence of Periodic Solutions for Odd Order Ordinary Differential Equations via the Homotopy Method

This paper deals with the problems of finding periodic solutions for the third order ordinary differential equations of the form (1) where T is a fixed positive number and f satisfies some additional conditions which will be stated later.The periodicity problem has been one of main topics in the qualitative theory of ordinary     

A Method to Prove Convergence of Some Nonlinear Differential Equation's Form Solution

Wewillrecommendamethodtoprovetheconvergenceofsomenonlineardifferentiale-quationsformsolution,whichareunsolvableonlybyregularways.Themethodcanbeappliedtobothordinaryandpartialdifferentialequations.Firstwecanapplyittotheproofofthefollowingnonlinearpartialdifferentialequationsystem,ssolution.(refertoL1j)Iftherealtermofmatric{aj.},scharacteristicvaluehasthesamesign,whiIetherealtermofmartic{bij},characteristichasoppositeone,orisO,functionX,(x,,..',xkl%,.-',9),X,(x1,..',x.,R,..',P)isanalysticbyva…     

Definite Conditions and Asymptotic Solution of One Nonlinear Partial Differential Equation in Morphogenesis

One reaction-diffusion equationhas been presented in the study of Morphogenesis.In this paper,reasonable de-finite conditions of the equation are proposed and the asymptotic form of itssolution is obtained by using perturbation method.So the existence of solution ofthis probtem is solved.     

Existence and Uniqueness of Solution for a Fourth Order Differential－Difference Equation

ＥｘｉｓｔｅｎｃｅａｎｄＵｎｉｑｕｅｎｅｓｓｏｆＳｏｌｕｔｉｏｎｆｏｒａＦｏｕｒｔｈＯｒｄｅｒＤｉｆｆｅｒｅｎｔｉａｌ－Ｄｉｆｆｅｒｅｎｃｅ　ＥｑｕａｔｉｏｎＷａｎｇＰｅｉｇｕａｎｇ（王培光）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，Ｈｅｂｅ...     

Stability Analysis of an Optimal Control Problem for a Hyperbolic Equation

In this paper, we derive some results concerning the continuous dependence on parameters of optimal solutions to an optimal control problem that involves a quasilinear hyperbolic differential equation with a boundary condition and a nonlinear integral functional of action; continuous dependence of such kind is sometimes referred to as stability or sensitivity. To present a sufficient condition for the continuous dependence, we use the Kuratowski–Painlevé upper limit of a sequence of sets. Also offered is a technical interpretation of the results.     

Numerical Solution of the Lippmann–Schwinger Equation by Approximate Approximations

A new method for the numerical solution of volume integral equations is proposed and applied to a Lippmann–Schwinger type equation in diffraction theory. The approximate solution is represented as a linear combination of the scaled and shifted Gaussian. We prove spectral convergence of the method up to some negligible saturation error. The theoretical results are confirmed by a numerical experiment.     

Nonlinear Frequency Analysis of FGM Pipes Based on the Homotopy Method北大核心CSCD

Based on the homotopy analysis method, the nonlinear vibration of porous functionally graded material (FGM) conveying pipes under generalized boundary conditions was studied. Based on the power-law distribution of the FGM and the Voigt model, the physical properties of the porous pipe material were described. Under the Euler-Bernoulli beam theory and the von Kármán nonlinear theory, and by means of Hamilton’s variational principle, the dynamic control equations and generalized boundary conditions for porous FGM conveying pipes were established. The homotopy analysis method was used to solve the nonlinear vibration characteristics of the porous FGM conveying pipe under generalized boundary conditions. The numerical results show that, the translation spring has little effect on the critical velocity of instability, while the rotation spring increases the critical velocity of instability, making the system more stable; in the nonlinear system, the viscoelastic coefficient does not change the critical velocity; the pipe length, the power-law exponent and the porosity all influence the nonlinear free vibration of the porous FGM conveying pipe. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

2π-periodic Solution for a Kind High Order Delay Duffing Equation

In this paper, the high order delay Duffing equation ax(2n) bx g(x(t-τ)) =p(t) are considered,using the theory of coincidence degree, the sufficient condition for its there being at least a 2π-periodic solution is obtained.     

Differential equations satisfied by Eisenstein series of level 2

Ramanujan’s differential equations for the classical Eisenstein series are of great importance to many areas in number theory and special functions. H.H. Chan recently demonstrated that these differential equations can be derived from the triple product identity and the quintuple product identity in an elementary manner. In this article, we extend this method in a uniform manner to derive corresponding differential equations for the Eisenstein series of level 2. Several applications of these differential equations are also given.