二维Newton-Boussinesq方程周期初值问题经典解的整体存在性

研究一类二维Newton-Boussinesq方程的周期初值问题,将方程转化为积分方程,用不动点原理得到解的局部存在性,用能量估计的方法证明经典解的整体存在性．     

Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturbation method

In this paper He’s homotopy perturbation method has been adapted to calculate higher-order approximate periodic solutions for a nonlinear oscillator with discontinuity for which the elastic force term is an anti-symmetric and quadratic term. We find that He’s homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Just one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 0.73% for all values of oscillation amplitude, while this relative error is as low as 0.040% when the second iteration is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance method reveals that the former is very effective and convenient.     

发展型p-Laplacian 方程组的周期解

In this paper, the authors study the existence of periodic solutions to an evolution p-Laplacian system. The authors prove a comparison principle of the system in general form. Then the existence of periodic solutions to the system of evolution p-Laplacian equations is obtained with the help of the comparison principle and the monotone iteration technique.     

Frequency dependent iteration method for forced nonlinear oscillators

A new iteration method for nonlinear vibrations has been developed by decomposing the periodic solution in two parts corresponding to low and high harmonics. For a nonlinear forced oscillator, the iteration schema is proposed with different formulations for these two parts. Then, the schema is deduced by using the harmonic balance technique. This method has proven to converge to the periodic solutions provided that a convergence condition is satisfied. The convergence is also demonstrated analytically for linear oscillators. Moreover, the new method has been applied to Duffing oscillators as an example. The numerical results show that each iteration schema converges in a domain of the excitation frequency and it can converge to different solutions of the nonlinear oscillator.     

The existence of periodic solutions for nonlinear beam equations on by a para‐differential method

This paper focuses on the construction of periodic solutions of nonlinear beam equations on the d‐dimensional tori. For a large set of frequencies, we demonstrate that an equivalent form of the nonlinear equations can be obtained by a para‐differential conjugation. Given the nonresonant conditions on each finite dimensional subspaces, it is shown that the periodic solutions can be constructed for the block diagonal equation by a classical iteration scheme.     

Construction of exact periodic wave and solitary wave solutions for the long–short wave resonance equations by VIM

In this paper, He’s variational iteration method is employed to construct periodic wave and solitary wave solutions for the long–short wave resonance equations. The chosen initial solution can be in soliton form with some unknown parameters, which can be determined in the solution procedure. Some examples are given. The results reveal that the method is very effective and convenient.     

Periodic wave solutions of coupled integrable dispersionless equations by residue harmonic balance

We introduce the residue harmonic balance method to generate periodic solutions for nonlinear evolution equations. A PDE is firstly transformed into an associated ODE by a wave transformation. The higher-order approximations to the angular frequency and periodic solution of the ODE are obtained analytically. To improve the accuracy of approximate solutions, the unbalanced residues appearing in harmonic balance procedure are iteratively considered by introducing an order parameter to keep track of the various orders of approximations and by solving linear equations. Finally, the periodic solutions of PDEs result. The proposed method has the advantage that the periodic solutions are represented by Fourier functions rather than the sophisticated implicit functions as appearing in most methods.     

The diffusive relaxation limit of non-isentropic Euler-Maxwell equations for plasmas

This paper concerns the non-isentropic Euler-Maxwell equations for plasmas with short momentum relaxation time. With the help of the Maxwell-type iteration, it is obtained that, as the relaxation time tends to zero, periodic initial-value problem of certain scaled non-isentropic Euler-Maxwell equations has unique smooth solutions existing in the time interval where the corresponding classical drift-diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the corresponding drift-diffusion model from the non-isentropic Euler-Maxwell equations.     

The asymptotic periodicity in a Schoener’s competitive model

This paper deals with a two species model with Schoener’s competitive interaction. The existence and the asymptotic behavior of T-periodic solutions for the periodic system of quasilinear parabolic equations under nonlinear boundary conditions are given by using upper and lower solutions and corresponding iteration. The numerical simulations are also presented to illustrate our result. It is shown that periodic solutions may exist if the inter-specific competition rates are weak.     

模糊随机微分方程解的存在唯一性

将实数空间上的随机微分方程推广到模糊数空间,即为模糊随机微分方程.本文用Picard迭代的方法证明了其解的存在唯一性定理,推广了现有文献的结果,并且给出Picard迭代近似解误差的估计式.     

Numerical implementation of iteration processes applied in the study of nonlinear boundary value problems of the theory of elasticity and filtration

Nonlinear elastic problems for hardening media are solved by applying the universal iteration process proposed by A.I. Koshelev in his works on the regularity of solutions to quasilinear elliptic and parabolic systems. This requires numerically solving a linear elliptic system at each step of the iteration procedure. The method is numerically implemented in the MATLAB environment by using its PDE Toolbox. A modification of the finite-element procedure is suggested in order to solve a linear algebraic system at each iteration step. The computer model is tested on simple examples. The same nonlinear problems are also solved by the method of elastic solutions, which consists in replacing the Laplace operator in the universal iteration process by the Lamé operator of linear elasticity. As is known, this iteration process converges to a weak solution of the nonlinear problem, provided that the displacements are fixed on the boundary. The method is tested on examples with stresses on the boundary. The third part of the paper is devoted to the nonlinear filtration problem. General properties of the iteration process for nonlinear parabolic systems have been studied by A.I. Koshelev and V.M. Chistyakov. The numerical implementation is based on slightly modified PDE Toolbox procedures. The program is tested on simple examples.     

Positive periodic solutions of parabolic evolution problems: a translation along trajectories approach

A translation along trajectories approach together with averaging procedure and topological degree are used to derive effective criteria for existence of periodic solutions for nonautonomous evolution equations with periodic perturbations. It is shown that a topologically nontrivial zero of the averaged right hand side is a source of periodic solutions for the equations with increased frequencies. Our setting involves equations on closed convex cones, therefore it enables us to study positive solutions of nonlinear parabolic partial differential equations.     

The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems

In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.

     

Time periodic solutions of the diffusive Nicholson blowflies equation with delay

This paper is concerned with the Nicholson blowflies equation with nonlinear diffusion and time delay subject to the homogeneous Dirichlet boundary condition in a bounded domain. We establish the existence of nontrivial periodic solutions of the time-periodic problem under general conditions by constructing a coupled upper–lower solution pair and by applying the Schauder fixed point theorem. The attractivity of the periodic solutions is also discussed by using the monotone iteration method.     

Existence of doubly periodic solutions for a class of telegraph system with indefinite weight

Using the method of monotone iteration and Schauder fixed point theorem, we establish the existence of doubly periodic solutions for a coupled nonlinear telegraph system with indefinite weight.     

DIRECT CONSTRUCT OF PERIODIC SOLUTIONS TO PERTURBED KLEIN-GORDON EQUATIONS BY NEWTON ITERATION

61. IntroductionNear an equilibrium state to a system of nonlinear ODEs, one can regard the nonlinearsystem as a perturbation of its linearization. It is well known that if none of the characteristicexponents of the linearized system has zero real part, then the phase portrait near theequilibrium of the nonlinear system is topologically equivalent to that of its linearization(cL [181). However, if characteristic exponents with zero real part appearl then the linearizedsystem alone does not c…     

一种有效的边界元/有限元混合法迭代算法及其在回转体自由扭振分析中的应用*

本文给出一种新的边界元/有限元混合法迭代算法,基本做法是将近似的固有频率值代入自由振动问题的基本解,按一般混合法列式,通过迭代逐步修正近似解的值.这种算法避开了一般边界元法需要求解非代数特征值问题的困难,同时数值结果的精度基本上不依赖于区域内单元网格的疏密程度,这都给实际计算带来很多方便.应用于回转体自由扭振问题的分析,得到令人满意的数值结果.     

Periodic solutions for a degenerate parabolic equation

In this work, we establish the existence of nontrivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms. The key is the using of Moser’s iteration technique and the theory of the Leray–Schauder degree.     

The singular solutions of a nonlinear evolution equation taking continuous part of the spectral data into account in inverse scattering method

A procedure for finding the solutions of the Vakhnenko–Parkes equation by means of the inverse scattering method is described. Both the bound state spectrum and the continuous spectrum are considered in the associated eigenvalue problem. The suggested special form of the singularity function gives rise to periodic solutions. The interaction of a soliton with a one-mode periodic wave is studied.     

The Periodic Solutions of a Nonhomogeneous String with Dirichlet-Neumann Condition

This paper deals with the existence of periodic solutions of a nonhomogeneous string with Dirichlet-Neumann condition. The authors consider the case that the period is irrational multiple of space length and prove that for some irrational number, zero is not the accumulation point of the spectrum of the associated linear operator. This result can be used to prove the existence of the periodic solution avoid using Nash-Moser iteration.