Polynomial solutions of certain differential equations

In this paper, the Chebyshev matrix method is applied generalisations of the Hermite, Laguerre, Legendre and Chebyshev differential equations which have polynomial solution. The method is based on taking the truncated Chebyshev series expansions of the functions in equation, and then substituting their matrix forms into the result equation. Thereby the given equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Chebyshev coefficients.     

Stability of ROW methods for non autonomous systems of ordinary differential equations

AN-stability of ROW methods is studied. The concept of LN-equivalent schemes belonging to different classes of one-step methods is introduced to do it. Ways to construct ROW methods with improved stability for linear nonautonomous and nonlinear problems are studied using the algebraic stability of singly diagonally implicit Runge-Kutta (SDIRK) methods. The existing SDIRK methods are shown to be inapplicable to construct LN-stable ROW methods for numerical integration of stiff systems of ordinary differential equations.     

Stability of the solutions of systems of second-order differential equations

Criteria for stability of the solutions of systems of linear and nonlinear ordinary second-order differential equations were proposed relying on the studies of the spectra and logarithmic norms of the families of specially constructed matrices. Obtained were the stability criteria for systems of linear second-order differential equations expressed in terms of the coefficients at unknown functions and their first derivatives.     

Non-trivial solutions for nonlinear fourth-order elastic beam equations

The existence of at least one non-trivial solution to a boundary value problem for fourth-order elastic beam equations, under a non-standard growth condition of the nonlinear term, is established. Our approach is based on a local minimum theorem for differentiable functionals.     

Stability of solutions of differential equations with retarded arguments

A criterion is given for the stability of trivial solutions of the differential equations with the variable coefficients and variable delays. The derivation is based on Lyapunov's direct method.     

Stability of solutions of differential equations under generalized pulse perturbations

Consideration was given to stability of the solution of the system of differential equations to the function of a locally bounded variation added to the right-hand side of the derivative (understood in the general sense). Simple attributes of stability and asymptotic stability were established for different classes of these equations.     

Numerical solution of time-fractional fourth-order partial differential equations

A quintic B-spline collocation technique is employed for the numerical solution of time-fractional fourth-order partial differential equations. These equations occur in many applications in real-life problems such as modelling of plates and thin beams, strain gradient elasticity and phase separation in binary mixtures, which are basic elements in engineering structures and are of great practical significance to civil, mechanical and aerospace engineering. The time-fractional derivative is described in the Caputo sense. Backward Euler scheme is used for time discretization and the quintic B-spline-based numerical method is used for space discretization. The stability and convergence properties related to the time discretization are discussed and theoretically proven. The given problem is solved with three different boundary conditions, including clamped-type condition, simply supported-type condition, and a transversely supported-type condition. Numerical results are considered to investigate the accuracy and efficiency of the proposed method.     

Adaptive mesh generation for singularly perturbed fourth-order ordinary differential equations

In this paper, we consider a singularly perturbed boundary-value problem for fourth-order ordinary differential equation (ODE) whose highest-order derivative is multiplied by a small perturbation parameter. To solve this ODE, we transform the differential equation into a coupled system of two singularly perturbed ODEs. The classical central difference scheme is used to discretize the system of ODEs on a nonuniform mesh which is generated by equidistribution of a positive monitor function. We have shown that the proposed technique provides first-order accuracy independent of the perturbation parameter. Numerical experiments are provided to validate the theoretical results.     

New solutions for ordinary differential equations

This paper introduces a new method for solving ordinary differential equations (ODEs) that enhances existing methods that are primarily based on finding integrating factors and/or point symmetries. The starting point of the new method is to find a non-invertible mapping that maps a given ODE to a related higher-order ODE that has an easily obtained integrating factor. As a consequence, the related higher-order ODE is integrated. Fixing the constant of integration, one then uses existing methods to solve the integrated ODE. By construction, each solution of the integrated ODE yields a solution of the given ODE. Moreover, it is shown when the general solution of an integrated ODE yields either the general solution or a family of particular solutions of the given ODE. As an example, new solutions are obtained for an important class of nonlinear oscillator equations. All solutions presented in this paper cannot be obtained using the current Maple ODE solver.     

Oscillation theorems for certain neutral functional differential equations

In this paper, we shall offer sufficient conditions for the oscillation of all solutions to neutral functional differential equations of mixed type of the form

,here p and q are periodic functions.     

Comparison results for fuzzy differential equations

In this paper, we present several comparison results for the solutions of fuzzy differential equations, by using the Hukuhara derivative. These results constitute the extension to the fuzzy context of some comparison results for the solutions of linear differential equations. Besides, some side-results about the Hukuhara difference and the partial orderings are given. We illustrate the applicability of the new results by showing several examples.     

Exact linearization of nonlinear ordinary autonomous differential equations

Many methods for finding exact solutions to nonlinear ordinary differential equations (ODE) are based on certain euristic rules. The author suggested a newexact linearization method that provides an algorithmic procedure for constructing exact solutions for some important classes of ODEs [1].     

Chaos expansion for the solutions of stochastic differential equations

In this note we generalize the Isobe–Sato formula for kernels of the Wiener–Ito chaos expansion to nonautonomous systems. Expansion of a transition density is obtained and some version of Wiener's famous “black-box” identification problem is solved.     

Construction of several second- and fourth-order geometric partial differential equations for space curves

Geometric partial differential equations for curves and surfaces are used in many fields, such as computational geometry, image processing and computer graphics. In this paper, a few differential operators defined on space curves are introduced. Based on these operators, several second-order and fourth-order geometric flows for evolving space curves are constructed. Some properties of the changing rates of the arc-length of the evolved curves and areas swept by the curves are discussed. Short-term and long-term behaviors of the evolved curves are illustrated.     

基于四阶偏微分方程平滑的图像分割新方法

在方向曲率模值描述图像平滑度的泛函基础上,推导出一种四阶偏微分方程(PDE)图像降噪模型,在有效降噪的同时,较好地保持了边缘。该方法处理结果为分段线性图像,且在目标边缘处梯度存在阶跃。利用降噪结果的这一特点,提出一种新测地线主动轮廓模型。实验表明,新模型不但稳定,轮廓提取能力强,而且收敛速度快。     

Stability of linear impulsive differential equations

The idea of the stability of linear impulsive differential equations at fixed moments is made more precise.     

Stability of regime-switching stochastic differential equations

The main result is reduction of the asymptotic stability problem for a stochastic differential equation (SDE) with sufficiently rapid Markovian switching to the analogous wellstudied problem for the ??averaged?? SDE without switching. Applications to the switching stabilization problem and to ordinary differential equations (ODE) with switching are also considered.