A third-order convergent iterative method for solving non-linear equations

In this paper, we present and analyse a new predictor-corrector iterative method for solving non-linear single variable equations. The convergence analysis establishes that the new method is cubically convergent. Numerical tests show that the method is comparable with the well-known existing methods and in many cases gives better results.     

Asymptotic behaviour of the solutions of second order functional differential equations

This paper presents integral criteria to determine the asymptotic behaviour of the solutions of second order nonlinear differential equations of the type y^″(x)+q(x)f(y(x))=0, with q(x)>0 and f(y) odd and positive for y>0, as x tends to +∞. It also compares them with the results obtained by Chanturia (1975) in [11] for the same problem.     

Numerical solution of non-linear elliptic boundary-value problems by isomorphic iterative methods

New implicit iterative methods are presented for the efficient numerical solution of non-linear elliptic boundary-value problems. Isomorphic iterative schemes in conjunction with preconditioning techniques are used for solving non-linear elliptic equations in two and three-space dimensions. The application of the derived methods on characteristic 2D and 3D non-linear boundary-value problems is discussed and numerical results are given.     

Solid modelling based on sixth order partial differential equations

Although solid modelling based on partial differential equations (PDEs) has many advantages, such existing methods can either only deal with simple cases or incur expensive computational overheads. To overcome these shortcomings, in this paper we present an efficient PDE based approach to creating and manipulating solid models. With trivariate partial differential equations, the idea is to formulate an accurate closed form solution to the PDEs subject to various complex boundary constraints. The analytical nature of this solution ensures both high computational efficiency and modelling flexibility. In addition, we will also discuss how different geometric shapes can be produced by making use of controls incorporated in the PDEs and the boundary constraint equations, including the surface functions, tangents and curvature in the boundary constraints, the shape control parameters and the sculpting forces. Two examples are included to demonstrate the applications of the proposed approach and solutions.     

Extended one-step methods for the numerical solution of ordinary differential equations

A class of one-step finite difference formulae for the numerical solution of first-order differential equations is considered. The accuracy and stability properties of these methods are investigated. By judicious choice of the coefficients in these formulae a method is derived which is both A-stable and third-order convergent. Moreover the new method is shown to be L-stable and so is appropriate for the solution of certain stiff equations. Numerical results are presented for several test problems.     

Stability by order stars for non-linear theta-methods based on means

The linear stability analysis of non-linear one-step methods based on means is studied by means of the concept of stability regions and order stars. Concretely, non-linear θ-methods based on harmonic, contraharmonic, quadratic, geometric, Heronian, centroidal and logarithmic means are considered. Their stability diagrams and order stars show their A-stability for θ≥1/2, and L-stability in some cases. Order stars in the Riemann surface are a requirement for non-linear one-step methods. The advantages and disadvantages of this technique are presented.     

Fourth-order two-step iterative methods for determining multiple zeros of non-linear equations

In this paper, we describe and analyse two two-step iterative methods for finding multiple zeros of non-linear equations. We prove that the methods have fourth-order convergence. The methods calculate the multiple zeros with high accuracy. These are the first two-step multiple zero finding methods. The numerical tests show their better performance in the case of algebraic as well as non-algebraic equations     

Study of general Taylor-like explicit methods in solving stiff ordinary differential equations

In this paper, we present a further study of Taylor-like explicit methods in solving stiff ordinary differential equations. We derive the general form for Taylor-like explicit methods in solving stiff differential equations. We also analyse the order of convergence and stability property for the general form. Moreover, we give its corresponding vector form via introducing a new definition of vector product and quotient in another article. The convergence and stability of the vector form are considered as well.     

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.     

Sextic C 1-spline collocation methods for solving delay differential equations

In this paper, we propose a global collocation method for the numerical solution of the delay differential equations (DDEs). The method presented is based on sextic C ¹-splines s(x) as an approximation to the exact solution y(x) of the DDEs. Convergence results shows that the error bounds ‖ s ^j ?y ^j ‖=O(h ⁶), j=0, 1, in the uniform norm. Moreover, the stability analysis properties of these methods have been studied. Numerical experiments will also be considered.     

微分方程的Hamilton化与解法

提出一种求解微分方程的力学方法.首先,将一类常微分方程化成一个Hamilton方程,在特殊情况下化成Hamilton原来的方程,在一般情况下化成带非保守力的Hamilton方程.其次,利用Hamilton系统的Noether理论求守恒量.如果找到足够多的守恒量,便找到了方程的解.最后,举例说明结果的应用.     

Eigenvalue problems of second order impulsive differential equations

This paper is concerned with an eigenvalue problem for second order differential equations with impulse. The existence of a countably infinite set of eigenvalues and eigenfunctions is proved.     

Discrete gradient methods and linear projection methods for preserving a conserved quantity of stochastic differential equations

Numerical methods preserving a conserved quantity for stochastic differential equations are considered. A class of discrete gradient methods based on the skew-gradient form is constructed, and the sufficient condition of convergence order 1 in the mean-square sense is given. Then a class of linear projection methods is constructed. The relationship of the two classes of methods for preserving a conserved quantity is proved, which is, the constructed linear projection methods can be considered as a subset of the constructed discrete gradient methods. Numerical experiments verify our theory and show the efficiency of proposed numerical methods.     

Quadrature methods for periodic singular and weakly singular Fredholm integral equations

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.     

Non-oscillation of solutions of difference equations of third order

In this paper, sufficient conditions in terms of coefficient functions are obtained for non-oscillation of all solutions of a class of linear homogeneous third order difference equations of the form

     

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.