A two-step method with phase-lag of order infinity for the numerical integration of second order periodic initial-value problem

A two-step method with phase-lag of order infinity is developed for the numerical integration of second order periodic initial-value problem. The method has algebraic order six. Extensive numerical testing indicates that the new method is generally more accurate than other two-step methods.     

A class of explicit two-step superstable methods for second-order linear initial value problems

A class of explicit two-step superstable methods of fourth algebraic order for the numerical solution of second-order linear initial value problems is presented in this article. We need Taylor expansion at an internal grid point and collocation formulae for the derivatives of the solution to derive a method and then modify it into a class of methods having the desired stability properties. Computational results are presented to demonstrate the applicability of the methods to some standard problems.     

Two-step explicit methods for second-order IVPs with oscillatory solutions

In this paper, we present a class of non-conventional two-step explicit methods for solving the second-order initial value problems (IVPs) with periodic or oscillatory solutions. The methods have second algebraic order, a large interval of periodicity and high phase-lag order. For multi-dimensional problems, we give the vector form of the methods with the aid of a special vector operation. Some numerical results are reported to illustrate the efficiency of our methods.     

A novel method for nonlinear singular fourth order four-point boundary value problems

In this paper, a novel method is proposed for solving nonlinear singular fourth order four-point boundary value problems (BVPs) by combining advantages of the homotopy perturbed method (HPM) and the reproducing kernel method (RKM). Some numerical examples are presented to illustrate the strength of the method.     

A fourth order multiderivative method for linear second order boundary value problems

Fourth order methods are developed and analysed for the numerical solution of linear second order boundary value problems.

The methods are developed by replacing the exponential terms in a three-point recurrence relation by Padé approximants.

The derivations of second order and sixth order methods from the recurrence relation are outlined briefly.

One method is tested on two problems from the literature, one of which is mildly nonlinear.     

A new high efficient and high accurate Obrechkoff four-step method for the periodic nonlinear undamped Duffing's equation

Based on the idea of the previous Obrechkoff's two-step method, a new kind of four-step numerical method with free parameters is developed for the second order initial-value problems with oscillation solutions. By using high-order derivatives and apropos first-order derivative formula, the new method has greatly improved the accuracy of the numerical solution. Although this is a multistep method, it still has a remarkably wide interval of periodicity, . The numerical test to the well known problem, the nonlinear undamped Duffing's equation forced by a harmonic function, shows that the new method gives the solution with four to five orders higher than those by the previous Obrechkoff's two-step method. The ultimate accuracy of the new method can reach about 5×¹⁰⁻¹³, which is much better than the one the previous method could. Furthermore, the new method shows the great superiority in efficiency due to a reasonable arrangement of the structure. To finish the same computational task, the new method can take only about 20% CPU time consumed by the previous method. By using the new method, one can find a better ‘exact’ solution to this problem, reducing the error tolerance of the one widely used method (¹⁰⁻¹¹), to below 10⁻¹⁴.     

Optimality of an error estimate of a one-step numerical integration method for certain initial value problems

In a recent paper, an error estimate of a one-step numerical method, originated from the Lanczos tau method, for initial value problems for first order linear ordinary differential equations with polynomial coefficients, was obtained, based on the error of the Lanczos econo-mization process. Numerical results then revealed that the estimate gives, correctly, the order of the tau approximant being sought. In the present paper we further establish that the error estimate is optimum with respect to the integration of the error equation. Numerical examples are included for completeness.     

A new type of solution method for the generalized linear complementarity problem over a polyhedral cone

This paper addresses the generalized linear complementarity problem （GLCP） over a polyhedral cone. To solve the problem, we first equivalently convert the problem into an affine variational inequalities problem over a closed polyhedral cone, and then propose a new type of method to solve the GLCP based on the error bound estimation. The global and R-linear convergence rate is established. The numerical experiments show the efficiency of the method.     

A novel extended precise integration method based on Fourier series expansion for the H2-norm of linear time-varying periodic systems

A new reliable algorithm for computing the H₂-norm of linear time-varying periodic (LTP) systems via the periodic Lyapunov differential equation (PLDE) is proposed. By taking full advantage of the periodicity, the transition matrix of the underlying LTP system associated with the PLDE is effectively computed by developing a novel extended precise integration method based on Fourier series expansion, where the time-consuming work for the computation of the matrix exponential and its related integrals in every sub-interval is avoided. Then, a highly accurate and efficient algorithm for the PLDE is derived using the block form of the transition matrix. Thus, the H₂-norm is evaluated by solving a simple first-order ordinary differential equation. Finally, two numerical examples are presented and compared with other algorithms to verify the numerical accuracy and efficiency of the proposed algorithm.     

Solvability of nonlocal boundary value problems for ordinary differential equation of higher order with a -Laplacian

Nonlocal boundary value problems at resonance for a higher order nonlinear differential equation with a p-Laplacian are considered in this paper. By using a new continuation theorem, some existence results are obtained for such boundary value problems. An explicit example is also given in this paper to illustrate the main results.