Quintic B-spline collocation method for second order mixed boundary value problem

In this paper, we study a new quintic B-spline collocation method for linear and nonlinear second order mixed boundary value problems. Convergence is studied and the method is fourth order convergent. Numerical examples are also given to demonstrate the higher accuracy and efficiency of our method.     

Comparison of collocation methods for the solution of second order non-linear boundary value problems

This article concerns the application of cubic spline collocation tau-method for solving non-linear second order ordinary differential equations. Three collocation methods [Taiwo, O.A., 1986, A computational method for ordinary differential equations and error estimation. MSc dissertation, University of Ilorin, Nigeria (unpublished); Taiwo, O.A., 2002, Exponential fitting for the solution of two point boundary value problem with cubic spline collocation tau-method. International Journal of Computer Mathematics, 79(3), 229–306.] are discussed and applied to some second order non-linear problems. They are standard collocation, perturbed collocation, and exponentially fitted collocation. Numerical examples are given to illustrate the accuracy, efficiency and computational cost.     

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 collocation method for boundary value problems of differential equations with functional arguments

A collocation procedure with polynomial and piecewise polynomial approximation is considered for second order functional differential equations with two side-conditions. The piecewise polynomials are taken in the classC ¹ and reduce to polynomials of increasing degree on each interval of a suitable assigned partition. Appropriate choices of the partition are made, according to the jump discontinuities in the derivatives caused by the functional argument, in order to optimize the rate of convergence.     

Sixth order non-dissipative C 1 spline collocation method for oscillatory ordinary initial value problems

In this paper we construct a global method, based on quintic C ¹-spline, for the integration of first order ordinary initial value problems (IVPs) including stiff equations and those possessing oscillatory solutions as well. The method will be shown to be of order six and in particular is A-stable. Attention is also paid for the phase error (or dispersion) and it is proved that the method is dispersive and has dispersion order six with small phase-lag (compared with the extant methods having the same order (cf. [7])). Moreover, the method may be regarded as a continuous extension of the closed four-panel Newton–Cotes formula (NC4) (typically it is a continuous extension of an implicit Runge–Kutta method). In additiona priori error estimates, in the uniform norm, together with illustrative test examples will also be presented.     

Sextic spline collocation methods for nonlinear fifth-order boundary value problems

In this paper, two sextic-spline collocation methods are developed and analysed for approximating solutions of nonlinear fifth-order boundary-value problems. The first method uses a spline interpolant and the second one is based on a spline quasi-interpolant, which are constructed from sextic splines. They are both proved to be second-order convergent. Numerical results confirm the order of convergence predicted by the analysis. It has been observed that the methods developed in this paper are better than the others given in the literature.     

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 sixth order method for the integration of two point boundary value problems

A finite difference method for obtaining sixth order accurate approximation to the solution of the two-point linear boundary value problem is given. The convergence of the method is proved. Numerical results for a typical problem are tabulated and in each case the observed error is compared with its theoretical estimate. The numerical results are compared with those obtained from an earlier method of the author and the method of Noumerov.     

A modified lobatto collocation for linear boundary value problems of differential-algebraic equations

The Lobatto collocation method is modified for efficiently solving linear boundary value problems of differential-algebraic equations with index 1. The stability and superconvergence of this method are established. Numerical implementations are discussed and a numerical example is given.     

A numerical method for boundary value problems

This paper demonstrates the utility, generality and simplicity of a computational method of solving systems of ordinary differential equations. The central idea of the method revolves around the ability to generate a numerical approximation of the general solution of systems of linear differential equations. The idea of obtaining a numerical approximation to the general solution leads to changes in the traditional approaches for solving boundary value problems. The method is extended to nonlinear boundary value problems and to boundary value problems with a boundary condition given at infinity. The estimation of unknown parameters in a dynamical system is also treated.     

Solution of some boundary value problems in applied mechanics by the collocation least square method

The conventional simple but crude method of collocation is greatly improved by a least square augmentation. Simplicity in application and good accuracy of the proposed collocation least square scheme is demonstrated by the solution of some complex problems in applied mechanics.Example solutions of such problems include the linear and nonlinear analyses of isotropic plates, orthotropic plates and plates on elastic foundations. Numerical and graphical results are presented and compared with existing solutions. For the problems considered herein, the present method proves to be much less laborious than other numerical methods frequently employed by previous investigators.     

Galerkin collocation for an improved boundary element method for a plane mixed boundary value problem

We present the numerical implementation of a boundary element approximation for the plane mixed boundary value problem of the Laplacian. The performed Galerkin procedure is based on the direct boundary integral method. Its accuracy is improved by using appropriate singularity functions as additional test and trial functions besides quadratic splines. We analize the consistency error of the used numerical integrations and present asymptotic error estimates for the fully discretized numerical scheme which are of the same optimal orders as the Galerkin errors.     

求解对流扩散方程的混合三次B-样条配点法

通过对三次B-样条和三次三角B-样条基函数引入权因子[ω],给出了对流扩散方程的混合三次B-样条配点法。对对流扩散方程空间离散采用混合三次B-样条配点法和时间离散采用向前有限差分,引入参数[θ],建立差分格式。对差分格式的稳定性进行分析,得到稳定性条件。数值实验表明所构造方法的有效性,并且适当调整权因子[ω]和参数[θ]的值,可提高计算的精度。     

A general mesh independence principle for Newton's method applied to second order boundary value problems

Recent work has established that for certain classes of nonlinear boundary value problems, the number of Newton iterations applied to the related standard discrete problem for a given tolerance is independent of the mesh size when the mesh is sufficiently fine. This paper develops an extension of the mesh independence principle by relaxing the assumption on the differential equation, its boundary conditions, and the related difference approximation.     

Quintic splines method for second-order boundary value problems

Abstract

We develop a numerical method for computing smooth approximations to the solution of a system of second-order boundary value problems associated with obstacle, unilateral and contact problems based on uniform mesh quintic splines. It is shown that this method gives better approximations than those produced by other collocation, finite-difference and spline methods. A numerical example is given to illustrate the applicability of the new method.     

A parameter uniform B-spline collocation method for solving singularly perturbed turning point problem having twin boundary layers

A numerical scheme is proposed to solve singularly perturbed two-point boundary value problems with a turning point exhibiting twin boundary layers. The scheme comprises B-spline collocation method on a non-uniform mesh of Shishkin type. Asymptotic bounds are established for the derivative of the analytical solution of a turning point problem. The present method is boundary layer resolving as well as second-order accurate in the maximum norm. A brief analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter ? by decomposing the solution into smooth and singular components. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects.     

An efficient analytical approach for solving fourth order boundary value problems

Based on the homotopy analysis method (HAM), an efficient approach is proposed for obtaining approximate series solutions to fourth order two-point boundary value problems. We apply the approach to a linear problem which involves a parameter c and cannot be solved by other analytical methods for large values of c, and obtain convergent series solutions which agree very well with the exact solution, no matter how large the value of c is. Consequently, we give an affirmative answer to the open problem proposed by Momani and Noor in 2007 [S. Momani, M.A. Noor, Numerical comparison of methods for solving a special fourth-order boundary value problem, Appl. Math. Comput. 191 (2007) 218-224]. We also apply the approach to a nonlinear problem, and obtain convergent series solutions which agree very well with the numerical solution given by the Runge-Kutta-Fehlberg 4-5 technique.     

Numerical techniques for solving systems of second order boundary value problems

In this paper, we compare some collocation, finite difference and spline methods for solving a system of second order boundary value problems associated with obstacle problems. An example is given to illustrate the efficiency of these results.