E.H. Twizell

The numerical solution of fifth-order boundary value problems with sixth-degree B-spline functions

Abstract The numerical solution of fifth-order, nonlinear boundary-value problems with two-point boundary conditions is considered. A sixth-degree B-spline approximation is used to construct the numerical method, the coefficients of which are detailed in a table. The method is tested on two problems, one linear and one nonlinear.

Second-order,L0-stable methods for the heat equation with time-dependent boundary conditions

A family of second-order,L0-stable methods is developed and analysed for the numerical solution of the simple heat equation with time-dependent boundary conditions. Methods of the family need only real arithmetic in their implementation. In a series of numerical experiments no oscillations, which are a feature of some results obtained usingA0-stable methods, are observed in the computed solutions. Splitting techniques for first- and second-order hyperbolic problems are also considered.

Numerical methods for the solution of special sixth-order boundary-value problems

A family of numerical methods is developed for the solution of special nonlinear sixth-order boundary-value problems. Methods with second-, fourth-, sixth- and eighth-order convergence are contained in the family. Global extrapolation procedures on two and three grids, which increase the order of convergence, are outlined.

A rational cubic spline based on function values

Abstract A rational spline based on function values only is constructed, which is of the same order as a cubic spline. The rational spline may be used to control the position and shape of a curve or surface. The approximation properties of this spline are studied.

The numerical solution of third-order boundary-value problems with fourth-degree & B-spline functions

Third-order linear and non-linear boundary-value problems are solved using fourth-degree B-splines. The convergence of the method is discussed. The method is tested on two problems from the literature

Non polynomial spline approach to the solution of a system of third-order boundary-value problems

We use a quartic spline equivalent nonpolynomial spline functions to develop a numerical method for computing approximations to the solution of a system of third-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the present method gives approximations which are better than those produced by other collocation, finite-difference, and spline methods. A numerical example is given to illustrate the applicability and efficiency of the new method.

A finite-difference method for solving the cubic Schro¨dinger equation

A family of finite-difference methods is used to transform the initial/boundary-value problem associated with the nonlinear Schrodinger equation into a first-order, linear, initial-value problem. Numerical methods are developed by replacing the time and space derivatives by central-difference replacements. The resulting finite-difference methods are analysed for local truncation, errors, stability and convergence. The results of a number of numerical experiments are given.

Numerical methods for special nonlinear boundary-value problems of order 2 m

Abstract General algorithms based on transforming the problem into a system of m second-order problems, and on solving the problem as a (2m)th-order problem are proposed for obtaining the solution of the special nonlinear boundary-value problem y(2m)(x) = f(x, y), X0 ⩽ x ⩽ X1, y(2i)X0) = A2i, y(2i)(X1 = B2i, i = 0, 1,…, m - 1 and m ⩾ 1. The former is to be preferred to the latter, as it does not suffer from rounding errors and also it is easy and inexpensive to implement. The latter is found to be vulnerable to word-length problems when it is used to solve high-order boundary-value problems. Beyond using finite-difference methods, an alternative method, based on the assumption that the desired solution can be adequately represented by a finite polynomial, is also outlined for solving the special nonlinear boundary-value problem. Comparisons with the proposed equivalent low-order form show that the results obtained are less accurate and, apparently, at more computational expense. A simple example is carried out on the special nonlinear eighth-order boundary-value problem to illustrate the results given by the methods. Moreover, numerical results for the special nonlinear tenth-order boundary-value problem using the equivalent low-order method are also reported.

NUMERICAL METHODS FOR EIGHTH-, TENTH- AND TWELFTH-ORDER EIGENVALUE PROBLEMS ARISING IN THERMAL INSTABILITY

Second-order finite-difference methods are developed for the numerical solutions of the eighth-, tenth- and twelfth-order eigenvalue problems arising in the study of the effect of rotation on a horizontal layer of fluid heated from below. Instability setting-in as overstability may be modelled by an eighth-order ordinary differential equation. When a uniform magnetic field also acts across the fluid in the same direction as gravity, instability setting-in as ordinary convection may be modelled by a tenth-order differential equation, while instability setting-in as overstability may be modelled by a twelfth-order differential equation. The numerical methods are developed by making direct replacements of the derivatives in the differential equations and then by computing the eigenvalues, which may incorporate Rayleigh number, horizontal wave speed and a time constant, from the resulting algebraic eigenvalue problem. The eigenvalues are also computed by writing the differential equations as systems of second-order differential equations and then using second- and fourth-order methods to obtain the eigenvalues. Numerical results obtained using the two approaches are compared with estimates appearing in the literature.

A new C2 rational interpolation based on function values and constrained control of the interpolant curves

In this paper a new method is developed to create a high-order smoothness interpolation using values of the function being interpolated. This is a kind of rational cubic interpolation with quadratic denominator. This rational spline not only belongs to C^2 in the interpolating interval, but could also be used to constrain the shape of the interpolant curve such as to force it to be in the given region, all because of the selectable parameters in the rational spline itself. The more important achievement mathematically of this method is that the uniqueness of the interpolating function for the given data would be replaced by the uniqueness of the interpolating curve for the given data and the selected parameters.