A P-stable linear multistep method for solving general third order initial value problems of ordinary differential equations without first reducing the problems into a system of first order equations is considered. The approach for the development of this method is essentially based on collocation of the differential system generated from a basis function. A predictor for the evaluation of $y_{n+k}$ for an odd $k\ge 3$ in the main method is also proposed. The two resulting methods, the corrector and the predictor are P-stable for $k = 3$ . These as a block are tested on a number of problems to show their efficiency. When the methods (the corrector and the predictor) are evaluated at $x = x_{n+3}$ identical schemes are obtained as special cases of the methods, while the set of first and second derivatives obtained from the corrector are different from those obtained from the predictor.

A P-Stable Linear Multistep Method For Solving General Third Order Ordinary Differential Equations

A class of numerical methods is proposed for solving general third-order ordinary differential equations directly by collocation at the grid points x = x n+j , i = 0(1)k and at an off grid point x = x n+u , where k is the step number of the method and u is an arbitrary rational number in (x n , x n+k ). A predictor of order 2k − 1 is also proposed to cater for y n+k in the main method. Taylor series expansion is employed for the calculation of y n+1, y n+2, y n+u and their higher derivatives. Evaluation of the resulting method at x = x n+k for any value of u in the specified open interval yields a particular discrete scheme as a special case of the method. The efficiency of the method is tested on some general initial value problems of third-order ordinary differential equations.

A class of hybrid collocation methods for third-order ordinary differential equations

In this paper, a new approach based on least squares support vector machines (LS-SVMs) is proposed for solving linear and nonlinear ordinary differential equations (ODEs). The approximate solution is presented in closed form by means of LS-SVMs, whose parameters are adjusted to minimize an appropriate error function. For the linear and nonlinear cases, these parameters are obtained by solving a system of linear and nonlinear equations, respectively. The method is well suited to solving mildly stiff, nonstiff, and singular ODEs with initial and boundary conditions. Numerical results demonstrate the efficiency of the proposed method over existing methods.

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Approximate Solutions to Ordinary Differential Equations Using Least Squares Support Vector Machines

In this paper, we propose a linear multistep method of order 5 that is self-starting for the direct solution of the general second-order initial value problem (IVP). The method is derived by the interpolation and collocation of the assumed approximate solution and its second derivative at x=xn+j, j=1, 2,..., r-1, and x=xn+j, j=1, 2,..., s-1, respectively, where r and s are the number of interpolation and collocation points, respectively. The interpolation and collocation procedures lead to a system of (r+s) equations involving (r+s) unknown coefficients, which are determined by the matrix inversion approach. The resulting coefficients are used to construct the approximate solution from which multiple finite difference methods (MFDMs) are obtained and simultaneously applied to provide a direct solution to IVPs. In particular, the method is implemented without the need for either predictors or starting values from other methods. Numerical examples are given to illustrate the efficiency of the method.

A self-starting linear multistep method for a direct solution of the general second-order initial value problem

Solving second order initial value problems by a hybrid multistep method without predictors

An important factor among others that should be considered in developing a numerical integrator for the solution of ordinary differential equations is the prudent management of computer time, which depends essentially on the number of functions to be evaluated per iteration. Recognising the importance of this factor, this article therefore proposes a class of four-step methods each with three functions evaluation per iteration. The procedure which yields a system of equations for stepnumber k ≥ 4 is based on collocation of the differential system at selected grid points which ensures the symmetry of the methods. The methods are compared for accuracy and computer time with an existing four-step method containing five functions evaluation per iteration (see Awoyemi, 1999a). Three other Predictors are similarly proposed for use in the main method.

D. O. Awoyemi

Papers

A new sixth-order algorithm for general second order ordinary differential equations

A class of hybrid collocation methods for third-order ordinary differential equations