Samuel N. Jator

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

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=x n+j , j=1, 2, …, r−1, and x=x n+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.

Trigonometrically fitted block Numerov type method for y'= f(x, y, y')

A trigonometrically fitted block Numerov type method (TBNM), is proposed for solving y′′ = f(x, y, y′) directly without reducing it to an equivalent first order system. This is achieved by constructing a continuous representation of the trigonometrically fitted Numerov method (CTNM) and using it to generate the well known trigonometrically fitted Numerov method (TNUM) and three new additional methods, which are combined and applied in block form as simultaneous numerical integrators. The stability property of the TBNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

Block hybrid method using trigonometric basis for initial value problems with oscillating solutions

A continuous hybrid method using trigonometric basis (CHMTB) with one ‘off-step’ point is developed and used to produce two discrete hybrid methods which are simultaneously applied as numerical integrators by assembling them into a block hybrid method with trigonometric basis (BHMTB) for solving oscillatory initial value problems (IVPs). The stability property of the BHMTB is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

Continuous block backward differentiation formula for solving stiff ordinary differential equations

In this paper, we consider an implicit Continuous Block Backward Differentiation formula (CBBDF) for solving Ordinary Differential Equations (ODEs). A block of p new values at each step which simultaneously provide the approximate solutions for the ODEs is derived, where p is the number of points. A performance comparison of the continuous block methods is made with existing methods. Numerical results indicate that the CBBDF is more efficient in improving the number of integration steps with better accuracy.

A high order B-spline collocation method for linear boundary value problems

Abstract Collocation methods are investigated because of their simplicity and inherent efficiency for applications to linear boundary value problems on [ a , b ] . The objective of the present research is obtaining numerical solution of the boundary value problems for d th order linear boundary value problem by a B-spline collocation method using B-splines of order k and their index of regularity is m , d - 1 ⩽ m ⩽ k - 2 . The collocation points in our method form a strictly increasing sequence of points in [ a , b ] , each interior j th collocation point belongs to the interior of the compact support of corresponding j th B-spline basis element, and the number of B-spline basis elements equals the number of collocation points. The order of accuracy of the proposed method is shown to be optimal. The mathematical properties of this collocation method are less well established , primarily because the order of accuracy depends on the regularity and order of B-spline basis and location of the collocation points. The error analysis is through the Green’s function approach than the matrix approach. We compare the efficiency and accuracy of our method to nodal and orthogonal collocation methods as applied to linear ordinary differential equations with boundary conditions. Our collocation method like Greville collocation method is more convenient than nodal or orthogonal collocation because exactly the correct number of collocation points is available. The Greville and Botella collocation methods are special cases of our collocation method.

An algorithm for second order initial and boundary value problems with an automatic error estimate based on a third derivative method

A third derivative method (TDM) with continuous coefficients is derived and used to obtain a main and additional methods, which are simultaneously applied to provide all approximations on the entire interval for initial and boundary value problems of the form y′′ = f(x, y, y′). The convergence analysis of the method is discussed. An algorithm involving the TDMs is developed and equipped with an automatic error estimate based on the double mesh principle. Numerical experiments are performed to show efficiency and accuracy advantages.

A continuous two-step method of order 8 with a block extension for y″=f(x,y,y′)

Abstract We derive a piecewise continuous hybrid third derivative approximation (CHTDA) that is defined for all values of the independent variable on the range of interest. This continuous approximation has the ability to provide a continuous solution between all the grid points with a uniform accuracy comparable to that obtained at the grid points. Furthermore, the CHTDA is used to produce several implicit hybrid third derivative formulas (IHTDFs), which are expressed in block form and applied as a block hybrid third derivative algorithm (BHTDA) for the numerical solution of second order initial value problems. The stability of the block extension is discussed. Numerical examples are given to illustrate the accuracy and efficiency of the BHTDA.

A Self Starting Block Adams Methods for Solving Stiff Ordinary Differential Equation

A self starting multistep method with continuous coefficient is developed through interpolation and collocation procedures and used to obtain the Adams-type methods that are assembled into block matrix equation for solving initial value problems (IVPs) with emphasis on stiff ordinary differential equations. The methods are numerical integrators which are combined to simultaneously provide the approximate solution for IVPs. The stability of the resulting block methods is discussed and the superiority of the block methods over existing ones, such as the boundary value methods and the standard Adams methods is established numerically.

Implicit third derivative Runge-Kutta-Nyström method with trigonometric coefficients

The paper presents a trigonometrically-fitted implicit third derivative Runge-Kutta-Nystöm method (TTRKNM) whose coefficients depend on the frequency and stepsize for periodic initial value problems. The TTRKNM is a pair of methods which is obtained from its continuous version and applied to produce simultaneous approximations to the solution and its first derivative at each point in the interval of interest. A discussion of the stability property of the method is given. Numerical experiments are performed to demonstrate the accuracy and efficiency of the method.

On a family of trigonometrically fitted extended backward differentiation formulas for stiff and oscillatory initial value problems

A Class of Continuous Extended Backward Differentiation Formula using Trigonometric Basis functions (CEBDTB) with one superfuture point is constructed and used to generate a family of Trigonometrically Fitted Extended Backward Differentiation Formula (TFEBDF) and other discrete methods as by-products. This family of discrete schemes together with the TFEBDF are simultaneously applied as numerical integrators by assembling them into a Block Trigonometrically Fitted Extended Backward Differentiation Method (BTFEBDM). The error analysis, stability properties, and implementation of the BTFEBDM are discussed, and the class of methods is shown to be suitable for oscillatory and/or stiff Initial Value Problems (IVPs). The performance of the method is demonstrated on some numerical examples to show its accuracy and computational efficiency.