A Six-Step Continuous Multistep Method For The Solution Of General Fourth Order Initial Value Problems Of Ordinary Differential Equations

In this paper, continuous Linear Multistep Method (LMM) for the direct solution of fourth order initial value problems in ordinary differential equation is derived. The study provides the use of both collocation and interpolation techniques to obtain the schemes. Direct form of power series is used as basis function for approximation. An order six symmetric and zero-stable method is obtained. To implement our method, predictors of the same order of accuracy as the main method were developed using Taylor’s series algorithm.  This implementation strategy is found to be efficient and more accurate as the result has shown in the numerical experiments. The result obtained confirmed the superiority of our method over existing schemes Keywords: Direct method; Fourth order; interpolation; collocation multistep methods,;power series; approximate solutions

Three Steps Hybrid Block Method for the Solution of General Second Order Ordinary Differential Equations

Block method is adopted in this paper for the direct solution of second order ordinary differential equations. The method is derived by collocation and interpolation of power series approximate solution to give a continuous hybrid linear multistep method which is implemented in block method to derive the independent solution at selected grid points. The properties of the derived scheme were investigated and found to be zero-stable, consistent and convergent. The efficiency of the derived method was tested and found to compare favourably with the existing methods

Analysis of a new numerical approach to solutions of heat conduction equations arising from heat diffusion

In this work, a new numerical finite difference scheme with the aim of obtaining a new numerical scheme that will be used to solve for the solution of Partial Differential Equations (PDE) arising from heat conduction equation is developed. This is significant because in recent times there is a growing interest in literatures to obtain a continuous numerical method for solving PDE. The numerical accuracy of this new approach is also studied. Detailed numerical results have shown that the new method provides better results than the known explicit finite difference method by Schmidt. And in terms of stability, the new scheme has been able to clearly shown that it is more stable than the old Schmidt explicit method. There is no semi-discretization involved and no reduction of PDE to a system of ODEs in the new approach, but rather a system of algebraic equations is directly obtained. MATLAB software wasused to solve for the desired solutions and the results obtained has shown that the method is near exact solutions

A Robust Implicit Optimal Order Formula for Direct Integration of Second Order Orbital Problems

In this paper, a robust implicit formula of optimal order for direct integration of general second order orbital problems of ordinary differential equations (ODEs) is proposed. This method is considered capable avoiding the computational burden and wastage in computer time in connection with the method of reduction to first order systems. The integration algorithms and analysis of the basic properties are based on the adoption of Taylor’s expansion and Dahlquist stability model test. The resultant integration formula is of order ten and it is zero-stable, consistent, convergent and symmetric. The numerical implementation of the method to orbital and two-body problems demonstrates increased accuracy with the same computational effort on comparison with similar second order formulas. Keywords: Optimal-order, Zero-stability, Convergence, Consistent, IVPs, Predictor-corrector, Error constant, Symmetric

Parallel algorithm with spectral convergence for nonlinear integro-differential equations

We discuss a numerical algorithm for solving nonlinear integro-differential equations, and illustrate our findings for the particular case of Volterra type equations. The algorithm combines a perturbation approach meant to render a linearized version of the problem and a spectral method where unknown functions are expanded in terms of Chebyshev polynomials (El-gendi's method). This approach is shown to be suitable for the calculation of two-point Green functions required in next to leading order studies of time-dependent quantum field theory.Comment: 15 pages, 9 figure

An Order-seven Implicit Symmetric Scheme Applied to Second Order Initial Value Problems of Differential Equations

In this paper, a five-step predictor-corrector method of algebraic order seven is presented for solving second order initial value problems of ordinary differential equations directly without reduction to first order systems. Analysis of the basic properties of the method is considered and found to be consistent, zero-stable and symmetric. Some sample linear and nonlinear problems are solved to demonstrate the applicability of the method. It is observed that the present method approximates the exact solution well when compared with the two existing schemes that solved the same set of problems. Keywords: zero-stability, convergence, consistent, predictor-corrector, error constant, symmetri

A Family of Implicit Higher Order Methods for the Numerical Integration of Second Order Differential Equations

A family of higher order implicit methods with k steps is constructed, which exactly integrate the initial value problems of second order ordinary differential equations directly without reformulation to first order systems. Implicit methods with step numbers  are considered. For these methods, a study of local truncation error is made with their basic properties. Error and step length control based on Richardson extrapolation technique is carried out. Illustrative examples are solved with the aid of MATLAB package. Findings from the analysis of the basic properties of the methods show that they are consistent, symmetric and zero-stable. The results obtained from numerical examples show that these methods are much more efficient and accurate on comparison. These methods are preferable to some existing methods owing to the fact that they are efficient and simple in terms of derivation and computation Keywords: Error constant, implicit methods, Order of accuracy, Zero-Stability, Symmetr