scholarly journals Collocation Techniques for Block Methods for the Direct Solution of Higher Order Initial Value Problems of Ordinary Differential Equations

2017 ◽  
Vol 3 (4) ◽  
pp. 39
Author(s):  
Kamoh Nathaniel Mahwash
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Higher Order ◽  
Direct Solution ◽  
Initial Value ◽  
Block Methods

A Collocation Method for Direct Numerical Integration of Initial Value Problems in Higher Order Ordinary Differential Equations

2011 ◽  
Vol 57 (2) ◽  
pp. 311-321
Author(s):  
R. Adeniyi ◽  
M. Alabi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Higher Order ◽  
Initial Value ◽  
Integration Schemes ◽  
Collocation Technique ◽  
Direct Numerical Integration

A Collocation Method for Direct Numerical Integration of Initial Value Problems in Higher Order Ordinary Differential Equations This paper concerns the solution of initial value problems (IVPs) in ordinary differential equations (ODEs) of orders higher than unity. The Chebyshev polynomials is hereby adopted as basis function in a multi-step collocation technique for the derivation of continuous integration schemes for direct solution of these ODEs without recourse to the conventional approach of first reducing such to their equivalent first order differential systems.

scholarly journals A Quantitative Comparison of Numerical Method for Solving Stiff Ordinary Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
S. A. M. Yatim ◽  
Z. B. Ibrahim ◽  
K. I. Othman ◽  
M. B. Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Computation Time ◽  
Quantitative Comparison ◽  
Initial Value ◽  
Step Size ◽  
Stiff Ordinary Differential Equations ◽  
Block Methods ◽  
Variable Step

We derive a variable step of the implicit block methods based on the backward differentiation formulae (BDF) for solving stiff initial value problems (IVPs). A simplified strategy in controlling the step size is proposed with the aim of optimizing the performance in terms of precision and computation time. The numerical results obtained support the enhancement of the method proposed as compared to MATLAB's suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

scholarly journals Solving third order ordinary differential equations directly using hybrid numerical models

2020 ◽  
pp. 69-76
Author(s):  
J. O. Kuboye ◽  
O. F. Quadri ◽  
O. R. Elusakin
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Numerical Models ◽  
Gaussian Elimination ◽  
Initial Value ◽  
Third Order ◽  
Block Methods ◽  
Grid Points

In this work, numerical methods for solving third order initial value problems of ordinary differential equations are developed. Multi-step collocation is used in deriving the methods, where power series approximate solution is employed as a basis function. Gaussian elimination approach is considered in finding the unknown variables $a_j, j=0(1)8$ in interpolation and collocation equations, which are substituted into the power series to give the continuous implicit schemes. The discrete schemes and its derivatives are derived by evaluating the grid and non-grid points. These schemes are arranged in a matrix form to produce block methods. The order of the developed methods are found to be six. The numerical results proved the efficiency of the methods over the existing methods.

scholarly journals The Treatment of Second Order Ordinary Differential Equations Using Equidistant One-step Block Hybrid

2019 ◽  
pp. 1-9
Author(s):  
Y. Skwame ◽  
J. Sabo ◽  
M. Mathew
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Second Order ◽  
Direct Solution ◽  
Block Method ◽  
Initial Value ◽  
One Step ◽  
Better Than

A general one-step hybrid block method with equidistant of order 6 has been successfully developed for the direct solution of second order IVPs in this article. Numerical analysis shows that the developed method is consistent and zero-stable which implies its convergence. The analysis of the new method is examined on two highly and mildly stiff second-order initial value problems to illustrate the efficiency of the method. It is obvious that our method performs better than the existing method within which we compare our result with. Hence, the approach is an adequate one for solving special second order IVPs.

scholarly journals A Family of Stiffly Stable Second Derivative Block Methods for Initial Value Problems in Ordinary Differential Equations

2019 ◽  
pp. 221-239 ◽  
Author(s):  
Samuel A. Ajayi ◽  
Kingsley O. Muka ◽  
Oluwasegun M. Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Absolute Stability ◽  
Truncation Error ◽  
Second Derivative ◽  
Initial Value ◽  
First Order ◽  
Stiff Ordinary Differential Equations ◽  
Block Methods

In this paper, we present a family of stiffly stable second derivative block methods (SDBMs) suitable for solving first-order stiff ordinary differential equations (ODEs). The methods proposed herein are consistent and zero stable, hence, they are convergent. Furthermore, we investigate the local truncation error and the region of absolute stability of the SDBMs. A flowchart, describing this procedure is illustrated. Some of the developed schemes are shown to be A-stable and L-stable, while some are found to be A()-stable. The numerical results show that our SDBMs are stiffly stable and give better approximations than the existing methods in the literature.

scholarly journals Optimization of one-step hybrid method for direct solution of fifth order ordinary differential equations of initial value problems

2021 ◽  
Vol 9 (1) ◽  
pp. 239-249
Author(s):  
Roseline Bosede Ogunrinde ◽  
Ololade Funmilayo Fayose ◽  
Taiwo Stephen Fayose
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Method ◽  
Basis Function ◽  
Initial Value Problems ◽  
Test Problems ◽  
Direct Solution ◽  
Initial Value ◽  
Unknown Parameters ◽  
Fifth Order

This paper focuses on the derivation, analysis and implementation of a hybrid method by optimizing the order of the method by introduction of six-hybrid points for direct solution of fifth order ordinary differential equations of initial value problems (IVPs). Power series was used as the basis function for the solution of the IVP. The basis function was interpolated at some selected hybrid points whereas the fifth derivative of the approximate solution was collocated at all the interval of integration of the method to generate a system of linear equations for the determination of the unknown parameters. The derived method was tested for consistency, zero stability, convergence and absolute stability. The method was tested with two linear test problems to confirm its accuracy and usability. The comparison of the results with some existing methods shows the superiority of the accuracy of the method.

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