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.

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 Study of Numerical Solution of Fourth Order Ordinary Differential Equations by fifth order Runge-Kutta Method

2019 ◽  
pp. 230-238
Author(s):  
Najmuddin Ahamad ◽  
Shiv Charan
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Initial Value Problems ◽  
Computational Cost ◽  
Kutta Method ◽  
Initial Value ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Fifth Order

In this paper we present fifth order Runge-Kutta method (RK5) for solving initial value problems of fourth order ordinary differential equations. In this study RK5 method is quite efficient and practically well suited for solving boundary value problems. All mathematical calculation performed by MATLAB software for better accuracy and result. The result obtained, from numerical examples, shows that this method more efficient and accurate. These methods are preferable to some existing methods because of their simplicity, accuracy and less computational cost involved.

A PHASE-FITTED AND AMPLIFICATION-FITTED EXPLICIT TWO-STEP HYBRID METHOD FOR SECOND-ORDER PERIODIC INITIAL VALUE PROBLEMS

2006 ◽  
Vol 17 (05) ◽  
pp. 663-675 ◽  
Author(s):  
HANS VAN DE VYVER
Keyword(s):  
Hybrid Method ◽  
Initial Value Problems ◽  
Numerical Results ◽  
New Method ◽  
Test Problems ◽  
Phase Lag ◽  
Initial Value ◽  
Periodic Initial Value Problems ◽  
Free Parameters ◽  
Fifth Order

In this paper a phase-fitted and amplification-fitted explicit two-step hybrid method is introduced. The construction is based on a modification of a fifth-order dissipative method recently developed by Franco.19 Two free parameters are added in order to nullify the phase-lag and the amplification. Numerical results obtained for well-known test problems show the efficiency of the new method when it is compared with other existing codes.

Sign in / Sign up

Export Citation Format

Share Document