scholarly journals Modified Hybrid Method with Four Stages for Second Order Ordinary Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1028
Author(s):  
Faieza Samat ◽  
Eddie Shahril Ismail
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Method ◽  
Hybrid Methods ◽  
High Accuracy ◽  
Parameter Selection ◽  
Second Order ◽  
Duffing Equation ◽  
Numerical Comparisons ◽  
Phase Properties

A modified explicit hybrid method with four stages is presented, with the first stage exactly integrating exp(wx), while the remaining stages exactly integrate sin(wx) and cos(wx). Special attention is paid to the phase properties of the method during the process of parameter selection. Numerical comparisons of the proposed and existing hybrid methods for several second-order problems show that the proposed method gives high accuracy in solving the Duffing equation and Kramarz’s system.

scholarly journals A Four-Stage Fifth-Order Trigonometrically Fitted Semi-Implicit Hybrid Method for Solving Second-Order Delay Differential Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Sufia Zulfa Ahmad ◽  
Fudziah Ismail ◽  
Norazak Senu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Method ◽  
Delay Differential Equations ◽  
Scientific Literature ◽  
Second Order ◽  
New Method ◽  
Implicit Methods ◽  
Delay Differential ◽  
Fifth Order

We derived a two-step, four-stage, and fifth-order semi-implicit hybrid method which can be used for solving special second-order ordinary differential equations. The method is then trigonometrically fitted so that it is suitable for solving problems which are oscillatory in nature. The methods are then used for solving oscillatory delay differential equations. Numerical results clearly show the efficiency of the new method when compared to the existing explicit and implicit methods in the scientific literature.

scholarly journals An Order Four Continuous Numerical Method for Solving General Second Order Ordinary Differential Equations

2021 ◽  
pp. 42-47
Author(s):  
Friday Obarhua ◽  
Oluwasemire John Adegboro
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Initial Value Problems ◽  
Hybrid Methods ◽  
Second Order ◽  
Initial Value ◽  
Hybrid Numerical Method ◽  
Order Four ◽  
Finite Domains

Continuous hybrid methods are now recognized as efficient numerical methods for problems whose solutions have finite domains or cannot be solved analytically. In this work, the continuous hybrid numerical method for the solution of general second order initial value problems of ordinary differential equations is considered. The method of collocation of the differential system arising from the approximate solution to the problem is adopted using the power series as a basis function. The method is zero stable, consistent, convergent. It is suitable for both non-stiff and mildly-stiff problems and results were found to compete favorably with the existing methods in terms of accuracy.

scholarly journals Boundary Value Methods for Second-Order PDEs via the Lanczos-Chebyshev Reduction Technique

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
T. A. Biala ◽  
S. N. Jator ◽  
R. B. Adeniyi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
High Accuracy ◽  
Second Order ◽  
Reduction Technique ◽  
Boundary Value Methods

In this paper, we study the performance of Boundary Value Methods (BVMs) on second-order PDEs. The PDEs are transformed into a system of second-order ordinary differential equations (ODEs) using the Lanczos-Chebyshev reduction technique. The conditions under which the BVMs converge and the computational complexities of the algorithms are discussed. Numerical illustrations are given to show the simplicity and high accuracy of the approach.

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