Trigonometrically fitted multi-step hybrid methods for oscillatory special second-order initial value problems

2017 ◽  
Vol 95 (5) ◽  
pp. 979-997 ◽  
Author(s):  
Jiyong Li ◽  
Ming Lu ◽  
Xuli Qi
Keyword(s):  
Initial Value Problems ◽  
Hybrid Methods ◽  
Second Order ◽  
Initial Value

scholarly journals Direct Solution of Second Order Ordinary Differential Equations Using a Class of Hybrid Block Methods

2020 ◽  
Vol 5 (2) ◽  
Author(s):  
Emmanuel A Areo ◽  
Nosimot O Adeyanju ◽  
Sunday J Kayode
Keyword(s):  
Initial Value Problems ◽  
Hybrid Methods ◽  
Multistep Method ◽  
Second Order ◽  
Block Method ◽  
Initial Value ◽  
Order Error ◽  
Block Methods ◽  
Grid Points ◽  
Step Number

This research proposes the derivation of a class of hybrid methods for solution of second order initial value problems (IVPs) in block mode. Continuous linear multistep method of two cases with step number k = 4 is developed by interpolating the basis function at certain grid points and collocating the differential system at both grid and off-grid points. The basic properties of the method including order, error constant, zero stability, consistency and convergence were investigated. In order to examine the accuracy of the methods, some differential problems of order two were solved and results generated show a better performance when comparison is made with some current methods.Keywords- Block Method, Hybrid Points, Initial Value Problems, Power Series, Second Order 

scholarly journals Optimized Hybrid Methods for Solving Oscillatory Second Order Initial Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
N. Senu ◽  
F. Ismail ◽  
S. Z. Ahmad ◽  
M. Suleiman
Keyword(s):  
Hybrid Method ◽  
Initial Value Problems ◽  
Scientific Literature ◽  
Hybrid Methods ◽  
Second Order ◽  
Numerical Results ◽  
Phase Lag ◽  
Initial Value ◽  
Oscillatory Initial Value Problems

Two-step optimized hybrid methods of order five and order six are developed for the integration of second order oscillatory initial value problems. The optimized hybrid method (OHMs) are based on the existing nonzero dissipative hybrid methods. Phase-lag, dissipation or amplification error, and the differentiation of the phase-lag relations are required to obtain the methods. Phase-fitted methods based on the same nonzero dissipative hybrid methods are also constructed. Numerical results show that OHMs are more accurate compared to the phase-fitted methods and some well-known methods appeared in the scientific literature in solving oscillating second order initial value problems. It is also found that the nonzero dissipative hybrid methods are more suitable to be optimized than phase-fitted methods.

Numerical simulation of second-order initial-value problems using a new class of variable coefficients and two-step semi-hybrid methods

SIMULATION ◽  
2021 ◽  
pp. 003754972098082
Author(s):  
Ali Shokri ◽  
Mohammad Mehdizadeh Khalsaraei ◽  
Hamid Mohammad-Sedighi ◽  
Ali Atashyar
Keyword(s):  
Numerical Simulation ◽  
Initial Value Problems ◽  
Hybrid Methods ◽  
Variable Coefficients ◽  
Second Order ◽  
Initial Value ◽  
Hybrid Schemes ◽  
New Class ◽  
New Family ◽  
Algebraic Order

In this paper, a new family of two-step semi-hybrid schemes of the 12th algebraic order is proposed for the numerical simulation of initial-value problems of second-order ordinary differential equations. The proposed methods are symmetric and belong to the family of multiderivative methods. Each method of the new family appears to be hybrid, but after implementing the hybrid terms, it will continue as a multiderivative method. Therefore, the designation semi-hybrid is used. The consistency, convergence, stability, and periodicity of the methods are investigated and analyzed. In order to show the accuracy, consistency, convergence, and stability of the proposed family, it was tested on some well-known problems, such as the undamped Duffing’s equation. The simulation results demonstrate the efficiency and advantages of the proposed method compared to the currently available methods.

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 Delta-Nabla Type Maximum Principles for Second-Order Dynamic Equations on Time Scales and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-28
Author(s):  
Jiang Zhu ◽  
Dongmei Liu
Keyword(s):  
Time Scales ◽  
Initial Value Problems ◽  
Second Order ◽  
Dynamic Equations ◽  
Maximum Principles ◽  
Uniqueness Theorems ◽  
Initial Value ◽  
Approximation Theorems ◽  
Construction Techniques ◽  
Linear And Nonlinear

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.

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