Integrating Oscillatory General Second-Order Initial Value Problems Using a Block Hybrid Method of Order 11

In some cases, high-order methods are known to provide greater accuracy with larger step-sizes than lower order methods. Hence, in this paper, we present a Block Hybrid Method (BHM) of order 11 for directly solving systems of general second-order initial value problems (IVPs), including Hamiltonian systems and partial differential equations (PDEs), which arise in multiple areas of science and engineering. The BHM is formulated from a continuous scheme based on a hybrid method of a linear multistep type with several off-grid points and then implemented in a block-by-block manner. The properties of the BHM are discussed and the performance of the method is demonstrated on some numerical examples. In particular, the superiority of the BHM over the Generalized Adams Method (GAM) of order 11 is established numerically.

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DISSIPATIVE TRIGONOMETRICALLY-FITTED METHODS FOR SECOND ORDER IVPs WITH OSCILLATING SOLUTION

International Journal of Modern Physics C ◽

10.1142/s0129183102003954 ◽

2002 ◽

Vol 13 (10) ◽

pp. 1333-1345 ◽

Cited By ~ 16

Author(s):

T. E. SIMOS

Keyword(s):

Hybrid Method ◽

Initial Value Problems ◽

Oscillating Solution ◽

Second Order ◽

Initial Value ◽

Step Method ◽

Numerical Examples ◽

Oscillating Solutions ◽

Trigonometrical Fitting ◽

Trigonometrically Fitted Methods

In this paper a dissipative trigonometrically-fitted two-step explicit hybrid method is developed. This method is based on a dissipative explicit two-step method developed recently by Papageorgiou, Tsitouras and Famelis.6 Numerical examples show that the procedure of trigonometrical fitting is the only way in one to produce efficient dissipative methods for the numerical solution of second order initial value problems (IVPs) with oscillating solutions.

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

FUOYE Journal of Engineering and Technology ◽

10.46792/fuoyejet.v5i2.537 ◽

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

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An explicit hybrid method of Numerov type for second-order periodic initial-value problems

Journal of Computational and Applied Mathematics ◽

10.1016/0377-0427(94)00011-o ◽

1995 ◽

Vol 59 (1) ◽

pp. 79-90 ◽

Cited By ~ 21

Author(s):

J.M. Franco

Keyword(s):

Hybrid Method ◽

Initial Value Problems ◽

Second Order ◽

Initial Value ◽

Periodic Initial Value Problems

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Numerical Simulation of One Step Block Method for Treatment of Second Order Forced Motions in Mass-Spring Systems

Asian Journal of Research and Reviews in Physics ◽

10.9734/ajr2p/2021/v5i230157 ◽

2021 ◽

pp. 1-11

Author(s):

J. Sabo ◽

T. Y. Kyagya ◽

W. J. Vashawa

Keyword(s):

Numerical Simulation ◽

Initial Value Problems ◽

Second Order ◽

Block Method ◽

Initial Value ◽

Mathematical Problems ◽

One Step ◽

Grid Points ◽

The One ◽

Mass Spring

This paper discuss the numerical simulation of one step block method for treatment of second order forced motions in mass-spring systems of initial value problems. The one step block method has been developed with the introduction of off-mesh point at both grid and off- grid points using interpolation and collocation procedure to increase computational burden which may jeopardize the accuracy of the method in terms of error. The basic properties of the one step block method was established and numerical analysis shown that the one step block method was found to be consistent, convergent and zero-stable. The one step block method was simulated on three highly stiff mathematical problems to validate the accuracy of the block method without reduction, and obviously the results shown are more accurate over the existing method in literature.

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Efficient k-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly

Mathematics ◽

10.3390/math8101752 ◽

2020 ◽

Vol 8 (10) ◽

pp. 1752

Author(s):

Higinio Ramos ◽

Samuel N. Jator ◽

Mark I. Modebei

Keyword(s):

Computational Efficiency ◽

Initial Value Problems ◽

Numerical Experiments ◽

Computing Time ◽

Approximate Solutions ◽

Second Order ◽

Initial Value ◽

Special Equation ◽

Block Methods ◽

Grid Points

There are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of k-step block methods for directly solving general second-order initial-value problems. Each of these methods consists of a set of 2k multi-step formulas (although we will see that this number can be reduced to k+1 in case of a special equation) that provides approximate solutions at k grid points at once. The usual way to obtain these formulas is by using collocation and interpolation at different points, which are not all necessarily in the mesh (it may also be considered intra-step or off-step points). An important issue is that for each k, all of them are essentially the same method, although they can adopt different formulations. Nevertheless, the performance of those formulations is not the same. The analysis of the methods presented give some clues as how to select the most appropriate ones in terms of computational efficiency. The numerical experiments show that using the proposed formulations, the computing time can be reduced to less than half.

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New Algorithm for First order Stiff Initial Value Problems

Fasciculi Mathematici ◽

10.1515/fascmath-2017-0002 ◽

2017 ◽

Vol 58 (1) ◽

pp. 19-28 ◽

Cited By ~ 1

Author(s):

A. O. Adesanya ◽

R. O. Onsachi ◽

M. R. Odekunle

Keyword(s):

Initial Value Problems ◽

Numerical Experiments ◽

System Of Nonlinear Equations ◽

Initial Value ◽

Unknown Parameters ◽

First Order ◽

Continuous Scheme ◽

Discrete Methods ◽

Grid Points ◽

The Stability

AbstractIn this paper, we consider the development and implementation of algorithms for the solution of stiff first order initial value problems. Method of interpolation and collocation of basis function to give system of nonlinear equations which is solved for the unknown parameters to give a continuous scheme that is evaluated at selected grid points to give discrete methods. The stability properties of the method is verified and numerical experiments show that the new method is efficient in handling stiff problems.

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Optimized Hybrid Methods for Solving Oscillatory Second Order Initial Value Problems

Discrete Dynamics in Nature and Society ◽

10.1155/2015/217578 ◽

2015 ◽

Vol 2015 ◽

pp. 1-11 ◽

Cited By ~ 1

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.

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