An Accuracy-preserving Block Hybrid Algorithm for the Integration of Second-order Physical Systems with Oscillatory Solutions

Authors

  • Joshua Sunday
    Department of Mathematics, University of Jos, Jos 930003, Nigeria
    https://orcid.org/0000-0001-6304-4965
  • Joel N. Ndam
    Department of Mathematics, University of Jos, Jos 930003, Nigeria
  • Lydia J. Kwari
    Department of Mathematics, Federal College of Education, Pankshin 933105, Nigeria

Keywords:

Accuracy-preserving, algorithm, block hybrid method, oscillation, physical systems, second-order

Abstract

It is a known fact that in most cases, to integrate an oscillatory problem, higher order A-stable methods are often needed. This is because such problems are characterized by stiffness, chaos and damping, thus making them tedious to solve. However, in this research, an accuracy-preserving relatively lower order Block Hybrid Algorithm (BHA) is proposed for solution of second-order physical systems with oscillatory solutions. The sixth order algorithm was derived using interpolation and collocation of power series within a single step interval [tn; tn+1]. In order to circumvent the Dahlquist-barrier and also obtain an accuracy-preserving algorithm, four o-step points were incorporated within the single step interval. A number of special cases of oscillatory problems were solved using the proposed method and the results obtained clearly showed that it outperformed other existing methods we compared our results with even though the BHA is of lower order relative to such methods. Some of the second-order physical systems considered were the Kepler, Bessel and damped problems. Some important properties of the BHA were also analyzed and the results of the analysis showed that it is consistent, zero-stable and convergent 

Dimensions

H. Soraya, “Pendulum with aerodynamic and viscous damping”, Journal of Applied Information, Communication and Technology 3 (2016) 43.

S. N. Jator & K. L. King, “Integrating oscillatory general second-order initial value problems using a block hybrid method of order 11”, Mathematical Problems in Engineering (2018) 3750274.

D. V. V. Wend, “Uniqueness of solution of ordinary differential equations”, The American Mathematical Monthly 74 (1967) 948.

L. Brugnano and D. Trigiante, Solving ODEs by Multistep Initial and Boundary Value Methods, Gordon Breach, Amsterdam, (1998).

M. I. Modebei, O. O. Olaiya & I. P. Ngwongwo, “Computational study of some three-step hybrid integrators for solution of third order ordinary differential equations”, Journal of Nigerian Society of Physical Sciences

(2021) 459.

F. O. Obarhua & O. J. Adegboro, “Order four continuous numerical method for solving general second order ordinary differential equations”, Journal of Nigerian Society of Physical Sciences 3 (2021) 42.

E. Hairer, “A one-step method of order 10 for y00 = f (x; y)”, IMA Journal of Numerical Analysis 2 (1982) 83.

R. D’Ambrosio, M. Ferro & B. Paternoster, “Two-step hybrid collocation methods fory00 = f (x; y)”, Applied Mathematics Letters 22 (2009) 1076.

S. D. Yakubu, Y. A. Yahaya & K. O. Lawal,”3-point block hybrid linear multistep methods for solution of special second order ordinary differential equations”, Journal of the Nigerian Mathematical Society 40 (2021) 149.

F. O. Obarhua & S. J. Kayode, “Continuous explicit hybrid method for solving second rder ordinary differential equations”, Pure and Applied Mathematics Journal 9 (2020) 26.

R. Allogmany & F. Ismail, “Direct solution of using three point block method of order eight with applications”, Journal of King Saud University-Science 33 (2021) 101337.

B. Y. Guo & J. P. Yan, “Legendre-Gauss collocation method for initial value problems of second order ordinary differential equations”, Applied Numerical Mathematics 59 (2009) 1386.

S. N. Jator, “Solving second order initial value problems by hybrid method without predictor”, Appl. Math. Comput. 217 (2010) 4036.

H. Ramos, Z. Kalogiratou, T. Monovasilis & T. E. Simos, “A trigonometrically fitted optimized two-step hybrid block method for solving initialvalue problems of the form y00 = f (x; y; y0) with oscillatory solutions”,

Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2014.

F. F. Ngwane & S. N. Jator, “Block hybrid method using trigonometric basis for initial value problems with oscillating solutions”, Numerical Algorithms 63 (2013) 713.

B. Wang, A. Iserles & X. Wu, “Arbitrary-Order Trigonometric Fourier Collocation Methods for Multi-Frequency Oscillatory Systems”, Foundations of Computational Mathematics 16 (2016) 158.

S. N. Jator & H. B. Oladejo, “Block Nystrom method for singular differential equations of the Lane-Emden type and problems with highly oscillatory solutions”, International Journal of Applied and Computational Mathematics 3 (2017) 1385.

J. Sunday, “Optimized two-step second derivative methods for the solutions of stiff ystems”, Journal of Physics Communications 6 (2022) 055016.

H. Ramos, S. Mehta & J. Vigo-Aguiar, “A unified approach for the development of k-step block Falkner-type methods for solving general second order initial value problems in ODEs”, Journal of Computational and Applied

Mathematics 318 (2017) 550.

O. Adeyeye & O. Zurmi, “Maximal order block method for the solution of second order ordinary differential equations”, IAENG International Journal of Applied Mathematics 46 (2016) 4.

J. Sunday, G. M. Kumleng, N. M. Kamoh, J. A. Kwanamu, Y. Skwame & O. Sarjiyus, “Implicit four-point hybrid block integrator for the simulations of stiff models”, Journal of the Nigerian Society of Physical Sciences

(2022) 287.

V. J. Shaalini & S. E. Fadugba, “A new multistep method for solving delay differential equations using Lagrange interpolation”, Journal of Nigerian Society of Physical Sciences 3 (2021) 159.

M. Kida, S. Adamu, O. O. Aduroja & T. P. Pantuvo, “Numerical solution of stiff and oscillatory problems using third derivative trigonometrically fitted block method”, Journal of Nigerian Society of Physical Sciences 4 (2022) 34.

S. O. Fatunla, “Numerical integrators for stiff and highly oscillatory differential equations”, Mathematics of Computation 34 (1980) 373.

J. D. Lambert, Numerical methods for ordinary differential systems: The initial value problem, JohnWiley & Sons LTD, United Kingdom (1991).

J. D. Lambert, Computational methods in ordinary differential equations, John Wiley & Sons, Inc., New York (1973).

L. J. Kwari, J. Sunday, J. N. Ndam & A. A. James, “On the numerical approximation and simulation of damped and undamped Duffing oscillators”, Science Forum (Journal of Pure and Applied Sciences) 21 (2021) 503.

M. R. Odekunle, A. O. Adesanya & J. Sunday, “4-Point Block Method for the Direct Integration of First-Order Ordinary Differential Equations”, International Journal of Engineering Research and Applications 5 (2012) 1182.

R. H. Ibrahim & A. H. Saleh, “Re-evaluation solution methods for Kepler’s equation of an elliptical orbit”, Iraqi Journal of Science 60 (2019) 2269.

R. I. Abdulganiy, H. Ramos, A. O. Akinfenwa & S. A. Okunuga, “A functionally-fitted block numerov method for solving second-order initial value problems with oscillatory solutions”, Mediterr. J. Math 18 (2021) 259.

S. O. Fatunla, “Numerical integrators for stiff and highly oscillatory differential equations”, Math Comput. 34 (1980) 373.

S. Faydaoglu & T. Ozis, “Periodic solutions for certain non-smooth oscillators with high nonlinearities” Appl. Comput. Math. 20 (2021) 366.

F. Bakhtiari-Nejad & M. Nazari, “Nonlinear vibration analysis of isotropic cantilever plate with visco-elastic laminate”, Nonlinear Dynamics 56 (2009) 325.

J. Sunday, A. Shokri & D. Marian, “Variable Step Hybrid Block Method for the Approximation of Kepler Problem”, Fractal and Fractional 6 (2022) 343.

J. Guckenheimer & P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer-Verlag, (1983).

P. Harihara & N. D. Childs, “Solving problems in dynamics and vibrations using MATLAB”, Department of Mechanical Engineering, Texas A and M University College Station (2020) 1.

B. S. H. Kashkari & M. I. Syam, “Optimization of one-step block method with three hybrid points for solving first-order ordinary differential equations”, Results in Physics 12 (2019) 592.

J. Sunday, A. Shokri, R. O. Akinola, K. V. Joshua & K. Nonlaopon, “A convergence-preserving non-standard finite difference scheme for the solutions of singular Lane-Emden equations”, Results in Physics 42 (2022) 106031.

S. S. Shyam, B. Omar, A. Hijaz & C. Yu-Ming, “Second order differential equation: oscillation theorems and applications”, Mathematical Problems in Engineering (2020) 1.

L. Tongxing, V. R. Yuriy & T. Shuhong, “Oscillation of second-order nonlinear differential equations with damping”, Mathematica Slovaca 64 (2014) 1227.

J. Sunday, A. Shokri, J. A. Kwanamu & K. Nonlaopon, “Numerical Integration of Stiff Differential Systems using Non-Fixed Step-Size Strategy”, Symmetry 14 (2022) 1575.

F. J. Agocs, W. J. Handley, A. N. Lasenby & M. P. Hobson, “Efficient method for solving highly oscillatory ordinary differential equations with applications to physical systems”, Physical Review Research 2 (2020) 1.

P. F. Rowat & A. I. Selverston, “Modeling the gastric mill central pattern generator of the lobster with a relaxation-oscillator network”, Journal of Neurophysiology 70 (1993) 1030.

S. Nourazar & A. Mirzabeigy, “Approximate solution for nonlinear Duffing oscillator with damping e ect using the modified differential transform method”, Scientia Iranica B 20 (2013) 364.

J. Sunday, C. Chigozie, E. O. Omole & J. B. Gwong, “A pair of three-step hybrid block methods for the solutions of linear and nonlinear first-order systems”, European Journal of Mathematics and Statistics 3 (2022) 14.

Published

2023-01-14

How to Cite

An Accuracy-preserving Block Hybrid Algorithm for the Integration of Second-order Physical Systems with Oscillatory Solutions. (2023). Journal of the Nigerian Society of Physical Sciences, 5(1), 1017. https://doi.org/10.46481/jnsps.2023.1017

Issue

Volume 5, Issue 1, February 2023

Section

Original Research
Copyright & Licensing

Copyright (c) 2023 Joshua Sunday, Joel N. Ndam, Lydia J. Kwari

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

An Accuracy-preserving Block Hybrid Algorithm for the Integration of Second-order Physical Systems with Oscillatory Solutions. (2023). Journal of the Nigerian Society of Physical Sciences, 5(1), 1017. https://doi.org/10.46481/jnsps.2023.1017

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