The Simulation of One-Step Algorithms for Treating Higher Order Initial Value Problems
Asian Research Journal of Mathematics,
Page 34-47
DOI:
10.9734/arjom/2021/v17i930329
Abstract
The simulation of one-step methods using interpolation and collocation for the treatment of higher order initial value problems is proposed in this paper. The new approach is derived using interpolation and collocation as a basic function through power series polynomial, where the basic properties are also analyzed. The derived method is used to treat some highly stiff linear problems. The new approach compute clearly showed that the method is reliable, efficient and gives faster convergence when compared with those in literature.
Keywords:
- Algorithms
- higher order initial value problems
- one-step
- and simulation
How to Cite
Sabo, J., Ayinde, A. M., Ishaq, A. A., & Ajileye, G. (2021). The Simulation of One-Step Algorithms for Treating Higher Order Initial Value Problems. Asian Research Journal of Mathematics, 17(9), 34-47. https://doi.org/10.9734/arjom/2021/v17i930329
References
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Ogunware BG, Awoyemi DO, Adoghe LO, Olanegan OO, Omole EO. Numerical treatment of general third order ordinary differential equations using Taylor series as predictor. Phys. Sci. Int. J. 2018;17(3):1-8.
Jikantoro YD, Ismail F, Senu N, Ibrahim ZB. A new integrator for special third order differential equations with application to thin film flow problem. Indian J. Pure Appl. Math. 2018;49(1):151-167.
Adeyeye O, Omar Z. New self-starting approach for solving special third order initial value problems. Int. J. Pure Appl. Math. 2018;118(3):511-517.
Awoyemi DO, Kayode, SJ, Adoghe, LO. A five-step p-stable method for the numerical integration of third order ordinary differential equations. Amer. J. Comput. Math. 2014;4:119-126.
Skwame Y, Dalatu PI, Sabo J, Mathew M. Numerical application of third derivative hybrid block methods on third order initial value problem of ordinary differential equations. Int. J. Stat. Appl. Math. 2019;4(6):90-100.
Yakusak NS, Emmanue S, John D, Taiwo EO. Orthogonal collocation technique for the construction of continuous Hybrid method for second order Initial Value Problems. IJEAM. 2015;2(1):177-181.
Taparki RM, Gurah D, Simon S. An implicit Runge-Kutta method for solution of third order initial value problem in ordinary differential equations. Int. J. Numer. Math. 2010;6:174-189.
Awoyemi DO, Kayode SJ, Adoghe LO. A five-step p-stable method for the numerical integration of third order ordinary differential equations, Amer. J. Comput. Math. 2014;4:119-126.
Spiegel RM. Theory and Problems of Advance Mathematics for Engineers and scientist, McGraw Hill, Inc. New York; 1971.
Lambert JD. Computational methods in ordinary differential equations. Introductory Mathematics for Scientists and Engineers. Wiley; 1973.
Fatunla SO. Numerical methods for initial value problems in ordinary differential equations. Academic press inc. Harcourt Brace Jovanovich Publishers, New York; 1988.
Sarafyan D. New algorithms for the continuous approximate solutions of ordinary differential equations and the upgrading of the order of the processes. Computers & Mathematics with Applications. 1990;20(1):77-100.
Awoyemi DO. A class of Continuous methods for general second order initial value problems in ordinary differential equations. International Journal of Computer Mathematics. 1999;72(1):29-37.
Dahlquist G. Convergence and stability in the numerical integration of ordinary differential equations. Mathematica Scandinavia. 1959;4:33-53.
Hall G, Suleiman M B. Stability of Adams-type formulae for second order ordinary differential equations. IMA Journal of Numerical Analysis. 1981;1(4):427-438.
Omar Z. Developing parallel 3-point implicit block method for solving second order ordinary differential equations directly. IJMS. 2004;11(1):91-103.
Kayode SJ. A class of maximal order linear multistep collocation methods for direct solution of ordinary differential equations. Unpublished doctoral dissertation, Federal University of Technology, Akure, Nigeria; 2004.
Sabo J, Skwame Y, Kyagya TY, Kwanamu JA. The direct simulation of third order linear problems on single step block method. Asian Journal of Research in Computer Science. 2021;12(2):1-12.
Adeniyi RB, Adeyefa EO. On Chebyshev collocation approach for continuous formulation of implicit hybrid block method for IVPs in second order ordinary differential equations. IOSR-J. Math. 2013;6(4):09-12.
Omar Z, Abdelrahim R. Application of single step with three generalized hybrid points block method for solving third order ordinary differential equations. J. Nonlinear Sci. Appl. 2016;9, 2705-2717.
Omar Z, Abdullahi YA, Kuboye JO. Predictor-Corrector block method of order seven for solving third order ordinary differential equations. Int. J. Math. Anal. 2016;10(5):223-235.
Kuboye JO, Omar Z. Numerical solution of third order ordinary differential equations using a seven-step block method. Int. J. Math. Anal. 2015;9(15):743-754.
Odekunle MR, Egwurube MO, Adesanya AO, Udo MO. Five steps block predictor-block corrector method for the solution of y''= f (x, y, y'). Appl. Math. 2014;5:1252-1266.
Ogunware BG, Awoyemi DO, Adoghe LO, Olanegan OO, Omole EO. Numerical treatment of general third order ordinary differential equations using Taylor series as predictor. Phys. Sci. Int. J. 2018;17(3):1-8.
Jikantoro YD, Ismail F, Senu N, Ibrahim ZB. A new integrator for special third order differential equations with application to thin film flow problem. Indian J. Pure Appl. Math. 2018;49(1):151-167.
Adeyeye O, Omar Z. New self-starting approach for solving special third order initial value problems. Int. J. Pure Appl. Math. 2018;118(3):511-517.
Awoyemi DO, Kayode, SJ, Adoghe, LO. A five-step p-stable method for the numerical integration of third order ordinary differential equations. Amer. J. Comput. Math. 2014;4:119-126.
Skwame Y, Dalatu PI, Sabo J, Mathew M. Numerical application of third derivative hybrid block methods on third order initial value problem of ordinary differential equations. Int. J. Stat. Appl. Math. 2019;4(6):90-100.
Yakusak NS, Emmanue S, John D, Taiwo EO. Orthogonal collocation technique for the construction of continuous Hybrid method for second order Initial Value Problems. IJEAM. 2015;2(1):177-181.
Taparki RM, Gurah D, Simon S. An implicit Runge-Kutta method for solution of third order initial value problem in ordinary differential equations. Int. J. Numer. Math. 2010;6:174-189.
Awoyemi DO, Kayode SJ, Adoghe LO. A five-step p-stable method for the numerical integration of third order ordinary differential equations, Amer. J. Comput. Math. 2014;4:119-126.
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