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**Implementation of four-point fully implicit block method for solving ordinary differential equations.**
*(English)*
Zbl 1114.65080

Summary: This paper describes the development of a four-point fully implicit block method for solving first order ordinary differential equations (ODEs) using variable step size. This method will estimate the solutions of initial value problems (IVPs) at four points simultaneously. The method developed is suitable for the numerical integration of non-stiff and mildly stiff differential systems. The performances of the four-point block method are compared in terms of maximum error, total number of steps and execution times to the non-block method 1PVSO of Z. Omar [Developing parallel block methods for solving higher order ODEs directly, Ph.D. Thesis, University Putra Malaysia, Malaysia (1999)].

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

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\textit{Z. A. Majid} and \textit{M. B. Suleiman}, Appl. Math. Comput. 184, No. 2, 514--522 (2007; Zbl 1114.65080)

### References:

[1] | Johnson, A. I.; Barney, J. R., Numerical solution of large systems of stiff ordinary differential equations, (Lapidus, L.; Schiesser, W. E., Modular Simulation Framework. Modular Simulation Framework, Numerical Methods for Differential Systems (1976), Academic Press Inc.: Academic Press Inc. New York), 97-124 |

[2] | Char, B. W.; Geddes, K. O.; Gonnet, G. H.; Leong, B. L.; Monagan, M. B.; Watt, S. M., First Leaves: A Tutorial Introduction to Maple V (1992), Waterloo Maple Publishing: Waterloo Maple Publishing Springer-Verlag · Zbl 0758.68037 |

[3] | Hairer, E.; Norsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0789.65048 |

[4] | Rosser, J. B., Runge Kutta for all season, SIAM Rev., 9, 417-452 (1976) · Zbl 0243.65041 |

[5] | Shampine, L. F.; Gordon, M. K., Computer Solution of Ordinary Differential Equations: The Initial Value Problem (1975), W.H. Freeman and Company: W.H. Freeman and Company San Francisco · Zbl 0347.65001 |

[6] | M.B. Suleiman, Generalised multistep Adams and backward differentiation methods for the solution of stiff and non-stiff ordinary differential equations, Ph.D. Thesis, University of Manchester, 1979.; M.B. Suleiman, Generalised multistep Adams and backward differentiation methods for the solution of stiff and non-stiff ordinary differential equations, Ph.D. Thesis, University of Manchester, 1979. |

[7] | Chu, M. T.; Hamilton, H., Parallel solution of ODEs by multiblock methods, SIAM J. Sci. Stat. Comput., 8, 342-354 (1987) · Zbl 0628.65054 |

[8] | Worland, P. B., Parallel method for the numerical solution of ordinary differential equations, IEEE Trans. Comput., 1045-1048 (1976) · Zbl 0359.65063 |

[9] | Z. Omar, Developing parallel block methods for solving higher order ODEs directly, Ph.D. Thesis, University Putra Malaysia, Malaysia, 1999.; Z. Omar, Developing parallel block methods for solving higher order ODEs directly, Ph.D. Thesis, University Putra Malaysia, Malaysia, 1999. |

[10] | Z.A. Majid, Parallel block methods for solving ordinary differential equations, Ph.D. Thesis, University Putra Malaysia, Malaysia, 2004.; Z.A. Majid, Parallel block methods for solving ordinary differential equations, Ph.D. Thesis, University Putra Malaysia, Malaysia, 2004. |

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