Implicit Second Derivative Hybrid Linear Multistep Method with Nested Predictors for Ordinary Differential Equations
Keywords:
Linear multistep methods, hybrid, nesting, interpolation, collocation, boundary locus.Abstract
In this paper, we considered an implicit hybrid linear multistep method with nested hybrid predictors for solving first order initial value problems in ordinary differential equations. The derivation of the methods is based on interpolation and collocation approach using polynomial basis function. The region of absolute stability of the method is investigated using the boundary locus approach and the methods have been found to be stable for step-length
References
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[2] Butcher, J.C; A modified multistep method of numerical integration of ordinary Differential equations, J Ass. comput. Math; 1965, vol;12, pp.124-135.
[3] Butcher, J.C; A Transformed implicit Runge-Kutta Method, J. Ass. comput. Math., 1979, Vol.26, pp.731-738.
[4] Dahlquist, G. A; special stability problems for linear multistep methods, BIT. 1963, vol.3, pp. 27
[5] Donelson, J. & Hansen, E.; Cyclic Composite Multistep Predictor-Correctors Methods, SIAM, J. Num. Anal.,1971, Vol.8,pp.137-157.
[6] Enright,W.H., Second Derivative Multistep Methods for stiff ODE’s, SAIM. J. Num.Anal., 1974, vol.11, ISS.2 pp. 321-331.
[7] Enright, W.H., continuous numerical methods for ODE’s with defect control, J. computational. Appl. math., Vol.25, (2000), pp. 159-170.
[8] Esuabana I. M & Ekoro S. E.; Hybrid Linear Multistep Methods with Nested Hybrid Predictors for Solving Linear and Nonlinear Initial Value Problems in Ordinary Differential Equations, IISTE journal of Mathematical Theory and Modeling, 2017, vol. 7,iss. 11, pp. 77-88.
[9] Fatunla, S. O.; Numerical Methods for Initial Value Problems in Ordinary Differential Equations, New York: Academic Press, 1988.
[10] Gear, C. w.; Hybrid methods for IVP’s in ODEs, SIAM Journal on Numerical Analysis, vol.2,(1965), pp.69-86.
[11] Gragg, W. B & Shetter, H. J.; Generalised Multistep Predictor-Correctors methods, J. Assoc. Comput. Mach.,1964, Vol.11, pp.188-209.
[12] Hairer E. & Wanner G.; solving ordinary differential equation 11: Stiff and Differential Algebraic problems, 2nd rv.Ed. springer-verlag, New York,1996.
[13] Higham, D. J.; Higham, N.J. –MATLAB Guide, Society of industrial and applied Mathematics (SIAM), Philadelphia, PA, 2000.
[14] Ikhile, M. N. O & Okuonghae, R. I.; Stiffly Stable Continuous Extension of Second Derivative Linear Multistep Method with an off-step point for IVPs in ODEs, J. Nig. Assoc., Math. Phys., 2007, vol. 11, pp. 175-190.
[15] Lambert, J. D.; Computational methods for Ordinary Differential Systems, Chichester; Wiley, 1973, pp.91.
[16] Okuonghae, R. I., & Ikhile M. N. O., A class of Hybrid Linear Multistep Methods With -Stable Properties for Stiff IVPs in ODEs, J. Num. Math. 2014, Vol. 8, iss. 4, pp. 441-469.
[17] Robertson, H. H, The solution of a set of reaction rate equation in: Numerical Analysis: An introduction (J. Walsh, Ed.), academic Press, New York, 1966, pp. 178-182.
[2] Butcher, J.C; A modified multistep method of numerical integration of ordinary Differential equations, J Ass. comput. Math; 1965, vol;12, pp.124-135.
[3] Butcher, J.C; A Transformed implicit Runge-Kutta Method, J. Ass. comput. Math., 1979, Vol.26, pp.731-738.
[4] Dahlquist, G. A; special stability problems for linear multistep methods, BIT. 1963, vol.3, pp. 27
[5] Donelson, J. & Hansen, E.; Cyclic Composite Multistep Predictor-Correctors Methods, SIAM, J. Num. Anal.,1971, Vol.8,pp.137-157.
[6] Enright,W.H., Second Derivative Multistep Methods for stiff ODE’s, SAIM. J. Num.Anal., 1974, vol.11, ISS.2 pp. 321-331.
[7] Enright, W.H., continuous numerical methods for ODE’s with defect control, J. computational. Appl. math., Vol.25, (2000), pp. 159-170.
[8] Esuabana I. M & Ekoro S. E.; Hybrid Linear Multistep Methods with Nested Hybrid Predictors for Solving Linear and Nonlinear Initial Value Problems in Ordinary Differential Equations, IISTE journal of Mathematical Theory and Modeling, 2017, vol. 7,iss. 11, pp. 77-88.
[9] Fatunla, S. O.; Numerical Methods for Initial Value Problems in Ordinary Differential Equations, New York: Academic Press, 1988.
[10] Gear, C. w.; Hybrid methods for IVP’s in ODEs, SIAM Journal on Numerical Analysis, vol.2,(1965), pp.69-86.
[11] Gragg, W. B & Shetter, H. J.; Generalised Multistep Predictor-Correctors methods, J. Assoc. Comput. Mach.,1964, Vol.11, pp.188-209.
[12] Hairer E. & Wanner G.; solving ordinary differential equation 11: Stiff and Differential Algebraic problems, 2nd rv.Ed. springer-verlag, New York,1996.
[13] Higham, D. J.; Higham, N.J. –MATLAB Guide, Society of industrial and applied Mathematics (SIAM), Philadelphia, PA, 2000.
[14] Ikhile, M. N. O & Okuonghae, R. I.; Stiffly Stable Continuous Extension of Second Derivative Linear Multistep Method with an off-step point for IVPs in ODEs, J. Nig. Assoc., Math. Phys., 2007, vol. 11, pp. 175-190.
[15] Lambert, J. D.; Computational methods for Ordinary Differential Systems, Chichester; Wiley, 1973, pp.91.
[16] Okuonghae, R. I., & Ikhile M. N. O., A class of Hybrid Linear Multistep Methods With -Stable Properties for Stiff IVPs in ODEs, J. Num. Math. 2014, Vol. 8, iss. 4, pp. 441-469.
[17] Robertson, H. H, The solution of a set of reaction rate equation in: Numerical Analysis: An introduction (J. Walsh, Ed.), academic Press, New York, 1966, pp. 178-182.
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Published
2018-05-11
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Ekoro, S. E., O. Ikhile, M. N., & Esuabana, I. M. (2018). Implicit Second Derivative Hybrid Linear Multistep Method with Nested Predictors for Ordinary Differential Equations. American Scientific Research Journal for Engineering, Technology, and Sciences, 42(1), 297–308. Retrieved from https://asrjetsjournal.org/index.php/American_Scientific_Journal/article/view/4050
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