A FAMILY OF EXPONENTIALLY FITTED MULTIDERIVATIVE METHOD FOR STIFF DIFFERENTIAL EQUATIONS
DOI:
https://doi.org/10.24297/jam.v13i2.5649Keywords:
Exponentially fitted, Multiderivative linear multistep, stiff differential equations, A-stable.Abstract
In this paper, an A-stable exponentially fitted predictor-corrector using multiderivative linear multistep method for solving stiff differential equations is developed. The method which is a two-step third derivative method of order five contains free parameters. The numerical stability analysis of the method was discussed, and found to be A-stable. Numerical examples are provided to show the efficiency of the method when compared with existing methods in the literature that have solved the set of problems.
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References
[1] Abhulimen, C.E and Okunuga, S.A (2008), “Exponentially fitted second derivative multipstep method for stiff initial value problem for ODE’sâ€. Journal of Engineering Science and Applications, 5, 36-47
[2] Abhulimen, C.E and Omeike, G.E. (2011). A sixth-order exponentially fitted scheme for the numerical solution of systems of ordinary differential equationsâ€. Journal of Applied Mathematics and Bioinformatics, vol. 1 no 1, 175-186
[3] Abhulimen, C.E and Otunta, F.O (2006): “A Sixth Order Multi Derivative Multistep Method for stiff System of ODEâ€. International Journal of Numerical.Maths. (ITNM) 2(1) 248-268.
[4] Abhulimen,C.E. (2009). “Exponential Fitting Predictor-Corrector formula for stiff systems of Ordinary Differential Equationsâ€.International Journal of Computational and Applied Mathematics.ISSN 1819-4966 volume 4 no 2, pp 115-126.
[5] Brown,T.L(1977): “Some Characteristics of implicit Multstep Multi-derivative Integration formulasâ€. SIAM J.Numerical Math. 34(1) 59-84.
[6] Cash, J.R. (1981): “On the Exponential Fitting of Composite Multiderivative Linear Multistep Methodsâ€. SIAM J. Numerical Annal 18(5), 808-821.
[7] Dahlquist, G. (1963): “A special Stability Problem for Linear Multistep Methodâ€. BIT 3:27-43
[8] Enright, W.H. (1974): “Second Derivative Multistep Methods for Stiff Ordinary Differential Equationsâ€, SIAM J. Numerical Analysis II, 321-331
[9] Enright, W.H. and Pryce,J.O(1983), “Two FORTRAN Packages for Assessing IV Methodsâ€. Technical Report No .16/83.Department of ComputerScience,University of Toronto, Canada.
[10] Jackson, L.W. and Kenue, S.K (1974): A Fourth –Order Exponentially Fitted Method, SIAM J. Numerical Annal. II, 965-978
[11] Lambert, J.D. (1973). Numerical Method in Ordinary Differential Equations with Initial Value Problem, John Wiley
[12] Liniger, W. and Willioughby, R.A. (1970): Efficient Integration methods for stiff systems of ordinary differential equations, SIAM J. NumerAnnal, 7, 47-65.
[13] Okunuga, S.A (1994). “Composite Multiderivative Linear Multistep methods for Equations from r Stiff IVPs in ODEsâ€.( Ph.D. Thesis), Dept. of Mathematics, University of Lagos Nigeria.
[14] Okunuga,S.A(1997). Fourth order composite two step method for stiff problems, Int. J. Comput. Math. 2 39-47.
[15] Otunta,F.Oand Abhulimen, C.E. (2005). “A 4th Order Exponentially Fitted Multiderivative method for Stiff IVPsâ€. Nigerian Association of Mathematical Physics. Vol. 9, pp. 295-306.
[16] Voss, D. (1988). “ A fifth order Exponentially Fitted Formulaâ€. SIAM. J Number.Anal.25, No 3.
[2] Abhulimen, C.E and Omeike, G.E. (2011). A sixth-order exponentially fitted scheme for the numerical solution of systems of ordinary differential equationsâ€. Journal of Applied Mathematics and Bioinformatics, vol. 1 no 1, 175-186
[3] Abhulimen, C.E and Otunta, F.O (2006): “A Sixth Order Multi Derivative Multistep Method for stiff System of ODEâ€. International Journal of Numerical.Maths. (ITNM) 2(1) 248-268.
[4] Abhulimen,C.E. (2009). “Exponential Fitting Predictor-Corrector formula for stiff systems of Ordinary Differential Equationsâ€.International Journal of Computational and Applied Mathematics.ISSN 1819-4966 volume 4 no 2, pp 115-126.
[5] Brown,T.L(1977): “Some Characteristics of implicit Multstep Multi-derivative Integration formulasâ€. SIAM J.Numerical Math. 34(1) 59-84.
[6] Cash, J.R. (1981): “On the Exponential Fitting of Composite Multiderivative Linear Multistep Methodsâ€. SIAM J. Numerical Annal 18(5), 808-821.
[7] Dahlquist, G. (1963): “A special Stability Problem for Linear Multistep Methodâ€. BIT 3:27-43
[8] Enright, W.H. (1974): “Second Derivative Multistep Methods for Stiff Ordinary Differential Equationsâ€, SIAM J. Numerical Analysis II, 321-331
[9] Enright, W.H. and Pryce,J.O(1983), “Two FORTRAN Packages for Assessing IV Methodsâ€. Technical Report No .16/83.Department of ComputerScience,University of Toronto, Canada.
[10] Jackson, L.W. and Kenue, S.K (1974): A Fourth –Order Exponentially Fitted Method, SIAM J. Numerical Annal. II, 965-978
[11] Lambert, J.D. (1973). Numerical Method in Ordinary Differential Equations with Initial Value Problem, John Wiley
[12] Liniger, W. and Willioughby, R.A. (1970): Efficient Integration methods for stiff systems of ordinary differential equations, SIAM J. NumerAnnal, 7, 47-65.
[13] Okunuga, S.A (1994). “Composite Multiderivative Linear Multistep methods for Equations from r Stiff IVPs in ODEsâ€.( Ph.D. Thesis), Dept. of Mathematics, University of Lagos Nigeria.
[14] Okunuga,S.A(1997). Fourth order composite two step method for stiff problems, Int. J. Comput. Math. 2 39-47.
[15] Otunta,F.Oand Abhulimen, C.E. (2005). “A 4th Order Exponentially Fitted Multiderivative method for Stiff IVPsâ€. Nigerian Association of Mathematical Physics. Vol. 9, pp. 295-306.
[16] Voss, D. (1988). “ A fifth order Exponentially Fitted Formulaâ€. SIAM. J Number.Anal.25, No 3.
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Published
2017-04-06
How to Cite
Cletus, A., & L a, U. (2017). A FAMILY OF EXPONENTIALLY FITTED MULTIDERIVATIVE METHOD FOR STIFF DIFFERENTIAL EQUATIONS. JOURNAL OF ADVANCES IN MATHEMATICS, 13(2), 7155–7162. https://doi.org/10.24297/jam.v13i2.5649
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