Moses Adebowale AKANBİ, Ashiribo Senapon WUSU

On A Four-Step Exponentially Fitted Scheme for The Solution of Stiff Differential Systems

Many problems that are often encountered in fields like engineering, mechanics, electronic, astrophysics, chemistry and control theory, yield initial value problems involving systems of ordinary differentialequations which exhibit a phenomenon which has come to be known as stiffness. In this work, a new four-stepexponentially-fitted predictor-corrector method involving the second derivative for solving system of stiff differential equations is constructed using a combination of the extended backward differentiation formula and the techniqueof exponential fitting. The constructed method is well-suited for systems with pronounced stiffness. The stabilityproperty of the constructed scheme is also considered. To investigate the accuracy of the constructed method, threestandard numerical examples with pronounced stiffness are considered. A comparison of the results obtained byimplementing the proposed methods on the numerical problems compared with those of existing standard methodshow that the constructed method is efficient and accurate for solving stiff systems of ordinary differential equations.

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[1] Abhulimen C.E., Otunta F.O., A sixth order multiderivative multistep methods for stiff systems of differential equations, International Journal of Numerical Mathematics (IJNM), 2(1)(2006), 248–268.

[2] Abhulimen C.E., A class of exponentially-fitted third derivative methods for solving stiff differential equations, International Journal of Physical Science, 3(8)(2008), 188–193.

[3] Abhulimen C.E., Omeike G.E., A sixth-order exponentially fitted scheme for the numerical solution of systems of ordinary diffferential equations, Journal of Applied Mathematics & Bioinformatics, 1(1)(2011), 175–186.

[4] Cash J.R., On the integration of stiff systems of O.D.E.s using extended backward differentiation formulae, SIAM J. Numer. Math., 34(1980), 235–246.

[5] Cash J.R., Second derivative extended backward differentiation formulas for the numerical solution of stiff systems, SIAM J. Numerical Anal., 18(1981), 21–36.

[6] Cash J.R., On exponentially fitting of composite multiderivative linear methods, SIAM J. Numerical Anal., 18(5)(1981), 808–821.

[7] Curtiss C.F. and Hirschfelder J.D., Integration of stiff equations, Proc. Nat. Acad. Sci., 38(1952), 235–243.

[8] Ehigie J.O, Okunuga S.A., Sofoluwe A.B, A class of exponentially fitted second derivative extended backward differentiation formula for solving stiff problems, Fasciculi Mathematici, 51(2013), 71–84.

[9] Enright W.H., Pryce J.D., Two Fortran Packages for Assessing IV methods, Technical Report 83(16), Department of Computer Science, University of Toronto, Canada, 1983.

[10] Jackson L.W., Kenue S.K., A fourth order exponentially fitted method, SIAM J. Numer. Anal., 11(1974), 965–978.

[11] Lambert J.D., Computational Methods in Ordinary Differential Equations, John Wiley & Sons, New York, 1973.

[12] Liniger W.S., Willoughby R.A., Efficient integration methods for stiff system of ODEs, SIAM J. Numerical Anal., 7(1970), 47–65.

[13] Okunuga S.A., A fourth order composite two-step method for stiff problems, International Journal of Computer Mathematics, 72(1)(1999), 39–47.

[14] Okunuga S.A., A class of multiderivative composite formula for stiff initial value problems, Advances in Modeling & Analysis, 35(2)(1999), 21–32.