Implicit Exponentially Fitted Hybrid Method for Special Second Order Initial Value Problems
Kaedah Penyuaian Eksponen Hibrid Tersirat bagi Masalah Nilai Awal Khas Peringkat Kedua
Keywords:
hybrid method, exponentially fitted, second order initial value problems, numerical solutionAbstract
An implicit exponentially fitted hybrid method is developed for solving special second order initial value problems. The coefficients of the new method are functions of step-size and the frequency of the problems. The stability region of the method is given. Numerical comparisons on several problems with exponential solutions demonstrate that the new method gives better accuracy compared to the existing method.
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References
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Brusa, L. and Nigro, L. 1980. A one-step method for direct integration of structural dynamic equations. Internat. J. Numer. Meth. Engrg. 15: 685-699.
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Chawla, M.M., Rao, P.S. and Neta, B. 1986. Two-step fourth-order P-stable methods with phase-lag of order six for y”= f(x,y). J. Comput. Appl.Math. 16: 233-236.
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Coleman, J.P. 2003. Order conditions for a class of two-step methods for y”= f(x,y). IMA J. Num. Analysis. 23: 197-220.
Dormand, J., El-Mikkawy, M.E.A. and Prince, P.J. 1987. Families of Runge-Kutta-Nystrom formula. IMA J. Num. Analysis. 7(2):235-250.
Fatunla, S.O. 1990. Block methods for second order ODEs. Internat. J. Comp. Math. 41: 55-63.
Fatunla, S.O., Ikhile, M.N.O. and Otunta, F.O. 1999. A class of P-stable linear multistep numerical methods. Internat. J. Comp. Math. 72: 1-13.
Franco, J.M. 2004. A class of explicit two-step hybrid methods for second-order IVPs. J. Comput. Appl. Math. 187: 41-57.
Hairer, E. 1979. Unconditionally stable methods for second order differential equations. Numer. Math. 32: 373-379.
Lambert, J.D. and Watson, I.A. 1976. Symmetric multistep methods for periodic initial-value problems. J. Ins. Math. Applics. 18: 189-202.
Simos, T.E. 1992. Explicit two-step methods with minimal phase-lag for the numerical integration of special second-order initial-value problems and their application to the one-dimensional Schrodinger equation. J. Comput. Appl. Math. 39: 89-94.
Simos, T. E. And Raptis, A.D. 1990. Numerov-type methods with minimal phase-lag for the numerical integration of the one-dimensional Schrodinger equation. Computing. 45: 175-181.
Tsitouras, Ch. 2006. Stage reduction on P-stable Numerov type methods of eighth order. J. Comput. Appl. Math. 191: 297-305.
Van der Houwen, P.J. and Sommeijer, B.P. 1987. Explicit Runge-Kutta (-Nystrom) methods with reduced phase errors for computing oscillating solutions. SIAM J. Numer. Anal. 24: 595-617.
Raptis, A. and Allison, A.C. 1978. Exponential-fitting methods for the numerical solution of the Schrodinger equation. Computer Physics Communications. vol. 14. pp. 1-5.
Coleman, J.P. and Izaru, L.Gr. 1996. P-stability and exponential-fitting methods for y′′ = f (x, y) . IMA Journal of Numerical Analysis. vol. 16. pp. 179-199.
Franco, J.M. 2004. Exponentially fitted explicit Runge-Kutta-Nystrom methods. Journal of Computational and Applied Mathematics. vol. 167. pp. 1-19.
Brusa, L. and Nigro, L. 1980. A one-step method for direct integration of structural dynamic equations. Internat. J. Numer. Meth. Engrg. 15: 685-699.
Cash, J.R. 1981. High order P-stable formulae for the numerical integration of periodic initial value problems. Numer. Math. 37: 355-370.
Chawla, M.M., Rao, P.S. and Neta, B. 1986. Two-step fourth-order P-stable methods with phase-lag of order six for y”= f(x,y). J. Comput. Appl.Math. 16: 233-236.
Chawla, M.M. and Rao, P.S. 1987. An explicit sixth-order method with phase-lag of order eight for y”= f(x,y). J. Comput. Appl. Math. 17: 365-368.
Coleman, J.P. 2003. Order conditions for a class of two-step methods for y”= f(x,y). IMA J. Num. Analysis. 23: 197-220.
Dormand, J., El-Mikkawy, M.E.A. and Prince, P.J. 1987. Families of Runge-Kutta-Nystrom formula. IMA J. Num. Analysis. 7(2):235-250.
Fatunla, S.O. 1990. Block methods for second order ODEs. Internat. J. Comp. Math. 41: 55-63.
Fatunla, S.O., Ikhile, M.N.O. and Otunta, F.O. 1999. A class of P-stable linear multistep numerical methods. Internat. J. Comp. Math. 72: 1-13.
Franco, J.M. 2004. A class of explicit two-step hybrid methods for second-order IVPs. J. Comput. Appl. Math. 187: 41-57.
Hairer, E. 1979. Unconditionally stable methods for second order differential equations. Numer. Math. 32: 373-379.
Lambert, J.D. and Watson, I.A. 1976. Symmetric multistep methods for periodic initial-value problems. J. Ins. Math. Applics. 18: 189-202.
Simos, T.E. 1992. Explicit two-step methods with minimal phase-lag for the numerical integration of special second-order initial-value problems and their application to the one-dimensional Schrodinger equation. J. Comput. Appl. Math. 39: 89-94.
Simos, T. E. And Raptis, A.D. 1990. Numerov-type methods with minimal phase-lag for the numerical integration of the one-dimensional Schrodinger equation. Computing. 45: 175-181.
Tsitouras, Ch. 2006. Stage reduction on P-stable Numerov type methods of eighth order. J. Comput. Appl. Math. 191: 297-305.
Van der Houwen, P.J. and Sommeijer, B.P. 1987. Explicit Runge-Kutta (-Nystrom) methods with reduced phase errors for computing oscillating solutions. SIAM J. Numer. Anal. 24: 595-617.
Raptis, A. and Allison, A.C. 1978. Exponential-fitting methods for the numerical solution of the Schrodinger equation. Computer Physics Communications. vol. 14. pp. 1-5.
Coleman, J.P. and Izaru, L.Gr. 1996. P-stability and exponential-fitting methods for y′′ = f (x, y) . IMA Journal of Numerical Analysis. vol. 16. pp. 179-199.
Franco, J.M. 2004. Exponentially fitted explicit Runge-Kutta-Nystrom methods. Journal of Computational and Applied Mathematics. vol. 167. pp. 1-19.
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Published
2013-12-18
How to Cite
Samat, F. (2013). Implicit Exponentially Fitted Hybrid Method for Special Second Order Initial Value Problems: Kaedah Penyuaian Eksponen Hibrid Tersirat bagi Masalah Nilai Awal Khas Peringkat Kedua. Journal of Science and Mathematics Letters, 1, 9–17. Retrieved from https://ojs.upsi.edu.my/index.php/JSML/article/view/4301
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