A HIGHER ORDER TRIGONOMETRICALLY-FITTED METHOD FOR SECOND ORDER NONLINEAR PERIODIC PROBLEMS
DOI:
https://doi.org/10.4314/jfas.v13i2.23Keywords:
Collocation Technique, Nonlinear Periodic Problems, Trigonometrically-FittedAbstract
This paper present a higher order, block implicit, four step method with trigonometric coefficients constructed via multistep collocation technique. The stability properties of the method is discussed. Numerical results obtained disclose that the new method is suitable for the integration of second order nonlinear periodic problems.
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References
Wend D. V. V. Uniqueness of Solution of Ordinary Differential Equations. American Mathematical Monthly, 1967, 74:948-950.
Wend D. V. V. Existence and Uniqueness of Solution of Ordinary Differential Equations. Proceedings of the American Mathematical Society, 1969, 23(1):27-33.
Stiefel, E. And Bettis, D. G. Stabilization of Cowell’s Methods. Mathematics of Computation, 1969, 23, 731-740.
D’Ambrosio, R., Esposito, E. and Paternoster, B. Parameter estimation in Exponentially-Fitted hybrid methods for second order differential problems. J. Math. Chem., 2012, 50,155-168.
D’Ambrosio, R., Esposito, E. and Paternoster, B. Exponentially-Fitted two-step Runge-Kutta methods: construction and parameter selection. Applied Mathematics and Computation, 2012, 218, 7468-7480.
Franco, J. M. An embedded pair of Exponentially-Fitted explicit Runge-Kutta methods. Journal of Computational and Applied Mathematics, 2002, 149, 407-414.
Ixaru, L. Gr., Berghe, G. V. and Meyer, H. D. Frequency evaluation in exponentially-fitted algorithms for ODEs. Journal of Computational and Applied Mathematics, 2002, 140, 423-434.
Brugnano, L. and Trigiante, D., Solving Differential Problem by Multistep Initial and Boundary Value Methods. Amsterdam: Gordon and Breach Science Publishers, 1998.
Lambert, J. D. and Mitchell, A.R. On the solution of y^' = f(x,y) by a class of high accuracy difference formula of low order, Z. Angew. Math. Phys., 1962, 13, 223–232.
Tsitouras, Ch. Explicit eight order two step methods with nine stages for integrating oscillatory problems. Int. J. Mod. Physc., 2006, 17, 861-876.
Jator, S. N., Akinfenwa, A. O., Okunuga, S. A. and Sofoluwe, A. B. High-order continuous third derivative formulas with block extension for y^''=f(x,y,y^' ). International Journal of Computer Mathematics, 2013, 90(9), 1899-1914.
Franco, J.M. Exponentially fitted explicit Runge-Kutta-Nyström methods. Journal of Computational and Applied Mathematics, (2004),167, 1-19.
Lambert, J. D. Computational methods in ordinary differential system, the initial value problem. New York: John Wiley and Sons, 1973.
Lambert, J. D. and Watson, I. A. Symmetric Multistep Methods for Periodic Initial Value Problems. Journal of Institute of Mathematics and its Applications, 1976, 18,189-202.
Nguyen, H. S., Sidje, R. B. and Cong, N. H. Analysis of trigonometric implicit Runge-Kutta methods. Journal of computational and Applied Mathematics, 2007, 198, 187-207.
Jator, S. N. Solving second order initial value problems by a hybrid multistep method without predictors. Applied Mathematics and Computation, 2010, 277, 4036-4046.
Jator, S. N. Implicit third derivative Runge-Kutta-Nyström method with trigonometric coefficients. Numerical Algorithms, 2015, 70(1), 133-150. Doi: 10.1007/s11075-014-9938-5
Ehigie, J. O., Zou, M., Hou, X., and You. X. On Modified TDRKN Methods for Second-Order Systems of Differential Equations, International Journal of Computer Mathematics, 2017. DOI: 10.1080/00207160.2017.1343943.
Duxbury, S. C. Mixed collocation methods for y"=f(x,y). Durham theses, Durham University, 1999.
Jain, M.K. and Aziz, T. Cubic Spline Solution of Two-Point Boundary Value Problems with Significant First Derivatives, Computer Methods in Applied Mechanics and Engineering, 1983, 39, 83-91.
Jator, S.N. and Li, J. An algorithm for second order initial and boundary valu problems with an automatic error estimate based on a third derivative method, Numer Algor, 2012, 59, 333-346.
Abdulganiy, R. I., Akinfenwa, O. A. and Okunuga, S. A. Maximal Order Block Trigonometrically Fitted Scheme for the Numerical Treatment of Second Order Initial Value Problem with Oscillating Solutions, International Journal of Mathematical Analysis and Optimization, 2017, 168 – 186.
Fatunla, S. O. Numerical methods for initial value problems in ordinary differential equation. United Kingdom Conference on: Academic Press Inc, 1988.
Fatunla, S. O. Block methods for second order ODEs. International Journal of Computer Mathematics, 1991, 41, 55-63.
Ndukum, P. L. Biala, T. A., Jator, S. N., and Adeniyi, R. B. On a family of Trigonometrically-Fitted extended backward differentiation formulas for stiff and oscillatory initial value problems. Numer Algor, 2016. DOI: 10. 1007/s11075-016-0148-1.
Ramos, H. & Vigo-Aguiar, J. On the frequency choice in trigonometrically fitted methods. Applied Mathematics Letters, 2010, 23, 1378-1381.
Ramos, H. & Vigo-Aguiar, J. A trigonometrically-fitted method with two frequencies, one for the solution and another one for the derivative. Computer Physics Communications, 2014, 185, 1230-1236.
Vigo-Aguiar, J. & Ramos, H. A strategy for selecting the frequency in trigonometrically-fitted methods based on the minimization of the local truncation error and the total energy error. J. Math Chem, 2014, 52, 1050-1058.
Vigo-Aguiar, J. & Ramos, H. On the choice of the frequency in trigonometrically fitted methods for periodic problems. Journal of computational and Applied Mathematics, 2015, 277, 94-105.
Ngwane, F. F. and Jator, S. N. A Family of Trigonometrically-Fitted Enright Second Derivative Methods for Stiff and Oscillatory Initial Value problems. Journal of Applied Mathematics, 2015, 1-17. DOI: 10.1155/2015/343295.
Jator, S.N. Trigonometric symmetric boundary value method for oscillating solutions including the sine-Gordon and Poisson equations. Applied & Interdisciplinary Mathematics, 2015, 3, 1-16.
Abdulganiy, R. I., Akinfenwa, O. A., Okunuga, S. A. and Oladimeji, G. O. A Robust Block Hybrid Trigonometric Method for the Numerical Integration of Oscillatory Second Order Nonlinear Initial Value Problems. AMSE JOURNALS-AMSE IIETA publication-2017-Series: Advances A, 2017, 54,497-518.
Fang, Y., and Wu, X. A trigonometrically fitted explicit Numerov-type method for second order initial value problems with oscillating solutions. Applied Numerical Mathematics, 2008, 58, 341-351.
Van Dooren, R., Stabilization of Cowell’s classical finite difference methods for numerical integration. Journal of Computational Physics, 1974, 16, 186-192, 1974.
Fang, Y., Song, Y. and Wu, X. A robust trigonometrically fitted embedded pair for perturbed oscillators, Journal of Computational and Applied Mathematics, 2009, 225(2), 347–355.
Franco, J. M. A class of explicit two-step hybrid methods for second-order IVPs, Journal of Computational and Applied Mathematics, 2006, 187,41-57.
Wend D. V. V. Existence and Uniqueness of Solution of Ordinary Differential Equations. Proceedings of the American Mathematical Society, 1969, 23(1):27-33.
Stiefel, E. And Bettis, D. G. Stabilization of Cowell’s Methods. Mathematics of Computation, 1969, 23, 731-740.
D’Ambrosio, R., Esposito, E. and Paternoster, B. Parameter estimation in Exponentially-Fitted hybrid methods for second order differential problems. J. Math. Chem., 2012, 50,155-168.
D’Ambrosio, R., Esposito, E. and Paternoster, B. Exponentially-Fitted two-step Runge-Kutta methods: construction and parameter selection. Applied Mathematics and Computation, 2012, 218, 7468-7480.
Franco, J. M. An embedded pair of Exponentially-Fitted explicit Runge-Kutta methods. Journal of Computational and Applied Mathematics, 2002, 149, 407-414.
Ixaru, L. Gr., Berghe, G. V. and Meyer, H. D. Frequency evaluation in exponentially-fitted algorithms for ODEs. Journal of Computational and Applied Mathematics, 2002, 140, 423-434.
Brugnano, L. and Trigiante, D., Solving Differential Problem by Multistep Initial and Boundary Value Methods. Amsterdam: Gordon and Breach Science Publishers, 1998.
Lambert, J. D. and Mitchell, A.R. On the solution of y^' = f(x,y) by a class of high accuracy difference formula of low order, Z. Angew. Math. Phys., 1962, 13, 223–232.
Tsitouras, Ch. Explicit eight order two step methods with nine stages for integrating oscillatory problems. Int. J. Mod. Physc., 2006, 17, 861-876.
Jator, S. N., Akinfenwa, A. O., Okunuga, S. A. and Sofoluwe, A. B. High-order continuous third derivative formulas with block extension for y^''=f(x,y,y^' ). International Journal of Computer Mathematics, 2013, 90(9), 1899-1914.
Franco, J.M. Exponentially fitted explicit Runge-Kutta-Nyström methods. Journal of Computational and Applied Mathematics, (2004),167, 1-19.
Lambert, J. D. Computational methods in ordinary differential system, the initial value problem. New York: John Wiley and Sons, 1973.
Lambert, J. D. and Watson, I. A. Symmetric Multistep Methods for Periodic Initial Value Problems. Journal of Institute of Mathematics and its Applications, 1976, 18,189-202.
Nguyen, H. S., Sidje, R. B. and Cong, N. H. Analysis of trigonometric implicit Runge-Kutta methods. Journal of computational and Applied Mathematics, 2007, 198, 187-207.
Jator, S. N. Solving second order initial value problems by a hybrid multistep method without predictors. Applied Mathematics and Computation, 2010, 277, 4036-4046.
Jator, S. N. Implicit third derivative Runge-Kutta-Nyström method with trigonometric coefficients. Numerical Algorithms, 2015, 70(1), 133-150. Doi: 10.1007/s11075-014-9938-5
Ehigie, J. O., Zou, M., Hou, X., and You. X. On Modified TDRKN Methods for Second-Order Systems of Differential Equations, International Journal of Computer Mathematics, 2017. DOI: 10.1080/00207160.2017.1343943.
Duxbury, S. C. Mixed collocation methods for y"=f(x,y). Durham theses, Durham University, 1999.
Jain, M.K. and Aziz, T. Cubic Spline Solution of Two-Point Boundary Value Problems with Significant First Derivatives, Computer Methods in Applied Mechanics and Engineering, 1983, 39, 83-91.
Jator, S.N. and Li, J. An algorithm for second order initial and boundary valu problems with an automatic error estimate based on a third derivative method, Numer Algor, 2012, 59, 333-346.
Abdulganiy, R. I., Akinfenwa, O. A. and Okunuga, S. A. Maximal Order Block Trigonometrically Fitted Scheme for the Numerical Treatment of Second Order Initial Value Problem with Oscillating Solutions, International Journal of Mathematical Analysis and Optimization, 2017, 168 – 186.
Fatunla, S. O. Numerical methods for initial value problems in ordinary differential equation. United Kingdom Conference on: Academic Press Inc, 1988.
Fatunla, S. O. Block methods for second order ODEs. International Journal of Computer Mathematics, 1991, 41, 55-63.
Ndukum, P. L. Biala, T. A., Jator, S. N., and Adeniyi, R. B. On a family of Trigonometrically-Fitted extended backward differentiation formulas for stiff and oscillatory initial value problems. Numer Algor, 2016. DOI: 10. 1007/s11075-016-0148-1.
Ramos, H. & Vigo-Aguiar, J. On the frequency choice in trigonometrically fitted methods. Applied Mathematics Letters, 2010, 23, 1378-1381.
Ramos, H. & Vigo-Aguiar, J. A trigonometrically-fitted method with two frequencies, one for the solution and another one for the derivative. Computer Physics Communications, 2014, 185, 1230-1236.
Vigo-Aguiar, J. & Ramos, H. A strategy for selecting the frequency in trigonometrically-fitted methods based on the minimization of the local truncation error and the total energy error. J. Math Chem, 2014, 52, 1050-1058.
Vigo-Aguiar, J. & Ramos, H. On the choice of the frequency in trigonometrically fitted methods for periodic problems. Journal of computational and Applied Mathematics, 2015, 277, 94-105.
Ngwane, F. F. and Jator, S. N. A Family of Trigonometrically-Fitted Enright Second Derivative Methods for Stiff and Oscillatory Initial Value problems. Journal of Applied Mathematics, 2015, 1-17. DOI: 10.1155/2015/343295.
Jator, S.N. Trigonometric symmetric boundary value method for oscillating solutions including the sine-Gordon and Poisson equations. Applied & Interdisciplinary Mathematics, 2015, 3, 1-16.
Abdulganiy, R. I., Akinfenwa, O. A., Okunuga, S. A. and Oladimeji, G. O. A Robust Block Hybrid Trigonometric Method for the Numerical Integration of Oscillatory Second Order Nonlinear Initial Value Problems. AMSE JOURNALS-AMSE IIETA publication-2017-Series: Advances A, 2017, 54,497-518.
Fang, Y., and Wu, X. A trigonometrically fitted explicit Numerov-type method for second order initial value problems with oscillating solutions. Applied Numerical Mathematics, 2008, 58, 341-351.
Van Dooren, R., Stabilization of Cowell’s classical finite difference methods for numerical integration. Journal of Computational Physics, 1974, 16, 186-192, 1974.
Fang, Y., Song, Y. and Wu, X. A robust trigonometrically fitted embedded pair for perturbed oscillators, Journal of Computational and Applied Mathematics, 2009, 225(2), 347–355.
Franco, J. M. A class of explicit two-step hybrid methods for second-order IVPs, Journal of Computational and Applied Mathematics, 2006, 187,41-57.
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Published
2021-04-25
How to Cite
ABDULGANIY, R. I.; AKINFENWA, O. A.; OSUNKAYODE, A. K.; OKUNUGA, S. A. A HIGHER ORDER TRIGONOMETRICALLY-FITTED METHOD FOR SECOND ORDER NONLINEAR PERIODIC PROBLEMS. Journal of Fundamental and Applied Sciences, [S. l.], v. 13, n. 2, p. 1056–1078, 2021. DOI: 10.4314/jfas.v13i2.23. Disponível em: https://www.jfas.info/index.php/JFAS/article/view/290. Acesso em: 13 may. 2026.
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