A new Block Hybrid Trigonometrically Fitted Method (BHTM) for the numerical integration of second order nonlinear initial value problems with oscillatory solutions is presented in this paper. The BHTM is based on multistep collocation method. The examination of the stability properties of the method shows that it is A-stable. Numerical experiments are carried out to show the accuracy and efficiency of the method on second order nonlinear initial value problems with oscillatory solutions.
Collocation, Hybrid, Nonlinear Second Order IVPs, Trigonometrically Fitted.
The authors would like to thank the Nigeria TETFUND for the research grant given towards this research work.
1. Akinfenwa, O. A., Akinnukawe, B., and Mudasiru, S. B., A family of Continuous Third Derivative Block Methods for solving stiff system of first order ordinary differential equations. Journal of the Nigerian Mathematical Society, vol. 34, pp.160-168, 2015.
2. Akinfenwa, O. A., Jator, S. N., and Yao, N. M., A continuous hybrid method for solving parabolic partial differential equations (PDE). AMSE Journals Series: Advances A, vol. 48, No. 1, pp. 17-27, 2011.
3. Archar, S. D., Symmetric multistep Obrechkoff methods with zero phase-lag for periodic initial value problems of second order differential equations. J. Appl. Math. Comput., vol. 218, pp. 2237-2248, 2011.
4. Awoyemi, D.O., A class of continuous methods for general second order initial value problem in ordinary differential equation. International Journal of Computer Mathematics, vol. 72, No. 1, pp. 29-37, 1999.
5. Brugnano, L. and Trigiante, D., Solving Differential Problem by Multistep Initial and Boundary Value Methods. Amsterdam: Gordon and Breach Science Publishers, 1998.
6. Coleman, J.P. and Duxbury, S.C., Mixed collocation methods for . Journal of computational and Applied Mathematics, vol. 126, pp. 47-75, 2000.
7. De Meyer, H., Vanthournout, J. And Vanden Berghe, G., On a new type of mixed interpolation, J. Comput. Appl. Math, vol. 30, pp. 55-69, 1990.
8. Duxbury, S.C., Mixed collocation methods for . Durham theses, Durham University, 1999.
9. Fang, Y., Song, Y. and Wu, X., A robust trigonometrically fitted embedded pair for perturbed oscillators,” Journal of Computational and Applied Mathematics, vol. 225, No. 2, pp. 347–355, 2009.
10. Fatunla S.O. Numerical methods for initial value problems in ordinary differential equation. United Kingdom: Academic Press Inc. 1988.
11. Gautschi, W., Numerical Integration of Ordinary Differential Equations Based on Trigonometric Polynomials, Numerische Mathematik, vol. 3, pp. 381-397, 1961.
12. Ixaru, L. Gr., Vanden Berghe, G. and De Meyer, H., Frequency evaluation in exponential fitting multistep algorithms for ODEs, J. Comput. Appl. Math., vol. 140, pp. 423-434, 2002.
13. Jator, S.N., Trigonometric symmetric boundary value method for oscillating solutions including the sine-Gordon and Poisson equations. Applied & Interdisciplinary Mathematics, vol. 3, pp. 1-16, 2016.
14. Jator, S. N., Swindell, S., & French, R. D., Trigonmetrically Fitted Block Numerov Type Method for . Numer Algor, vol. 62, pp. 13-26, 2013
15. Lambert J.D., Computational methods in ordinary differential system, the initial value problem. New York: John Wiley & Sons, 1973.
16. Milne, W. E., Numerical Solution of Differential Equations. John Wiley & Sons, 1953.
17. Ndukum, P. L. Biala, T. A., Jator, S. N., & Adeniyi, R. B., On a family of trigonometrically fitted extended backward differentiation formulas for stiff and oscillatory initial value problems. Numer Algor, vol 74, No 1, pp 267–287, 2017
18. Nguyen, H.S., Sidje, R.B. & Cong, N.H, Analysis of trigonometric implicit Runge-Kutta methods. Journal of computational and Applied Mathematics, vol. 198, pp. 187-207, 2007
19. Ngwane, F. F. & Jator, S.N., Block hybrid method using trigonometric basis for initial problems with oscillating solutions. Numerical Algorithm, vol. 63, pp. 713-725, 2003a.
20. Ngwane, F. F. & Jator, S. N., Solving Oscillatory Problems Using a Block Hybrid Trigonometrically Fitted Method with Two Off-Step Points. Texas State University. San Marcos, Electronic Journal of Differential Equation, vol. 20, pp. 119-132, 2003b
21. Ngwane, F. F. & Jator, S. N., Solving the Telegraph and Oscillatory Differential Equations by a Block Hybrid Trigonometrically Fitted Algorithm. Journal of Differential Equations, vol. 2015, pp. 1-15, 2015a.
22. Ngwane, F. F. & Jator, S. N., A Family of Trigonometrically Fitted Enright Second Derivative Methods for Stiff and Oscillatory Initial Value problems. Journal of Applied Mathematics, vol. 2015, pp. 1-17, 2015b.
23. Oluwatosin, E.A., Derivation of two step method for solving initial value problems of Ordinary Differential Equations. Continental J. Education Research, vol. 6, No. 1, pp. 39-44, 2013
24. Ozawa, K., A functional fitting Runge Kutta Method with variable coefficients, Japanese Journal of Industrial and Applied Mathematics, vol. 18, pp. 105-128, 2001.
25. Ramos, J. I., Piecewise-linearized methods for initial value problems with oscillating solutions, Applied Mathematics and Computation, vol. 181, pp. 123-146, 2006.
26. Ramos, H. and Singh, G., A tenth order A-stable two-step hybrid block method for solving initial value problems of ODE. Applied Mathematics and Computation, vol. 310, pp. 75-88, 2017.
27. Rosser, J. B., A Runge Kutta for season, SIAM, vol. 9, pp. 417-425, 1967
28. Senu, N., Suleimon, M., Ismail, F. & Othman, M., A New Diagonally Implicit Runge-Kutta-Nystrom Method for Periodic IVPs. WSEAS Transactions on Mathematics, vol. 9, No. 9, pp. 679-688, 2010.
29. Shokri, A. and Saadat, H., High Phase lag order trigonometrically fitted two step Obrechkoff methods for the numerical solution of periodic initial value problems. Numer. Algor., vol. 68,pp. 337-354, 2017.
30. Simos, T. E., A P-stable complete in Phase Obrechkoff trigonometrically fitted method for periodic initial value problems. Proc. R. Soc., vol. 441, pp. 283-289, 1993.
31. Simos, T. E., An Exponentially-Fitted Runge-Kutta Method for the Numerical Integration of Initial Value Problems with Periodic or Oscillating Solutions. Computer Physics Communications, vol. 115, pp. 1-8, 1998.
32. Simos, T. E., Exponentially-Fitted Runge-Kutta-Nyström Method for the Numerical Solution of Initial Value-Problems with Oscillating Solutions. Applied Mathematics Letters, vol. 15, pp. 217-225, 2002.
33. Vaden Berghe, G., De Meyer, H., Van Daele, M. & Van Hecke, T., Exponentially-fitted explicit Runge-Kutta methods, Comput. Phys. Comm., vol. 123, pp. 7-15, 1999.
34. Van Daele, M. and Vanden Berghe, G., P-stable exponentially fitted Obrechkoff methods of arbitrary order for second order differential equations. Numer. Algor., vol. 46, pp. 333-350, 2007.
35. Van Dooren, R., Stabilization of Cowell’s classical finite difference methods for numerical integration. Journal of Computational Physics, vol. 16, pp. 186-192, 1974.
36. Vanden Berhe G. and Van Daele, M., Exponentially-fitted Numerov methods, J. Comp. Appl. Math., vol. 200, pp. 140-153, 2007.
37. Vanthournout, J., Vaden Berghe, G. and De Meyer, H., Families of backward differentiation methods based in a new type of mixed interpolation, computers math. Applic., vol. 20, No. 11, pp. 19-30, 1990.
38. Wang, Z., Zhao, D., Dai, Y, and Wu, D, An improved trigonometrically fitted p-stable Obrechkoff method for periodic initial value problems. Proc. R. Soc., vol. 461, pp. 1639-1658, 2005.
39. Yakusak, N. S. and Adeniyi, R. B., A Four-Step Hybrid Block Method for First Order Initial Value Problems in Ordinary Differential Equations. AMSE Journals Series: Advances A, vol. 52, No. 1, pp. 17-30, 2015.