Back
 AJCM  Vol.3 No.2 , June 2013
Four Steps Continuous Method for the Solution of y″= f (x, y, y′)
Abstract: This paper proposes a continuous block method for the solution of second order ordinary differential equation. Collocation and interpolation of the power series approximate solution are adopted to derive a continuous implicit linear multistep method. Continuous block method is used to derive the independent solution which is evaluated at selected grid points to generate the discrete block method. The order, consistency, zero stability and stability region are investigated. The new method was found to compare favourably with the existing methods in term of accuracy.
Cite this paper: A. Adesanya, M. Odekunle and M. Udoh, "Four Steps Continuous Method for the Solution of y″= f (x, y, y′)," American Journal of Computational Mathematics, Vol. 3 No. 2, 2013, pp. 169-174. doi: 10.4236/ajcm.2013.32025.
References

[1]   D. O. Awoyemi, “A New Sixth Order-Order Algorithm for the General Second Order Ordinary Differential Equation,” International Journal of Computer Mathematics, Vol. 77, No. 1, pp. 117-124. doi:10.1080/00207160108805054

[2]   D. O. Awoyemi and S. J. Kayode, “A Maximal Order Collocation Method for Direct Solution of Initial Value Problems of General Second Order Ordinary Differential Equation,” Proceedings of the Conference Organised by the National Mathematical Centre, Abuja, 2005.

[3]   S. O. Adee, P. Onumanyi, U. V. serisena and Y. A. Yahaya, “A Note on Starting Numerov’s Method More Accurately by an Hybrid Formula of Order Four for an Initial Value Problem,” Journal of Computational and Applied Mathematics, Vol. 175, No. 4, 2005, pp. 269-273.

[4]   D. O. Awoyemi and M. O. Idowu, “A Class of Hybrid Collocation Method for Third Order Ordinary Differential Equation,” International Journal of Computer Mathematics, Vol. 82, No. 10, 2005, pp. 1287-1293. doi:10.1080/00207160500112902

[5]   A. O. Adesanya, T. A. Anake and M. O. Udoh, “Improved Continuous Method for Direct Solution of General Second Order Ordinary Differential Equations,” Journal of Nigerian Association of Mathematical Physics, Vol. 13, 2008, pp. 59-62.

[6]   S. N. Jator, “A Sixth Order Linear Multistep Method for Direct Solution of ,” International Journal of Pure and Applied Mathematics, Vol. 40, No. 1, 2007, pp. 457-472.

[7]   S. N. Jator and J. Li, “A Self Starting Linear Multistep Method for the Direct Solution of the General Second Order Initial Value Problems,” International Journal of Computer Mathematics, Vol. 86, No. 5, 2009, pp. 817-836. doi:10.1080/00207160701708250

[8]   R. D’Ambrosio, M. Ferro and B. Paternoster, “Two-Steps Hybrid Method for ,” Journal of Applied Mathematics Letters, Vol. 22, No. 7, 2009, pp. 1076-1080. doi:10.1016/j.aml.2009.01.017

[9]   I. Fudziah, H. K. Yap and O. Mohamad, “Explicit and Implicit 3-Point Block Method for Solving Special Second Order Ordinary Differential Equation Directly,” International Journal of Mathematical Analysis, Vol. 3, No. 5, 2009, pp. 239-254.

[10]   Y. A. Yahaya and A. M. Badmus, “A Class of Collocation Methods for General Second Order Differential Equation,” African Journal of Math and Computer Research, Vol. 2, No. 4, 2009, pp. 69-71.

[11]   D. O. Awoyemi, E. A. Adebile, A. O. Adesanya and T. A. Anake, “Modified Block Method for the Direct Solution of Second Order Ordinary Differential Equation,” International Journal of Applied Mathematics and Computation, Vol. 3, No. 3, 2011, pp. 181-188.

[12]   S. Mehrkanoon, “A Direct Variable Step Block Multistep Method for Solving General Third Order ODEs,” Journal of Numerical Algorithms, Vol. 57, No. 1, 2011, pp. 53-66. doi:10.1007/s11075-010-9413-x

[13]   S. Abbas, “Derivation of a new Block Method Similar to the Block Trapezoidal Rule for the Numerical Solution of First Order IVPs,” Science Echoes, Vol. 2, 2006, pp. 10-24.

[14]   A. O. Adesanya, “Block Method for Higher Order Ordinary Differential Equations,” Ph.D. Thesis, University of Technology, Akure, 2011, Unpublished.

 
 
Top