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AJCM> Vol.4 No.4, September 2014
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A New Eighth Order Implicit Block Algorithms for the Direct Solution of Second Order Ordinary Differential Equations

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DOI: 10.4236/ajcm.2014.44032    3,160 Downloads   3,907 Views   Citations
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Ademola M. Badmus1

Affiliation(s)

1Department of Mathematics, Nigerian Defence Academy, Kaduna, Nigeria.

ABSTRACT

This paper focuses on derivation of a uniform order 8 implicit block method for the direct solution of general second order differential equations through continuous coefficients of Linear Multi-step Method (LMM). The continuous formulation and its first derivatives were evaluated at some selected grid and off grid points to obtain our proposed method. The superiority of the method over the existing methods is established numerically.

KEYWORDS

Uniform Order, Second Order Initial Value Problem, Implicit Block Algorithms, Zero Stable

Cite this paper

Badmus, A. (2014) A New Eighth Order Implicit Block Algorithms for the Direct Solution of Second Order Ordinary Differential Equations. American Journal of Computational Mathematics, 4, 376-386. doi: 10.4236/ajcm.2014.44032.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Yahaya, Y.A. and Badmus, A.M. (2009) A Class of Collocation Methods for General Second Order Ordinary Differential Equations. African Journal of Mathematics and Computer Science Research, 2, 69-72.
[2] Badmus, A.M. and Ekpenyong, F.E. (2013) High Order Block Method for Direct Solution of General Second Order Ordinary Differential Equations. Academy Journal of Science and Engineering, Nigerian Defence Academy Kaduna, 7, 36-42.
[3] Badmus, A.M. and Yahaya, Y.A. (2009) An Accurate Uniform Order 6 Block Method for Direct Solution of General Second Order Ordinary Differential Equations. The Pacific Journal of Science and Technology, 10, 248-254.
[4] Jator, S.N. (2007) A Class of Initial Value Methods for Direct Solution of Second Order Initial Value Problems. 4th International Conference of Applied Mathematics and Computing, Plovdiv, 12-18 August 2007.
[5] Lambert, J.D. (1973) Computational Methods in Ordinary Differential Equations. John Wiley and Sons, New York, 278.
[6] Lambert, J.D. (1991) Numerical Method for Ordinary Differential Systems. John Wiley and Sons, New York, 293.
[7] Awoyemi, D.O. (1999) A Class of Continuous Method for General Second Order IVPs in Ordinary Differential Equations. International Journal of Computer Mathematics, 72, 29-37. http://dx.doi.org/10.1080/00207169908804832
[8] Jator, S.N. (2001) Improvements in Adams-Moulton Methods for the First Initial Value Problems. Journal of the Tennessee Academy of Science, 76, 57-60.
[9] Onumanyi, P., Awoyemi, D.O., Jator, S.N. and Siriseria, U.W. (1994) New Linear Multistep Methods with Continuous Coefficients for First Order IVPs. Journal of Nigeria Mathematics Society, 13, 37-51.
[10] Awoyemi, D.O. (2003) A P-Stable Linear Multistep Method for Direct Solution of General Third Order Ordinary Differential Equation. International Journal of Computer Mathematics, 80, 987-993.
http://dx.doi.org/10.1080/0020716031000079572
[11] Vigo-Angular, J. and Ramos, H. (2006) Variable Step-Size Implementation of Multi-Step Methods . Journal of Computation and Applied Mathematics, 92, 114-131.
http://dx.doi.org/10.1016/j.cam.2005.04.043
[12] Fatunla, S.O. (1991) Block Method for Second Order Differential Equations. International Journal of Computer Mathematics, 41, 55-63. http://dx.doi.org/10.1080/00207169108804026
[13] Yahaya, Y.A. (2004) Some Theories and Applications of Continuous LMM for Ordinary Differential Equations. PhD Thesis (Unpublished), University of Jos, Nigeria.
[14] Badmus, A.M. and Mshelia, D.W. (2012) Uniform Order Zero Stable k-Step Block Methods for Initial Value Problems of Ordinary Differential Equations. Journal of Nigerian Association of Mathematical Physics, 20, 65-74.
[15] Badmus, A.M. (2014) An Efficient Seven Point Block Method for Direct Solution of General Second Order Ordinary Differential Equations . British Journal of Mathematics and Computer Science, 4, 2840-2852.
http://dx.doi.org/10.9734/BJMCS/2014/6749

  
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