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2 – Block 3 - Point Modified Numerov Block Methods for Solving Ordinary Differential Equations
In this paper, linear multistep technique using power series as the basis function is used to develop the block methods which are suitable for generating direct solution of the special second order ordinary differential equations of the form y′′ = f(x,y), a < = x < = b with associated initial or boundary conditions. The continuaous hybrid formulations enable us to differentiate and evaluate at some grids and off – grid points to obtain two different three discrete schemes, each of order (4,4,4)T, which were used in block form for parallel or sequential solutions of the problems. The computational burden and computer time wastage involved in the usual reduction of second order problem into system of first order equations are avoided by this approach. Furthermore, a stability analysis and efficiency of the block method are tested on linear and non-linear ordinary differential equations whose solutions are oscillatory or nearly periodic in nature, and the results obtained compared favourably with the exact solution.
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[1] Fatunla S.O. (1991). Block Method for Second Order Initial Value Problem. International Journal of Computer Mathematics, England. Vol. 4, pp 55 63.
[2] Fatunla S.O. (1994). Higher Order parallel Methods for Second Order ODEs. Proceedings of the fifth international conference on scientific computing, pp 61 67.
[3] Lambert J.D. (1973). Computational Methods in Ordinary Differential Equations. John Willey and Sons, New York, USA.
[4] Awoyemi D.O. (1998). A class of Continuous Stormer Cowell Type Methods for Special Second Order Ordinary Differential Equations. Journal of Nigerian Mathematical Society Vol. 5, Nos. 1 & 2, pp100 108.
[5] Yahaya Y.A and Mohammed U. (2010). A 5 Step Block Method for Special Second Order Ordinary Differential Equations. Journal of Nigerian Mathematical Society. Vol. 29, pp 113 126.
[6] Yahaya Y.A and Adegboye, Z.A. (2008). A family of 4 step Block Methods for Special Second Order in Ordinary Differential Equations. Proceedings Mathematical Association of Nigeria, pp 23 32.
[7] Fudziah I. Yap L. K. and Mohammad O. (2009). Explicit and Implicit 3 point Block Methods for Solving Special Second Order Ordinary Differential Equations Directly. International Journal of math. Analysis, Vol. 3, pp 239 254.
[8] Onumanyi P., Awoyemi D.O, Jator S.N and Sirisena U.W.(1994). New linear Multistep with Continuous Coefficient for first order initial value problems. Journal of Mathematical Society, 13, pp 37 51.
[9] Hairer E. Norsett S.P. and Wanner G. (1993). Solving Ordinary Differential Equations I, Non-stiff problems, 2nd edition, Berlin, New York Springer-verlag. ISBN 978 3 540 56670 0.
[10] Aladeselu N. A. (2007). Improved family of block methods for I.V.P. Journal of the Nigeria Association of Mathematical Physics. Vol. 11, pp 153 158.
[11] Henrici P. (1962). Discrete Variable Methods for ODEs. John Willey New York U.S.A.
[12] Adeboye K.R. (2000). A superconvergent H2 Galerkin method for the solution of Boundary Value Problems. Proceedings National Mathematical Centre, Abuja, Nigeria. Vol. 1 no.1 pp 118 124, ISBN 978 35488 0 2.
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