Collocation Techniques for Block Methods for the Direct Solution of Higher Order Initial Value Problems of Ordinary Differential Equations
International Journal on Data Science and Technology
Volume 3, Issue 4, July 2017, Pages: 39-44
Received: Jun. 1, 2017;
Accepted: Aug. 21, 2017;
Published: Sep. 26, 2017
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Kamoh Nathaniel Mahwash, Department of Mathematics/Statistics Bingham University, Karu, Nigeria
Awari Yohanna Sani, Department of Mathematics/Statistics Bingham University, Karu, Nigeria
Chun Pamson Bentse, Department of Mathematics/Statistics Plateau State University Bokkos, Bokkos, Nigeria
This paper presents the derivation techniques of block method for solving higher order initial value problems of ordinary differential equations directly. The method was developed via interpolation and collocation of the shifted Legendre polynomials as basis function. The method is capable of providing the numerical solution at several points simultaneously.
Kamoh Nathaniel Mahwash,
Awari Yohanna Sani,
Chun Pamson Bentse,
Collocation Techniques for Block Methods for the Direct Solution of Higher Order Initial Value Problems of Ordinary Differential Equations, International Journal on Data Science and Technology.
Vol. 3, No. 4,
2017, pp. 39-44.
Adesanya, A. O., Anake, T. A., Bishop, S. A. and Osilagun, J. A. (2009). Two Steps Block Method for the Solution of General Second Order Initial Value Problems of Ordinary Differential Equation. ASSET 8(1): 59-68.
Adesanya, A. O., Anake, T. A. and Udoh, M. O. (2008). Improved Continuous Method for Direct Solution of General Second Order Ordinary Differential Equations. Journal of the Nigeria Association of Mathematical Physics.13: 59-62.
Anake, T. A. (2011). Continuous Implicit Hybrid One –Step Methods for the Solution of Initial Value Problems of Second Order Ordinary Differential Equations. PhD Thesis, Covenant University, Ota, Ogun State, Nigeria.
Awarri, Y. S., Chima, E. E., Kamoh, N. M. and Oladele, F. L. (2014). A Family of Implicit Uniformly Accurate Order Block Integrators for the Solution of Second Order Differential Equations; International Journal of Mathematics and Statistics Invention (IJMSI).2: 34-46.
Areo, E. A., Ademiluyi, R. A. and Babatola, P. O. (2008). Accurate Collocation Multistep Method for integration of first order ordinary differential equations. Journal of Modern Mathematics and Statistics. 2(1): 1-6.
Awoyemi, D. O. (1991). A Class of Continuous Linear Multistep Method for General Second Order Initial Value Problems in Ordinary Differential Equations. International Journal of Computer Mathematics.72: 29-37.
Awoyemi, D. O and Kayode, S. J. (2005). An Implicit Collocation Method for Direct Solution of Second Order Ordinary Differential Equations; Journal of Nigeria Association of Mathematical Physics.24:70-78.
Awoyemi, D. O. (2001). A New Sixth-Order Algorithm for General Second Order Ordinary Differential Equation. International Journal of computational mathematics. 77: 117-124.
Serisina, U. W., Kumleng, G. M. and Yahaya, Y. A. (2004). A New Butcher Type Two-Step Block Hybrid Multistep Method for Accurate and Efficient Parallel Solution of Ordinary Differential Equations. Abacus Mathematics Series31:1-7.
Fatunla, S. O. (1991). Block Method for Second Order Ordinary Differential Equations. International Journal of Computer Mathematics, 41:(1&2), 55-63.
Fatunla, S. O. (1995). A Class of Block Method for Second Order Initial Value Problems. International Journal of Computer Mathematics, 55:(1&2), 119-133.
Lambert, J. D. (1991). Numerical Methods for Ordinary Differential Equations. John Wiley and Sons New York, 293pp.
Lambert, J. D. (1973). Computational Methods in Ordinary Differential Equations. New York: John Wiley and Sons.
Okunuga, S. A. and Ohigie, J. (2009). A New Derivation of Continuous Multistep Methods Using Power Series as Basis Function. Journal of Modern Mathematics and Statistics 3(2): 43-50.
Okunuga, S. A and Onumanyi, P. (1985). An Accurate Collocation Method for Solving Ordinary Differential Equations. AMSE, France, 4(4): 45-48.
Onumanyi, P. A. Woyemi, D. O., Jator, S. N. and Sirisena, U. W. (1994). New Linear Multistep Methods with Continuous Coefficients for First Order Initial Value Problems. Journal of Nigeria Mathematics Society.13:37-51.
Onumanyi, P and Yusuph, Y. (2002). New Multiple FDMs Through Multistep Collocation for y^''=f(x,y). Abacus, 29: 92-100.
Onumanyi, P and Okunuga, S. A. (1986). Collocation Tau Method for the Solution of Ordinary Differential Equations. Abacus, 17: 51-60.
Onumanyi, P., Oladele, J. O., Adeniyi, R. B and Awoyemi, D. O. (1993). Derivation of Finite Method by Collocation. Abacus, 23(2): 72-83.
Onumanyi, P., Sirisena, W. U. & Jator, S. N. (1999). Continuous Finite Difference Approximations for Solving Differential Equations. International Journal of Computer and Mathematics, 72, 15-27.
Owolabi, K. M. (2015) ARebust Optimal Order Formula for Direct Integration of Second Order Orbital Problems. Advances in Physics Theories and Applications 42: p21-26.
Warren, W. S and Zill, D. G. (2013). Differential Equations with Boundary- Value Problem; Eight edition books/Cole, Cengage Learning.
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(4): 069-072.