**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**31995

##### On the Efficiency of Five Step Approximation Method for the Solution of General Third Order Ordinary Differential Equations

**Authors:**
N. M. Kamoh,
M. C. Soomiyol

**Abstract:**

In this work, a five step continuous method for the solution of third order ordinary differential equations was developed in block form using collocation and interpolation techniques of the shifted Legendre polynomial basis function. The method was found to be zero-stable, consistent and convergent. The application of the method in solving third order initial value problem of ordinary differential equations revealed that the method compared favorably with existing methods.

**Keywords:**
Shifted Legendre polynomials,
third order block method,
discrete method,
convergent.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.3298890

**References:**

[1] 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, Department of Mathematics, Covenant University, Ota, Ogun State, Nigeria .170pp

[2] Awari, 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.H. Poor, An Introduction to Signal Detection and Estimation. New York: Springer-Verlag, 1985, ch. 4.

[3] Awoyemi1, D. O., Kayode, S. J. and Adoghe, L. O. (2014) A Five-Step P-Stable Method for the Numerical Integration of Third Order Ordinary Differential Equations, American Journal of Computational Mathematics 4: 119-126

[4] Awoyemi, D.O. and Idowu, O. (2005) A class hybrid collocation methods for third order of ordinary differential equations, International Journal of Computer Mathematics 82: 1287-1293

[5] Badmus, A.M. and Yahaya,Y. A. (2009) Some multi derivative hybrid block methods for solution of general third order ordinary differential equations, Nigerian Journal of Scientific Research, 8: 103-107.

[6] Henrici, P. (1962). Discrete Variable Methods in Ordinary Differential Equations, New York: John Willey and Sons.

[7] Jator, S. N. (2008) On the numerical integration of third order boundary value problems by a linear multistep method, International Journal of Pure and Applied Mathematics, 46(3): 375-388.

[8] Kamoh N. M., Gyemang, D. G. and Soomiyol, M. C. (2017) On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations, Pure and Applied Mathematics Journal. 6(5):137-143

[9] Kuboye, J. O. and Omar, Z (2015) Numerical Solution of Third Order Ordinary Differential Equations Using a Seven-Step Block Method, International Journal of Mathematical Analysis 9(15):743 – 754

[10] Mohammmed, U., Jiya, M. and Mohammed, A. A. (2010) A class of six step block method for solution of general second order ordinary differential equations, Pacific Journal of Science and Technology, 11(2): 273-277

[11] Olabode, B.T (2007) Some Linear Multistep Methods for Special and General Third Order Initial Value Problems in Ordinary Differential Equations, Ph.D. Thesis, the Department of Mathematical Sciences The Federal University of Technology, Akure.

[12] Olabode, B.T. and Yusuph, Y. (2009) A new block method for special third order ordinary differential equations Journal of Mathematics and Statistics, 5(3): 167-170

[13] Okunuga, S. A. and Ehigie, 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.

[14] Owolabi, K. M. (2015). A Rebust Optimal Order Formula for Direct Integration of Second Order Orbital Problems Advances in Physics Theories and Applications, 42: 21-26

[15] Sagir, A.M. (2014) On the approximate solution of continuous coefficients for solving third order ordinary differential equations, International Journal of Mathematical, Computational Science and Engineering, 8(3): 39-43.

[16] https://www.definitions.net/definition/differential%20equation

[17] Zill, D. G. and Warren, W. S. (2013) Differential Equations with Boundary- Value Problems. Cengage Learning, books/Cole, eight edition, 664pp

[18] Awoyemi, D. O., Kayode, S. J. and Adoghe, L. O. (2014) A Five-Step P-Stable Method for the Numerical Integration of Third Order Ordinary Differential Equations. American Journal of Computational Mathematics. 4: 119-126