• editor.aipublications@gmail.com
  • Track Your Paper
  • Contact Us
  • ISSN: 2456-866X

International Journal Of Chemistry, Mathematics And Physics(IJCMP)

A Class of Continuous Implicit Seventh-eight method for solving y’= f(x, y) using power series

E. O. Omole , O. A Jeremiah , L.O. Adoghe

International Journal of Chemistry, Mathematics And Physics(IJCMP), Vol-4,Issue-3, May - June 2020, Pages 39-50 , 10.22161/ijcmp.4.3.2

Download | Downloads : 2 | Total View : 515


In this article, we develop a continuous implicit seventh-eight method using interpolation and collocation of the approximate solution for the solution of y’ = f(x,y) with a constant step-size. The method uses power series as the approximate solution in the derivation of the method. The independent solution was then derived by adopting block integrator. The properties of the method was investigated and found to be zero stable, consistent and convergent. The integrator was tested on numerical examples ranging from linear problem, Prothero-Robinson Oscillatory problem, Growth Model and Sir Model. The results show that the computed solution is closer to the exact solution and also, the absolutes errors are minimal and uses lesser time for the computations.

Seventh-eight method, Continuous Implicit method, Power Series, y’ = f(x, y), P-stable, Growth Model, SIR model, Prothero-Robinson Oscillatory problem, Convergent.

[1] Olver P. J., Lecture Notes on Numerical Analysis, May, 2008, http://www. users.math.umn.edu/~olver/num.html
[2] P. Kandasamy, K. Thilagavathy and K. Gunavathy, Numerical Methods, S. Chand and Company, New Delhi-110055, 2005.
[3] Jain M. K, Iyengar S. R. K, and Jain R. K. “Numerical methods for scientific and engineering computation”. 5th Edition, New Age International, New Delhi, India, RK (2007).
[4] Fatunla, S. O, Block Method for Second Order Initial Value Problem. International Journal of Computer Mathematics, England. Vol. 4, 1991,pp 55 – 63.
[5] U.S.U. Aashikpelokhai, A class of nonlinear one-step rational integrator, PhD Thesis, University of Benin, Nigeria, 1991.
[6] Ibijola, E. A, Skwame, Y. and Kumleng, G., Formation of hybrid block method of higher step sizes, through the continuous multi-step collocation, Am. Journal Sci. Ind. Res., (2011), 2(2), 161-173.
[7] Onimanyi, P., Awoyemi, D. O., Jator, S. N. &Sirisena, U. W., New Linear Multistep Methods with continuous coefficients for first order initial value problems. (1994). J.Nig.Math.Soc.:
[8] Adeniyi, R. B. Adeyefa, E. O. and Alabi, M.O., A continuous formulation of someclassical initial value solvers by non-perturbed multistep collocation approach usingchebyshev Polynomial as basis function. Journal of the Nigerian Association of Mathematical Physics, (2006), 10:261-274.
[9] Brown, R. L., Some characteristics of implicit multistep Multi derivative Integration formulas. SIAM, J. on Numerical Analysis. (1977), 14, 982 – 993.
[10] Henrici, P., Discrete variable methods for ODEs. (1962), John Wiley, New York.
[11] Sunday J. and M.R. Odekunle, A New Numerical Integrator for the Solution of Initial Value Problems in Ordinary Differential Equations. Pacific Journal of Science and Technology, Vol. 13 N0. 1, 2012, pp. 221– 227.
[12] Lambert, J. D., Computational methods in ordinary differential equation, (1973), John Wiley & Sons Inc.
[13] Chollom, J. P., Ndam, J. N. and Kumleng, G. M., On some properties of the block linear multistep methods. Science World Journal, (2007), 2(3), 11 -- 17.
[14] Butcher, J. C., Numerical Methods for Ordinary differential systems. (2003), West Sussex, England: John Wiley & sons
[15] Shampine, L.F. and Watts, H.A., Block implicit one-step methods. Journal of Computer Maths. (1969), 23:731-740.
[16] Fatunla, S.O., A class of block methods for second order IVPs, Int. J. Comput. Math., (1994), 55: 119-133.
[17] Areo E. A and Omojola M. T., A new One-Twelfth Continous Block method for the solution of modelled problems of ordinary differential equations, American Journal of Computational Mathematics, (2015), 5, 447-457
[18] Olanegan, O. O.,Ogunware, B. G., Omole E. O. Oyinloye, T. S. and Enoch, B. T., some Variable Hybrids Linear Multistep Methods for Solving First Order Ordinary Differential Equations Using Taylor’s Series, IOSR Journal of Mathematics, (2015), 5 (1), PP 08-13
[19] Adeniran A. O. & Longe I. O. “Solving directly second order initial value problems with Lucas polynomial. Journal of Advances in Mathematics and Computer science, 32(4), 1-7, (2019). http://dx.doi.org/10.9734/jamcs/2019/v32i430152.
[20] Lukman M. A, Olaoluwa O. E. “A One-step extended block hybrid formulae for solving Orbital problems”, Moj App. Bionics&Biomechanics, (2019), 3(1):37-43, http://dx.doi.org/10.15406/mojabb.2019.03.00101.
[21] Sunday J., Skwame Y., and Huoma I. U., “Implicit One-Step Legendre Polynomial Hybrid Block Method for the Solution of First-Order Stiff Differential Equations”. B. J.ofMaths. & Comp. Sci. 8(6), 482-491, (2015). https://doi.org/10.9734/BJMCS/2015/16252
[22] Badmus A. M. “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, (2014). http://dx.doi.org/10.4236/ajcm.2014.44032
[23] Jator, S.N., Lee, L. “Implementing a seventh-order linear multistep method in a predictor-corrector mode or block mode: which is more efficient for the general second order initial value problem”. SpringerPlus 3(1), 1–8 (2014). https://doi.org/10.1186/2193-1801-3-447
[24] Adeyeye, O., Kayode, S.J. “ Two-step two-point hybrid methods for general second orderdifferential equations” . Afr. J. Math. Comput. Sci. Res. 6(10), 191–196 (2013). https://doi.org/10.5897/AJMCSR2013.0502
[25] Ayinde S. O., Ibijola E. A.. A New Numerical Method for Solving First Order Differential Equations. American Journal of Applied Mathematics and Statistics. 2015; 3(4):156-160. doi: 10.12691/ajams-3-4-4
[26] Kama, P. and Ibijola, E. A., On a New One – Step Method for Numerical Integration of Ordinary Differential Equtions. International Journal of Computer Mathematics, Vol.78, No. 3,4. (2000).
[27] E.A.Ibijola, On the convergence consistency and stability of a one step method for integration of ordinary differential equation, International Journal of Computer Mathematics, 73 (1998), 261-277.
[28] R.B.Ogunrinde, A new numerical scheme for the solution of initial value problem (IVP) in ordinary differential equations, Ph.D. Thesis, Ekiti State University, Ado Ekiti. (2010)