An efficient hybrid numerical scheme for solvinggeneral second order initial value problems (IVPs)
Keywords:Collocation, Consistency, Initial value problems, Interpolation, Region of Absolute stability, Stiff ordinary differential equation, Zero Stability.
The paper presents a one step hybrid numerical scheme with two off grid points for solving directly the general second order initial value problems of ordinary differential equations. The scheme is developed using collocation and interpolation technique. The proposed scheme is consistent, zero stable and of order four. This scheme can estimate the approximate solution at both step and off step points simultaneously by using variable step size. Numerical results are given to show the efficiency of the proposed scheme over the existing schemes.
 C. E. Abhulimen and S. A. Okunuga (2008): Exponentially Fitted Second Derivative Multistep Method for Stiff Initial Value Problem for ODEs, journal of Engineering science and Applications, 5, 36-49.
 M. O. Alabi, A. T. Oladipo and A. O. Adesanya (2008): Initial Value solvers for Second Order Ordinary Differential Equations Using Chebyshev Polynomial as Basis Functions, Journal of Modern Mathematics, 2(1), 18-27.
 Ali Shorki (2012): The Symmetric P-Stable Hybrid Obrenchkoff Methods for the numerical solution of second Order IVPS. J. Pure.Appl. Math. 5(1), 28-35.
 D. O. Awoyemi, A. O. Adesanya and S. N. Ogunyebi(2009): Construction of Self Starting Numerov Method for the Solution of Initial Value Problem of General Second Order Ordinary Differential Equation. Journ. Num. math 4(2), 267-278.
 R. A. Bun and Y.D. Vasil'Yev (1992): A Numerical Method for Solving Differential Equations of any Orders. Comp. Math. Phys., 32(3), 317-330.
 M. T. Chu and H. Hamilton (1987): Parallel Solution of Ordinary Differential Equations by Multiblock Methods. SIAM Journal of Scientific and Statistical Computations, 8, 342-553.
 Gurjinder Singh, V. Kanwar and Saurabh Bhatia (2013): Exponentially fitted variants of the two-step Adams-Bashforth method for the numerical integration of initial value problem. journal of application and applied mathematics, 8(2), 741-755.
 S. O. Fatunla (1991): Block methods for second order IVPs, Int.J. Comput.Maths., 41 55-63.
 P. Henrici (1962): Discrete Variable Methods in ODE. New York: John Wiley and Sons.
 S. N. Jator (2007): A sixth order linear multistep method for direct solution second order differential equation. International journal of pure and applied Mathematics 40(1), 457-472.
 S. N. Jator and Li, J. (2009): a self starting linear multistep method for a direct solution of the general second order initial value problems. Intern J. Comp. Math. 86(5), 827-836.
 J.D. Lambert (1973): Computational Methods in ODEs. New York: John Wiley.
 B. T. Olabode (2009): An accurate scheme by block method for the third order ordinary differential equation. Pacific journal of science and technology. 10(1), http: // www. okamaiuniversity.us/pjst.htm.
 G. Psihoyios and T. E. Simos (2003): Trigonometrically fitted predictor-corrector methods for IVPs with Oscillating solution. Journal of Computational and Applied Mathematics, 158 (2003) 135-144.
 E. H. Twizel and A. Q. M. Khaliq (1984): Mutiderivative Methods for Periodic IVPs. SIAM Journal of Numerical Analysis, 21, 111-121.
 J. Vigo-Aguilar and H. Ramos (2006): Variable Stepsize Implementation of Multistep Methods for y'' = f(x, y, y'). Journal of Computational and Applied Mathematics,192, 114-131.
 V. V. Wend (1969): Existence and Uniqueness of solution of ordinary differential equation, proceedings of the American Mathematical Society. 23(1), 23-27.
 Y.A. Yahaya and A. M. Badmus (2009): A Class of Collocation Methods for General Second Order Ordinary Differential Equations. African Journal of Mathematics and Computer Science Research, 2(04), 069-072 .