Optimized hybrid block integrator for the numerical solution of third order initial and boundary value problems


  • M. I. Modebei National Mathematical Centre, Abuja


A hybrid method for the numerical approximation of the solution of general third order initial and boundary value problems is derived via the collocation technique. This method considers three intra-step points which are adequately selected so as to optimize the local truncation errors of the main formulas. The new method is zero-stable, consistent and convergent. Numerical examples from literature shows the efficiency of this method in terms of the global errors obtained.

Author Biography

M. I. Modebei, National Mathematical Centre, Abuja

National Mathematical Centre, Abuja (Research/Ph.D. student)


E.O. Tuck, L.W. Schwartz, A numerical and asymptotic study of some third order ordinary differential equation relevant to draining and coating flows, SIAM Review, 32, No. 3 (1990), 453-549.

S. Valarmathi, N. Ramanujam, Boundary value technique for finding numerical solution to boundary value problems for third order singular perturbed ordinary differential equations, Intern. J. Computer Math., 79, No.

(2002), 747-763.

A. Tirmizi, E.H. Twizell, Siraj-Ul-Islam, A numerical method for third-order non-linear boundary-value problems in engineering, Intern. J. Computer Math. 82 (2005) 103-109.

J. Henderson and K. R. Prasad. Existence and uniqueness of solutions of three-point bound-

ary value problems on time scales. Nonlin. Studies, 8(2001), 1-12.

C. P. Gupta and V. Lakshmikantham. Existence and uniqueness theorems for a third-order

three point boundary value problem. Nonlinear Anal.: Theory, Meth. Appl., 16(11)(1991),


K. N. Murty and Y. S. Rao. A theory for existence and uniqueness of solutions to three-point

boundary value problems. J. Math. Anal. Appl., 167(1)(1992), 43-48.

H. Ramos, Z. Kalogiratou, Th. Monovasilis and T. E. Simos, An optimized two-step hybrid block method for solving

general second order initial-value problems, Numerical Algorithm, Springer (2015, )DOI 10.1007/s11075-015-0081-8

R. P. Agarwal. Boundary Value Problems for Higher Order Differential Equations. World

Scientific, Singapore, 1986.

S. Islam, M. A. Khan, I. A. Tirmizi and E. H. Twizell. Non-polynomial splines approach

to the solution of a system of third-order boundary-value problems. Appl. Math. Comp.,

(1)(2005), 152-163.

F. Gao and C. M. Chi. Solving third-order obstacle problems with quartic B-splines. Appl.

Math. Comp., 180(1)(2006), 270-274.

P. K. Pandey. Solving third-order Boundary Value Problems with Quartic Splines. Panday

SpringerPlus, 5(1)(2016), 1-10 .

R.K. Sahi, S.N. Jator, N.A. Khan, Continuous Fourth Derivative Method For Third-order Boundary Value Problems, International Journal of Pure and Applied Mathematics

Volume 85 No. 5 2013, 907-923

Ra'ft Abdelrahim, Numerical solution of third order boundary value problems

using one-step hybrid block method, Ain Shams Engineering Journal 10 (2019) 179–183

S. N. Jator, Novel Finite Difference Schemes For Third Order Boundary Value Problems International Journal of Pure and Applied Mathematics Volume 53 No. 1 2009, 37-54

Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publishers, Amsterdam (1998)

Modebei M. I., Adeniyi R. B., Jator S. N. and Ramos H. C. (2019) A block hybrid

integrator for numerically solving fourth-order Initial Value Problems Applied

Mathematics and Computation 346 680-694

B. S. H. Kashkari and S. Alqarni, (2019) Optimization of two-step block method with three hybrid

points for solving third order initial value problems, J. Nonlinear Sci. Appl., 12, (2019), 450-469

S.N. Jator (2009), Novel finite difference schemes for third-order boundary value problems, International Journal of Pure and Applied Mathematics, 53 (1), 37-54

Lambert J. D. (1973), 'Computational Methods in Ordinary Differential Equations', {it John Wiley, New York}.




How to Cite

Modebei, M. I. (2022). Optimized hybrid block integrator for the numerical solution of third order initial and boundary value problems. Journal of the Nigerian Mathematical Society, 41(1), 49–64. Retrieved from https://ojs.ictp.it/jnms/index.php/jnms/article/view/584