Efficient Hybrid Block Method For The Numerical Solution Of Second-order Partial Differential Problems via the Method of Lines

Olumide O. Olaiya; Rasaq A. Azeez; Mark I. Modebei

doi:10.46481/jnsps.2021.140

Authors

Olumide O. Olaiya
olaiyaolumide.o@gmail.com
Department of Mathematics Programme, National Mathematical Centre, Abuja, Nigeria
Rasaq A. Azeez Department of Mathematics, University of Abuja, Abuja, Nigeria
Mark I. Modebei Department of Mathematics Programme, National Mathematical Centre, Abuja, Nigeria

Keywords:

Initial Value Problem, Boundary Value Problem, Block method, Linear Multistep Method, Hybrid method, mehod od lines

Abstract

This study is therefore aimed at developing classes of efficient numerical integration schemes, for direct solution of second-order Partial Differential Equations (PDEs) with the aid of the method of lines. The power series polynomials were used as basis functions for trial solutions in the derivation of the proposed schemes via collocation and interpolation techniques at some appropriately chosen grid and off-grid points the derived
schemes are consistent, zero-stable and convergent. the proposed methods perform better in terms of accuracy than some existing methods in the literature.

Dimensions

REFERENCES

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

P. Onumanyi, U.W. Sirisena. & S.N Jator, “Continuous finite difference ap-

proximations for solving differential equations”, International. Journal of. Computer Mathematics. 72 (1999) 15-27. DOI: https://doi.org/10.1080/00207169908804831

P. Onumanyi, D. O. Awoyemi, S. N. Jator & U. W. Sirisena, “New linear

mutlistep methods with continuous coeffcients for first order initial value problems”,

Journal of. Nigeria. Mathematics. Society. 13, (1994), 37-51. DOI: https://doi.org/10.1016/0732-3123(94)90033-7

S. N. Jator “On the Hybrid Method with Three-Off Step Points for Initial Value Problems” International Journal of Mathematical Education in Science and Technology, 41, (2010), 110-118 http://dx.doi.org/10.1080/00207390903189203 DOI: https://doi.org/10.1080/00207390903189203

S. N. Jator “Trigonometric symmetric boundary value method for oscillating solutions including the sine-Gordon and Poisson equations” Cogent Mathematics 3 (2016), 1271269 DOI: https://doi.org/10.1080/23311835.2016.1271269

S. O. Fatunla, “Block Methods for second-order IVP”, International Journal of Mathematics 55, (1999)

Y. Yusuph & P. Onumanyi; “New multiple FDM through multistep collocation for y^''=f(x,y)” Proceedings of the conference, National Mathematical Centre, Abuja (2005).

Siraj-ul-Islam, Imran Aziz & Bozidar Sarler, “The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets”, Mathematical and Computer Modelling 52 (2010) 1577-1590. DOI: https://doi.org/10.1016/j.mcm.2010.06.023

J. Adewale, A. Olaide. & J. Sunday, “Continuous Block Method for the Solution of second rder Initial Value Problems of Ordinary Diff. Equation” International Journal of Pure and Applied mathematics 82, (2013), 405-416.

F.F. Ngwane & S. N. Jator, “L-Stable Block Hybrid Second Derivative Algorithm for Parabolic Partial Differential Equations”. American Journal of Computational Mathematics, 4, (2014) 87-92. http://dx.doi.org/10.4236/jasmi.2014.42008 DOI: https://doi.org/10.4236/jasmi.2014.42008

J. Vigo-Aguiar & H. Ramos, “A family of A-Stable Collocation Methods of Higher Order for Initial-Value Problems”. IMA Journal of Numerical Analysis, 27, (2007), 798-817. http://dx.doi.org/10.1093/imanum/drl040 DOI: https://doi.org/10.1093/imanum/drl040

J. R, Cash “Two New Finite Difference Schemes for Parabolic Equations”. SIAM Journal of Numerical Analysis, 21, (1984) 433-446. http://dx.doi.org/10.1137/0721032 DOI: https://doi.org/10.1137/0721032

S. N. Jator & J. Li, “An algorithm for Second Order Initial and Boundary Value Problems” Numerical Algorithm (2012), 59-67 DOI: https://doi.org/10.1007/s11075-011-9492-3

T. A. Biala, “Computational Study of the Boundary Value Methods and the Block Unification Methods for y''=f(x,y,y')”, Abstract and Applied Analysis 2016, Article ID 8465103, 14 pages http://dx.doi.org/10.1155/2016/8465103 DOI: https://doi.org/10.1155/2016/8465103

P. Henrici, “Discrete Variable Methods in Ordinary Differential Equations”, John Wiley, New York, 1962.

] J. D. Lambert, “Computational Methods in in Ordinary Differential Equations” :John Wiley and Sons, New York, NY, USA, 1973;

M. I. Modebei, R. B. Adeniyi, S. N. Jator & H. C. Ramos “A block hybrid integrator for numerically solving fourth-order Initial Value Problems” Applied Mathematics and Computation 346, (2019), 680-694 DOI: https://doi.org/10.1016/j.amc.2018.10.080