Numerical solution of eight order boundary value problems using Chebyshev polynomials

Document Type : Original Article


1 Department of Mathematics, Federal University of Agriculture, Abeokuta, Ogun State, Nigeria

2 Department of Mathematics, National Open University of Nigeria Jabi, Abuja, Nigeria

3 Department of Mathematics and Statistics, Osun State College of Technology Esa Oke, Osun State, Nigeria

4 Department of Mathematics, Osun State University, Osogbo, Nigeria

5 Department of Mathematics, Federal University of Technology Minna, Niger State, Nigeria.

6 Department of Mathematical and Computer Sciences, University of Medical Sciences, Ondo City, Ondo State


First-kind Chebyshev polynomials are used as the basis functions in this study to present the approximations to the eighth-order boundary-value problems. The problem is reduced using the suggested approach into a set of linear algebraic equations, which are then solved to determine the unknown constants. To demonstrate the application and effectiveness of the strategy, analytical results are provided using tables and graphs for three examples. The results obtained using the proposed method reveal that it is simple and outperforms comparable solutions in the literature.


Main Subjects

[1] A.F. Adebisi, T.A. Ojurongbe, K.A. Okunlola, O.J. Peter, Application of Chebyshev polynomial basis function on the solution of Volterra integro-differential equations using Galerkin method Mathematics and Computational Sciences, 2(1) 2021, 41-51.
[2] A.F. Adebisi, O.A. Uwaheren, O.E. Abolarin, R.M. Tayo, J.A. Adedeji, O.J. Peter, Solution of Typhoid Fever Model by Adomian Decomposition Method, Journal of Mathematical and Computer Science, 11(2) 2021, 1242-1255.
[3] A.F. Arshed, I. Hussain, Solution of sixth-order boundary value problems by Collocation method, International Journal of Physical Sciences, 7(43) 2012, 5729-5735.
[4] S.T. Ejaz, G. Mustafa, F. Khan, Subdivision schemes based collocation algorithms for solution of fourth order boundary value problems, Mathematical Problems in Engineering, Article ID 240138. 2015, 1-18.
[5] A. Golbabai, M. Javidi, Application of homotopy perturbation method for solving eighth order boundary value problems Applied Mathematics Computation, 191 2007, 334-346.
[6] M. Inc, D.J. Evans, An efficient approach to approximate solutions of eighth-order boundary value problems. International Journal of Computer Mathematics 81(6) 2004, 685-692.
[7] C.Y. Ishola, O.A. Taiwo, A.F. Adebisi, O.J. Peter, Numerical solution of two-dimensional Fredholm integro differential equations by Chebyshev integral operational matrix method, Journal of Applied Mathematics and Computational Mechanics, 21(1) 2022, 29-40.
[8] G. Kanwal, A. Ghaffar, M.M. Hafeezullah, S.A. Manan, M. Rizwan, G. Rahman, Numerical solution of 2-point boundary value problem by subdivision scheme, Communications Mathematics and Applications, 10(1) 2019, 19-29.
[9] K.N.S. Kasi Viswanadham, B. Sreenivasulu, Numerical Solution of Eighth Order Boundary Value Problems by Galerkin Method with Quintic B-splines. International Journal of Computer Applications 89(15) 2014, 7-13.
[10] A. Khalid, M.N. Naeem, Cubic B-spline solution of nonlinear sixth order boundary value problems, Punjab University Journal of Mathematics, 50(4) 2018, 91-103.
[11] A. Khalid, M.N. Naeem, P.Agarwal, A. Ghaffar, Z. Ullah, Z. Ullah, Numerical approximation for the solution of linear sixth order boundary value problems by cubic B-spline, Advances in Difference Equations, 492 2019, 1-16.
[12] F.G. Lang, X.P. Xu, A new cubic B-spline method for linear fifth order boundary value problems, Journal Applied Mathematics Computing, 36(1) 2011, 101-116.
[13] G.R. Liu, T.Y. Wu, Differential quadrature solutions of eighth-order boundary-value differential equations. Journal Computational, And Applied. Mathemtics, 145 2002, 223-235.
[14] E.J. Mamadu, I.N. Njoseh, Tau-collocation approximation approach for Solving first and second order ordinary differential equations, Journal of Applied Mathematics and Physics, 4 2016, 383-390.
[15] K.P. Murali, Fem based collocation method for solving eighth order boundary value problems using b-splines, ARPN Journal of Engineering and Applied Sciences, 11(23) 2016, 13594-13598.
[16] I.N. Njoseh, E.J. Mamadu, Numerical solutions of a generalized Nth Order Boundary Value Problems Using Power Series Approximation Method. Applied Mathematics, 7 2016, 1215-1224.
[17] T. Oyedepo, A.F. Adebisi, R.M. Tayo, J.A. Adedeji, M.A. Ayinde, O.J. Peter, Perturbed least squares technique for solving volterra fractional integro-differential equations based on constructed orthogonal polynomials, Journal of Mathematical and Computer Science, 11 2021, 203-218.
[18] O.J. Peter, F.A. Oguntolu, M.M. Ojo, A.O. Oyeniyi, R. Jan, I. Khan, Fractional Order Mathematical Model of Monkeypox Transmission Dynamics, Physica Scripta, 97(8) 2022, 1-26.
[19] O.J. Peter, A.S. Shaikh, M.O. Ibrahim, K.S. Nisar, D. Baleanu, I. Khan, A.I. Abioye. Analysis and Dynamics of Fractional Order Mathematical Model of COVID-19 in Nigeria Using Atangana-Baleanu Operator, Computers, Materials and Continua, 66(2) 2021, 1823-1848.
[20] O.J. Peter, A. Yusuf, M.M Ojo, S. Kumar, N. Kumari, F.A. Oguntolu, A Mathematical Model Analysis of Meningitis with Treatment and Vaccination in Fractional Derivatives. International Journal of Applied and Computational Mathematics, 8 2022, 117.
[21] O.J. Peter, A. Yusuf, K. Oshinubi, F.A, Oguntolu, A.A. Ibrahim, A.I. Abioye, T.A. Ayoola. Fractional Order of Pneumococcal Pneumonia Infection Model with Caputo Fabrizio Operator, Results in Physics, 29 2021, 104581.
[22] S.S. Shahid, M. Iftikhar, Variational Iteration Method for solution of Seventh Order Boundary Value Problem using Hes Polynomials, Journal of the Association of Arab Universities for Basic and Applied Sciences, 18 2015, 60-65.
[23] S.S. Siddiqi, G. Akram, Solution of eighth-order boundary value problems using the nonpolynomial spline technique, International Journal of Computer Mathematics, 84(3) 2007, 347-368.
[24] S.S. Siddiqi, A. Ghazala, Z. Sabahat , Solution of eighth order boundary value problems using Variational Iteration Technique, European Journal of Scientific Research, 30 2009, 361-379.
[25] I.A. Tirmizi, M.A. Khan, Non-polynomial splines approach to Trs6he solution of sixth-order boundary value problems, Applied Mathematics Computation, 195(1) 2008, 270-284.
[26] J. Tsetimi, K.O. Ogeh, A.B. Disu Modified variational iteration method with Chebyshev Polynomials for solving 12 the order Boundary value problems, Journal of Natural Sciences and Mathematics Research, 8(1) 2022, 44-51.
[27] I. Ullah, H. Khan, M.T. Rahim, Numerical solutions of fifth and sixth order nonlinear boundary value problems by Daftardar Jafari method, Journal Computational Engineering, Article ID 286039 2014,1-8.
[28] O.A. Uwaheren, A.F. Adebisi, C.Y. Ishola, M.T. Raji, A.O. Yekeem, O.J. Peter, Numerical Solution of Volterra integro-differential Equations by Akbari-Ganjis Method, BAREKENG: J. Math. App.,16(3) 2022, 1123-1130.
[29] O.A. Uwaheren, A.F. Adebisi, O.T. Olotu, M.O. Etuk, O.J. Peter, Legendre Galerkin Method for Solving Fractional Integro-Differential Equations of Fredholm Type, The Aligarh Bulletin of Mathematics, 40(1) 2021, 1-13.
[30] A.M. Wazwaz, The numerical solution of special eight-order boundary value problems by the modified decomposition method, Neural, Parallel Scientific Computations, 8(2) 2000, 133-146.