Reconstructed variational iteration algorithm via third-kind shifted Chebyshev polynomials for the numerical solution of seventh-order boundary value problems

Document Type : Original Article

Authors

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

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

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

4 Department of Maths and Statistics, Federal Polytechnic Bida, Bida, Nigeria.

5 Department of Mathematics, University of Ilorin, Kwara State, Nigeria

Abstract

The variational iteration algorithm using shifted Chebyshev polynomials of the third kind was used to obtain the numerical solution of seventh order Boundary Value Problems(PVBs) in this paper. The proposed method is made by constructing the shifted Chebyshev polynomials of the third kind for the given boundary value problems and used as a basis functions for the approximation. Numerical examples where also given to show the efficiency and reliability of the proposed method. Calculations were performed using maple 18 software.

Keywords


[1] G. C. Abanum, O. K. Oke, Modified Variational Iteration Method for Solution of Seventh Order Boundary Value Problem Using Canonical Polynomials, International Journal of Modern Mathematical Sciences, 17(1) 2019, 57-67
[2] G. Akram, S.S. Siddiqi, Solution of seventh order boundary value problems by variation of parameters method, Research Journal of Applied Sciences, Engineering and Technology, 5(1) 2013, 176-179.
[3] 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.
[4] A.F. Adebisi, K.A. Okunola, M.T. Raji, J.A. Adedeji, O.J. Peter, Galerkin and perturbed collocation methods for solving a class of linear fractional integro-differential equations, The Aligarh Bulletin of Mathematics, 40(2) 2021, 45-57.
[5] L. Bougoffa, R.C. Rach, A, Mennouni, An approximate method for solving a class of weakly-singular Volterra integro-differential equations, Applied Mathematics and Computation, 217(22) 2011, 8907-8913.
[6] H.N. Caglar, S.H. Caglar, E.H. Twizell, The numerical solution of fifth-order value problems with sixth degree B-spline function, Applied Mathematics Letters, 12(5) 1999, 25-30.
[7] M. Din, A. Yildirim, Solutions of Tenth and Ninth-order Boundary Value Problems by Modified Variational Iteration Method, Application and applied mathematics, 5(1) 2010, 11- 25.
[8] 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(2) 2016, 383-390.
[9] 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.
[10] M.A. Noor, S.T. Mohyud-Din, A new approach for solving fifth order boundary value problem, International Journal of Nonlinear Science, 9(4) 2010, 387-393.
[11] M.A. Noor, S.T. Mohyud-Din, Variational iteration decomposition method for eight order boundary value problem, Differential Equations and Nonlinear Mechanics, 2007.
[12] M. Noor, S.T. Mohyud-Din. Variational iteration method for fifth order boundary value problem using He’s polynomials, Mathematical Problems in Engineering, 2008.
[13] S.S. Shahid, M. Iftikhar, Variational iteration method for solution of seventh order boundary value problem using He’s polynomials, Journal of the Association of Arab Universities for Basic and Applied Sciences, 18 2015, 60-65.
[14] 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.
[15] 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.
[16] O.J. Peter, M.O. Ibrahim, Application of Variational Iteration Method in Solving Typhoid Fever Model, Knowledge and Control Systems Engineering (BdKCSE), Sofia, Bulgaria. 1(1) 2019, 1-5.
[17] O.J. Peter, O.A. Afolabi, F.A. Oguntolu, C.Y. Ishola, A. A. Victor, Solution of a Deterministic Mathematical Model of Typhoid Fever by Variational Iteration Method. Science World Journal, 13(2) 2018, 64-68.
[18] S.M. Reddy, Numerical Solution of Ninth Order Boundary Value Problems by Quintic B-splines International Journal of Engineering Inventions 5(7) 2016, 38-47.
[19] S.S. Siddiqi, M. Iftikhar, Solution of Seventh Order Boundary Value Problems by Variation of Parameters Method. Res. J. Appl. Sci., Engin. Tech., 5(1) 2013, 176–179.
[20] K.N.S. Kasi Viswanadham, S.M. Reddy, Numerical Solution of Seventh Order Boundary Value Problems by Petrov-Galerkin Method with Quintic B-splines as Basis Functions and Septic B-splines as Weight Functions. International Journal of Computer Applications, 12(5) 2015, 0975 – 8887.
[21] 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-Ganji’s Method, BAREKENG: J. Math. & App., 16(3) 2022, 1123-1130.
[22] 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.
[23] T. Oyedepo, O.A. Uwaheren, E.P. Okperhie, O. J. Peter. Solution of Fractional Integro-Differential Equation Using Modified Homotopy Perturbation Technique and Constructed Orthogonal Polynomials as Basis Functions, Journal of Science Technology and Education, 7(3) 2019, 157-164.
[24] 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-Ganji’s Method, BAREKENG: J. Math. & App., 16(3) 2022, 1123-1130.
Volume 3, Issue 4
December 2022
Pages 55-61
  • Receive Date: 26 November 2022
  • Revise Date: 09 December 2022
  • Accept Date: 25 December 2022
  • First Publish Date: 25 December 2022