# NUMERICAL SOLUTIONS OF THIRD ORDER FREDHOLM INTEGRO DIFFERENTIAL EQUATION VIA LINEAR MULTISTEP-QUADRATURE FORMULAE

• R. B. Ogunrinde
• A. A. Obayomi
• K. S. Olayemi
Keywords: Booles, Fredholm, Integro-differential equation, Simpson 3/8, Trapezoidal, Vieta-Pell-Lucas, Newton Cote Quadrature method

### Abstract

In this study, we present a 6-step linear multistep mehod in union with some newton-cote quadrature family for solving third order Linear Fredholm Integro-Differential Equation(LIDE). The schemes were derived using Vieta-Pell-Lucas Polynomial as the approximating function. The linear multi-step component is for the non integral part while the quadrature family is for the Integral part. The quaudrature methods Boole , Simpson 3/8 and Trapezoidal rule were separately combined with the linear multistep method. The qualititaive analysis of the scheme revealed that the method is consistence, stable and convergent. In order to further attest to the behavioural attribute of the methods, numerical experiments were carried out on some selected Initial Value Problems.The results from the tested problems and their absolute errors of  deviation revealed that the new method is very suitbale for solution to the tested problems.  The scheme when combined with Boole and Simpson 3/8 merhod, performed better with Tracendental function than when combined with Trapezoidal rule and vice –versa. The results further showed that the proposed method perform creditably well with lesser computional steps when compared with some existing methods when applied to the selected examples.

### References

Al-Timeme and Atifa (2003). Approximated methods for first order Volterra Integro-differential equations, M.Sc, Thesis, University of Technology.

Behrouz, R (2010) “Numerical Solution of the Linear Volterra Integro- Diiferential Equation by Homotopy Perturbation Method and Finite Difference Method”.World Applied Science Journal 9 (Special Issue of Applied Maths) :07-12

Bruner, H (1984) “Implicit Runge –Kutta Methods of Optimal Order for Volterra Integro-Differenetial Equations”,Mathematics of Computation,42 :(165),95-109 DOI: https://doi.org/10.1090/S0025-5718-1984-0725986-6

Darania, P. and Ali Ebadian (2007). A method for the numerical solution of the integro-differential equations,Applied Mathematics and Computation 188,657-668. DOI: https://doi.org/10.1016/j.amc.2006.10.046

Fatunla, S.O. (1991). A Block method for second order IVPs, International Journal of Computer Mathematics, Vol. 41, pp. 55-63. Available at http:dx.doi.org/10.1080/00207169108804026 DOI: https://doi.org/10.1080/00207169108804026

Feldstein, A and Sopka J.R. (1974)”Numerical Methods for Non-lnear Volterra Integro-Differential Equations”.SIAM Journal Numerical Analysis,11:826-846. DOI: https://doi.org/10.1137/0711067

Gegele, O.A,Evans, O.P.and Akoh, D. Numerical Solution of Higher Order Linear Fredholmn Integro-Differential Equations.Americal Journal of Engineering Research (AJER),e-ISSN:2320-0847,p-ISSN:2320-0936,volume-03,Issue-08,pp243-247.

Kamoh, N. M,Aboiyah,T. and Onah, E.S. (2017). On one investigating some quadrature rules for the solution of second order Volterra Integro-differential equations. IOSR Journal of Mathematics (IOSR-JM), Vol.13, e-ISSN:2278-5728.p-ISSN:2319-765X., Issue 5 Ver.iii.pp.45-50.www.iosrjournals.org, DOI:10.9790/5728-1305034550.

Kamoh, N.M., Gyemang, D.G. and Soomiyol, M.C. (2019). Comparing the efficiency of Simpson's 1/3 and Simpson's 3/8 rules for the numerical solution of first order Volterra Integro-differential equations, World Academy of Science. Engineering and Technology International Journal of Mathematical and Computational Sciences, Vol.,13. No5, ISNI:0000000091950263. pp.136-139.

Lambert, J.D. (1973). Computational Methods in Ordinary Differential Equations: John Wiley & Sons Incorporation New York.

Obayomi, A.A. (2012). A set of non-standard finite difference schemes for the solution of an equation of the type y^'=y(1-y^n ), International Journal of Pure and Applied Sciences and Technology, 12(20), pp. 34-42.

Obayomi, .A.A.,Ogunrinde R.B,(2015) Nonstandard discrete models for some initial value problems arising from nonlinear oscillator equations.Algorithmsresearch,4(1):8-13 DOI:10.5923/j.algorithms.20150401.02

Ogunrinde, R.B.(2010) A new numerical scheme for the solution of initial value problem (IVP) in ordinary differential equations.Ph.D Thesis,Ekiti State University,Ado-Ekiti.

Ogunrinde, R.B., Olayemi,K.S, Isah I.O. and Salawu A.S. (2020). A numerical solver for first order initial value problems of ordinary differential equation via the combination of Chebyshev polynomial and exponential function. Journal of Physical Sciences, ISSN2520-084X (online), Vol. 2, Issue No.1, pp17-32.

Salawu, A.S,Isah,I.O.,Olayemi, K.S.,Paul, R.V,. A comparative Study of Orthogonal polynomials for Numerical Solution of Ordinary Differential Equations. FUDMA Journal of Sciences(FJS). ISSN online:2616-1370,ISSN Print:2645-2944.Vol.,6 No.1,March,2022,pp 282-290.DOI:https://doi.org/1033003/fjs-2022-0601-898 DOI: https://doi.org/10.33003/fjs-2022-0601-898

Wazwaz, A.M. (1997). A First Course in Integral Equations. Singapore: World Scientific Publishing Company DOI: https://doi.org/10.1142/3444

Wazwaz, A.M. (2011). Linear and Non Linear Integral Equations Method and Applications. Higher Education Press, Beijing and Springer-Verlag Berlin Heieberg.

Yalcinbas and Sezer (2000) “The ApproximatenSolution of High Order Linear Volterra- Fredholm integro-Differential Equations in terms of Taylor Polynomials”. Applied Mathematics and Computation,112:291-308 DOI: https://doi.org/10.1016/S0096-3003(99)00059-4

Published
2023-06-03
How to Cite
Ogunrinde R. B., Obayomi A. A., & Olayemi K. S. (2023). NUMERICAL SOLUTIONS OF THIRD ORDER FREDHOLM INTEGRO DIFFERENTIAL EQUATION VIA LINEAR MULTISTEP-QUADRATURE FORMULAE. FUDMA JOURNAL OF SCIENCES, 7(3), 33 - 44. https://doi.org/10.33003/fjs-2023-0703-1785
Section
Research Articles