# A THREE-STEP INTERPOLATION TECHNIQUE WITH PERTURBATION TERM FOR DIRECT SOLUTION OF THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS

### Abstract

In this paper, we developed a new three-step method for numerical solution of third order ordinary differential equations. Interpolation and collocation methods were used by choosing interpolation points at steps points using power series, while collocation points at step points, using a combination of powers series and perturbation terms gotten from the Legendre polynomials, giving rise to a polynomial of degree and equations. All the analysis on the method derived shows that it is zero-stable, convergent and the region of stability is absolutely stable. Numerical examples were provided to test the performance of the method. Results obtained when compared with existing methods in the literature, shows that the method is accurate and efficient

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### References

2) Awoyemi D.O. (1999). A class of continuous linear method for general second order initial value problems in ordinary differential equations. International Journal of Comput. Maths vol. 72 pp. 29-37.

3) Awoyemi D.O. (2001). A new six order algorithm for general second order ordinary differential equation. International Journal of compt. Maths. Vol. 77 pp. 177-194.

4) Fatunla S.O. (1998). Numerical methods for initial value problems in ordinary differential equations. Academic press Inc. Harcount Brace Jovanovich publisher, New York.

5) Lambert J.D. (1973). Computational methods in ordinary differential equation. John Wiley & sons int.

6) Gout R.A., Hoskins R.F., Milier and Pratt M.J. (1973). Applicable Mathematics for engineers and scientists. Macmillan press Ltd. London.

7) Bruguano L, Trigiante D. (1998). Solving differential problems by multi-step initial and boundary value methods. Gordon and Breach sciences publishers, Amsterdam. Pp 280 – 299.

8) Anake T.A, Awoyemi, D.O. and Adesanya A.O. (2012). A one-step method for the solution of general second order ordinary differential equation. International Journal of science and technology vol. 2(4), pp 159-163.

9) Omar Z.B; Suleiman M.B. (2003). Parallel R-point implicit block method for solving higher order ordinary differential equation directly. Journal of ICT vol. 3(1) pp 53-66.

10) Omar Z.B, Suleiman M.B. (2005). Solving higher order ODEs directly using parallel 2-point explicitly block method Matematika.

11) Ogunware B.G; Omole E.O and Olanegan, O.O. (2015). Hybrid and non-hybrid implicit schemes schemes for solving third orders ODEs using block method as predictors. Mathematical theory and modelling vol. 5(3) pp. 10-25.

12) Abhulimen C.E and Aigbiremhon, A. (2018). Three-step block method for solving second order differential equation. International Journal of Mathematical Analysis and optimization: Theory and applications vol. 2018, pp 364-381.

13) Aigbiremhon, A.A and Ukpebor L.A. (2019). Four-steps collocation block method for solving second order differential equation. Nigerian Journal of mathematics and application vol. A, 28 pp. 18-37.

14) Badmus A.M; Yahaya Y.A. (2009). An accurate uniform order 6 block method for direct solution of general second order ordinary differential equation. Pacific Journal of science and technology vol. 10(2) pp. 248-254.

15) Olabode B.T. (2013). Block multistep method for the direct solution of third order of ordinary differential equations. FUTA Journal of Research in sciences. Vol. 2 pp. 194-200.

16) Adoghe L.O; Gbenga O.B and Omole E.O. (2016). A family of symmetric implicit higher order methods for the solution of third order initial values problems in ordinary differential equations. Theoretical mathematics & applications vol. 6 no 3 pp. 67-84.

17) Olabode B.T. (2007). Some linear multistep methods for special and general third order initial value problems. Ph.D thesis (unpublished). Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria.

18) Mohammed U. and Adeniyi R.B. (2014). A three step implicit hybrid linear multistep method for the solution of Third Order Ordinary Differential Equation General Mathematics Notes vol. 25, pp. 62-94.

19) Adesanya A.O. (2013). A new Hybrid Block Method for the solution of General Third Order Initial value problems in Ordinary Differential Equations. International Journal of Pure and Applied Mathematics vol. 86(2). Pp. 365-375.

20) Olabode B.T. (2009). An accurate scheme by block method for third order ordinary differential equations. The Pacific Journal of Sciences and Technology. Vol. 10(1) pp. 136-142.

21) Abualnaja K.M. (2015). A block procedure with linear multi-step methods using Legendre polynomials for solving ODEs. Journal of Applied Mathematics, vol. 16 pp. 717-732..

22) Jator S.N. (2007). A Sixth Order Linear Multistep method for the direct solution of ordinary differential equations. International Journal of pure and Applied Mathematics vol. 40(1), pp. 457-472.

23) Lambert J.D. (1991). Numerical method in Ordinary differential systems of Initial value problems. John Willey and Sons, New York.

24) Henrici P. (1962). Discrete variable method in ordinary Differential Equation. John Wiley and sons. New York.

25) Ogunware B. G., and Omole E. O. (2020). A Class of Irrational Linear Multistep Block

Method for the Direct Numerical Solution of Third Order Ordinary Differential

Equations. Turkish Journal of Analysis and Number Theory, 8(2): pp 21-27. doi:

10.12691/tjant-8-2-1.

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