Exponentially Fitted Diagonally Implicit EDITRK Method for Solving ODEs

Main Article Content

Firas A. Fawzi
Nour W. Jaleel

Abstract

This paper derives the EDITRK4 technique, which is an exponentially fitted diagonally implicit RK method for solving ODEs . This approach is intended to integrate exactly initial value problems (IVPs), their solutions consist of linear combinations of the group functions  and  for exponentially fitting  problems, with  being the problem’s major frequency utilized to improve the precision of the method. The modified  method EDITRK4 is a new three-stage fourth-order exponentially-fitted diagonally implicit approach for solving IVPs with functions that are exponential as solutions. Different forms of -order ODEs must be derived using the modified system, and when the same issue is reduced to a  framework of equations that can be solved using conventional RK approaches, numerical comparisons must be done. The findings show that the novel approach is more efficacious than previously published methods.

Article Details

How to Cite
Exponentially Fitted Diagonally Implicit EDITRK Method for Solving ODEs. (2023). Ibn AL-Haitham Journal For Pure and Applied Sciences, 36(1), 389-399. https://doi.org/10.30526/36.1.2883
Section
Mathematics
Author Biographies

Firas A. Fawzi, Department of Mathematics, Faculty of Computer Science and Mathematics, Tikrit University, Sallah AL-Deen, IRAQ

 

 

Nour W. Jaleel, Department of Mathematics,Faculty of Computer Science and Mathematics ,University of Tikrit

 

 

How to Cite

Exponentially Fitted Diagonally Implicit EDITRK Method for Solving ODEs. (2023). Ibn AL-Haitham Journal For Pure and Applied Sciences, 36(1), 389-399. https://doi.org/10.30526/36.1.2883

References

Myers, G. Thin Films with High Surface Tension, SIAM Rev. 1998, 40,441–462.

. Momoniat, E. Symmetries, First Integrals and Phase Planes of a Third-Order Ordinary Differential Equation from Thin Film Flow, Math. Comput. Model. 2009,49, 1-2, 215–225.

Duffy, B.R. ; Wilson, S.K. A Third-Order Differential Equation Arising in Thin-Film Flows and Relevant to Tanner’s law, Math. Lett.. 1997, 10, 63–68.

Paternoster, B, Runge-Kutta (-Nyström) Methods for ODEs with Periodic Solutions Based on Trigonometric Polynomials, Applied Numerical Mathematics. 1998, 28, 2– 4, 401– 412.

Berghe, G Vanden; De Meyer, H; Van Daele, M and Van Hecke, T, Exponentially Fitted Runge–Kutta Methods, Journal of Compu- tational and Applied Mathematics. 2000,125, 1– 2, 107–115.

Simos, TE . Exponentially Fitted Runge–Kutta Methods for the Numerical Solution of the Schrödinger Equation and Related Problems, Com- putational Materials Science. 2000, 18, 3– 4, 315–332.

Alshareeda, Firas Adel Fawzi, Runge-Kutta Type Methods for Solving Third-Order Ordinary Differential Equations and First-Order Oscillatory Problems, (Ph.D. thesis) , Universiti Putra Malaysia, 2017.

Fawzi, FA and Senu, N and Ismail, F, An Efficient of Direct Integrator of Runge-Kutta Type Method for Solving y''' = f (x, y, y'') with Application to Thin Film Flow Problem, International Journal of Pure and Applied Mathematics, 2018, 120, 27–50.

Mechee, M; Senu, N; Ismail, F ; Nikouravan, Bijan ; Siri, Z, A Three-Stage Fifth-Order Runge-Kutta Method for Directly Solving Special Third-Order Differential Equation with Application to Thin Film Flow Problem Mathematical Problems in Engineering, vol. 2013, 2013.

Mechee, M; Senu, N; Ismail, F; Nikouravan, Bijan ; Siri, Z, Exponentially Fitted and Trigonometrically Fitted Two-Derivative Runge-Kutta-Nyström Methods for Solving, Mathematical Problems in Engineering, vol. 2018, 2018.

Demba, MA; Senu, N and Ismail, F, Trigonometrically-Fitted Explicit Four-Stage Fourth-Order Runge–Kutta–Nyström Method for the Solution of Initial Value Problems with Oscillatory Behavior, Global Journal of Pure and Applied Mathematics. 2016, 12, 1, 67–80.

Fawzi, Firas A ; Jumaa, Mustafa H, The Implementations Special Third-Order Ordinary Differential Equations (ODE) for 5th-order 3rd- stage Diagonally Implicit Type Runge-Kutta Method (DITRKM), Ibn AL-Haitham Journal For Pure and Applied Sciences. 2022, 35, 1, 92–101.

Ghawadri, Nizam; Senu, Norazak; Adel Fawzi, Firas; Ismail, Fudziah and Ibrahim, Zarina Bibi, Diagonally Implicit Runge–Kutta Type Method for Directly Solving Special Fourth-Order Ordinary Differential Equations with Ill-Posed Problem of a Beam on Elastic Foundation, Algorithms. 2018, 12, 1, 10.

Gander, W. ; Gruntz, D. Derivation of Numerical Methods Using Computer Algebra, SIAM Review. 1999, 41, 3, 577–593.

Dormand, J. R. Numerical Methods for Differential Equations, A Computational Approach, CRC Press, Boca Raton, Fla, USA, 1996.

Sommeijer, Ben P, A Note on a Diagonally Implicit Runge-Kutta-Nyström Method, Journal of computational and applied mathematics. 1987, 19, 3, 395–399.

Moo, KW; Senu, N; Ismail, F ; Suleiman, M, A Zero-Dissipative Phase-Fitted Fourth Order Diagonally Implicit Runge-Kutta-Nyström Method for Solving Oscillatory Problems, Mathematical Problems in Engineering,vol. 2014, 2014.