Exponentially fitted Diagonally Implicit three-stage fifth-order RK Method for Solving ODEs.
DOI:
https://doi.org/10.29304/jqcm.2022.14.3.1056Keywords:
Numerical Methods, Exponentially fitted, Ordinary Differential Equations, Diagonal Implicit Type Runge Kutta Methods, Initial Value ProblemsAbstract
The EDITRK5 method. which this paper derives. is an exponentially fitted diagonally implicit RK method for solving ODEs with the equation With the help of the set functions and for exponentially fitting problems. this strategy is designed to integrate precise initial value problems (IVPs). The primary frequency of the issue. is used to increase the method's accuracy. The new approach For the purpose of solving IVPs using exponential functions as solutions. EDITRK5 is a novel three-stage five-order exponentially-fitted diagonally implicit method. When the same issue is reduced to the first-order framework of equations. which can be solved using traditional RK approaches. different forms of third-order ODEs must be constructed using the new system. and numerical comparisons must be made. The numerical results demonstrate that the new strategy is more effective than methods that have already been published.
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References
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[2] E. Momoniat. Symmetries. first integrals and phase planes of a third-order ordinary differential equation from thin film flow. Math. Comput. Model. 49. no. 1-2. pp. 215–225. 2009.
[3] T.G. Myers. Thin films with high surface tension. SIAM Rev. 40. no. 3. pp. 441–462. (1998).
[4] Paternoster. B. “Runge-Kutta (-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials". Applied Numerical Mathematics. vol. 28. no. 2– 4. pp.401– 412. 1998.
[5] Berghe. G Vanden and De Meyer. H and Van Daele. M and Van Hecke. T. “ Exponentially fitted Runge–Kutta methods". Journal of Compu- tational and Applied Mathematics. vol. 125. no. 1– 2. pp.107–115. 2000.
[6] Simos. TE “ Exponentially fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation and related problems". Com- putational Materials Science. vol. 18. no. 3– 4. pp.315–332. 2000.
[7] 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. vol. 120. pp. 27–50. 2018.
[8] Mechee. M and Senu. N and Ismail. F and Nikouravan. Bijan and 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.
[9] 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.
[10] Mohamed. Tahani Salama and Senu. Norazak and Ibrahim. Zarina Bibi and Nik Long. Nik Mohd Asri. “Exponentially Fitted and Trigonometrically Fitted Two-Derivative Runge-Kutta-Nyström Methods for Solving". Mathematical Problems in Engineering. vol. 2018. 2018.
[11] Demba. MA and 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. vol. 12. no. 1. pp.67–80. 2016.
[12] Ghawadri. Nizam and Senu. Norazak and Adel Fawzi. Firas and 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. vol. 12. no. 1. pp.10. 2018.
[13] Fawzi. Firas A and 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. vol. 35. no. 1. pp.92–101. 2022.
[14] W. Gander and D. Gruntz. “Derivation of numerical methods using computer algebra”. SIAM Review. vol. 41. no. 3. pp. 577–593. 1999.
[15] J. R. Dormand. “ Numerical Methods for Differential Equations”. A Computational Approach. CRC Press. Boca Raton. Fla. USA. 1996.
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Published
2022-10-28
How to Cite
Jaleel, N. W., & Fawzi, F. A. (2022). Exponentially fitted Diagonally Implicit three-stage fifth-order RK Method for Solving ODEs. Journal of Al-Qadisiyah for Computer Science and Mathematics, 14(3), Math Page 111–124. https://doi.org/10.29304/jqcm.2022.14.3.1056
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