An Improved Fifth-Order Runge-Kutta Method with Higher Accuracy and Efficiency for Solving Initial Value Problems
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Keywords:
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Fifth-Order Runge-Kutta Method, Taylor Series, Numerical Solution, Accuracy, Stability, Computational Efficiency
Abstract
Solving initial value problems (IVPs) in ordinary differential equations (ODEs) often requires numerical methods, with the fifth-order Runge-Kutta method being a widely used approach due to its balance between accuracy and computational efficiency. A novel and straight forward formula for the fifth order Runge-Kutta method is proposed, aiming to simplify calculations while maintaining high accuracy and stability. The method is derived using an optimized Taylor series expansion, leading to a more efficient formulation. Numerical experiments are conducted to compare the proposed method with existing fifth-order Runge-Kutta methods. The results showthat the proposed formula out performs existing methods in terms of accuracy, stability, and computational efficiency. This new formula provides a practical alternative for solving IVPs in ODEs with improved performance.
References
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Audu, K.J., A.R. Taiwo, and A.A. Soliu (2023). Assessment of Numerical Performance of Some Runge-Kutta Methods and New Iteration Method on First Order Differential Problems. Dutse Journal of Pure and Applied Sciences, 9(4a); 58–70.
Bosede, O.R., F.S. Emmanuel, and O.J. Temitayo (2012). On Some Numerical Methods for Solving Initial Value Problems in Ordinary Differential Equations. IOSR Journal of Mathematics (IOSRJM), 1(3); 25–31.
Chandru, M., R. Ponalagusamy, and P.J.A. Alphonse (2017). A New Fifth-Order Weighted Runge-Kutta Algorithm Based on Heronian Mean for Initial Value Problems in Ordinary Differential Equations. Journal of Applied Mathematics & Informatics, 35(1–2); 191–204.
Chapra, S.C. and R.P. Canale (2015). Numerical Methods for Engineers. McGraw-Hill Education.
Demba, M.A., N. Senu, and F. Ismail (2016). A 5(4) Embedded Pair of Explicit Trigonometrically-Fitted Runge-Kutta-Nyström Methods for the Numerical Solution of Oscillatory Initial Value Problems. Mathematical and Computational Applications, 21(4); 1–14.
Dennis, G.Z. (2012). A First Course in Differential Equations with Modeling Applications. Cengage Learning, Loyola Marymount University, 9th edition.
Donas, A., I. Famelis, P.C. Chu, and G. Galanis (2021). Optimization of the Navy’s Three-Dimensional Mine Impact Burial Prediction Simulation Model, Impact35, Using High Order Numerical Methods. Journal of Defense Modeling and Simulation, 20(2); 197–212.
Dukkipati, R.V. (2010). Numerical Methods. New Age International Limited Publisher.
Fardinah (2017). Solusi Persamaan Diferensial Biasa dengan Metode Runge-Kutta Orde Lima. Jurnal Matematika Dan Statistika Serta Aplikasinya, 5(1); 30–36 (in Indonesian).
Ghazal, Z.K. and K.A. Hussain (2021). Solving Oscillating Problems Using Modifying Runge-Kutta Methods. Ibn AL-Haitham Journal For Pure and Applied Sciences, 34(4); 58–67.
Goeken, D. and O. Johnson (1999). Fifth-Order Runge-Kutta with Higher Order Derivative Approximations. In Electronic Journal of Differential Equations, volume 02. University of Central Oklahoma.
Gumus, O.A., Q. Cui, G.M. Selvam, and A. Vianny (2022). Global Stability and Bifurcation Analysis of a Discrete Time SIR Epidemic Model. Miskolc Mathematical Notes, 23(1); 193–210.
Hossain, M.B., M.J. Hosain, M.M. Miah, and M.S. Alam (2017). A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP). Applied and Computational Mathematics, 6(6); 243–253.
Hussain, K.A. and W.J. Hasan (2023). Improved Runge-Kutta Method for Oscillatory Problem Solution Using Trigonometric Fitting Approach. Ibn AL-Haitham Journal For Pure and Applied Sciences, 36(1); 345–354.
Hussain, K.A., F. Ismail, N. Senu, F. Rabiei, and R. Ibrahim (2015). Integration for Special Third-Order Ordinary Differential Equations Using Improved Runge-Kutta Direct Method. Malaysian Journal of Science, 34(2); 172–179.
Kafle, J., B.K. Thakur, and I.B. Bhandari (2021). Application of Numerical Methods for the Analysis of Damped Parallel RLC Circuit. Journal of Institute of Science and Technology, 26(1); 28–34.
Kalogiratou, Z., T. Monovasilis, G. Psihoyios, and T.E. Simos (2014). Runge-Kutta Type Methods with Special Properties for the Numerical Integration of Ordinary Differential Equations. Physics Reports, 536(3); 75–146.
Kennedy, C.A. and M.H. Carpenter (2019). Higher-Order Additive Runge–Kutta Schemes for Ordinary Differential Equations. Applied Numerical Mathematics, 136; 183–205.
Kopecz, S. and A. Meister (2018). On Order Conditions for Modified Patankar–Runge–Kutta Schemes. Applied Numerical Mathematics, 123; 159–179.
Mathews, J.H. and K.D. Fink (2004). Numerical Methods Using MATLAB. Prentice-Hall, Inc., 3rd edition.
Mechee, M., N. Senu, F. Ismail, B. Nikouravan, and Z. Siri (2013). 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, 2013; 1–9.
Monovasilis, T., Z. Kalogiratou, and T.E. Simos (2016). Construction of Exponentially Fitted Symplectic Runge–Kutta–Nyström Methods from Partitioned Runge–Kutta Methods. Mediterranean Journal of Mathematics, 13(4); 2271–2285.
Monovasilis, T., Z. Kalogiratou, and T.E. Simos (2018). Trigonometrical Fitting Conditions for Two Derivative Runge-Kutta Methods. Numerical Algorithms, 79(3); 787–800.
Parhusip, H.A., S. Trihandaru, B. Aryo, A. Wicaksono, D. Indrajaya, Y. Sardjono, and O.P. Vyas (2022). Susceptible Vaccine Infected Removed (SVIR) Model for COVID-19 Cases in Indonesia. Science and Technology Indonesia, 7(3); 400–408.
Rabiei, F. and F. Ismail (2012). Fifth-Order Improved Runge-Kutta Method With Reduced Number of Function Evaluations. Australian Journal of Basic and Applied Sciences, 6(3); 97–105.
Ram, M. (2020). Recent Advances in Mathematics for Engineering. CRC Press, 1st edition.
Sukron, M., U. Habibah, and N. Hidayat (2021). Numerical Solution of Saint-Venant Equation Using Runge-Kutta Fourth-Order Method. Journal of Physics: Conference Series, 1872(1); 012036.
Suryaningrat, W., R. Ashgi, and S. Purwani (2020). Order Runge-Kutta with Extended Formulation for Solving Ordinary Differential Equations. International Journal of Global Operations Research, 1(4); 160–167.
Aliyi, K. and H. Muleta (2021). Numerical Method of the Line for Solving One Dimensional Initial-Boundary Singularly Perturbed Burger Equation. Indian Journal of Advanced Mathematics, 1(2); 4–14.
Audu, K.J., A.R. Taiwo, and A.A. Soliu (2023). Assessment of Numerical Performance of Some Runge-Kutta Methods and New Iteration Method on First Order Differential Problems. Dutse Journal of Pure and Applied Sciences, 9(4a); 58–70.
Bosede, O.R., F.S. Emmanuel, and O.J. Temitayo (2012). On Some Numerical Methods for Solving Initial Value Problems in Ordinary Differential Equations. IOSR Journal of Mathematics (IOSRJM), 1(3); 25–31.
Chandru, M., R. Ponalagusamy, and P.J.A. Alphonse (2017). A New Fifth-Order Weighted Runge-Kutta Algorithm Based on Heronian Mean for Initial Value Problems in Ordinary Differential Equations. Journal of Applied Mathematics & Informatics, 35(1–2); 191–204.
Chapra, S.C. and R.P. Canale (2015). Numerical Methods for Engineers. McGraw-Hill Education.
Demba, M.A., N. Senu, and F. Ismail (2016). A 5(4) Embedded Pair of Explicit Trigonometrically-Fitted Runge-Kutta-Nyström Methods for the Numerical Solution of Oscillatory Initial Value Problems. Mathematical and Computational Applications, 21(4); 1–14.
Dennis, G.Z. (2012). A First Course in Differential Equations with Modeling Applications. Cengage Learning, Loyola Marymount University, 9th edition.
Donas, A., I. Famelis, P.C. Chu, and G. Galanis (2021). Optimization of the Navy’s Three-Dimensional Mine Impact Burial Prediction Simulation Model, Impact35, Using High Order Numerical Methods. Journal of Defense Modeling and Simulation, 20(2); 197–212.
Dukkipati, R.V. (2010). Numerical Methods. New Age International Limited Publisher.
Fardinah (2017). Solusi Persamaan Diferensial Biasa dengan Metode Runge-Kutta Orde Lima. Jurnal Matematika Dan Statistika Serta Aplikasinya, 5(1); 30–36 (in Indonesian).
Ghazal, Z.K. and K.A. Hussain (2021). Solving Oscillating Problems Using Modifying Runge-Kutta Methods. Ibn AL-Haitham Journal For Pure and Applied Sciences, 34(4); 58–67.
Goeken, D. and O. Johnson (1999). Fifth-Order Runge-Kutta with Higher Order Derivative Approximations. In Electronic Journal of Differential Equations, volume 02. University of Central Oklahoma.
Gumus, O.A., Q. Cui, G.M. Selvam, and A. Vianny (2022). Global Stability and Bifurcation Analysis of a Discrete Time SIR Epidemic Model. Miskolc Mathematical Notes, 23(1); 193–210.
Hossain, M.B., M.J. Hosain, M.M. Miah, and M.S. Alam (2017). A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP). Applied and Computational Mathematics, 6(6); 243–253.
Hussain, K.A. and W.J. Hasan (2023). Improved Runge-Kutta Method for Oscillatory Problem Solution Using Trigonometric Fitting Approach. Ibn AL-Haitham Journal For Pure and Applied Sciences, 36(1); 345–354.
Hussain, K.A., F. Ismail, N. Senu, F. Rabiei, and R. Ibrahim (2015). Integration for Special Third-Order Ordinary Differential Equations Using Improved Runge-Kutta Direct Method. Malaysian Journal of Science, 34(2); 172–179.
Kafle, J., B.K. Thakur, and I.B. Bhandari (2021). Application of Numerical Methods for the Analysis of Damped Parallel RLC Circuit. Journal of Institute of Science and Technology, 26(1); 28–34.
Kalogiratou, Z., T. Monovasilis, G. Psihoyios, and T.E. Simos (2014). Runge-Kutta Type Methods with Special Properties for the Numerical Integration of Ordinary Differential Equations. Physics Reports, 536(3); 75–146.
Kennedy, C.A. and M.H. Carpenter (2019). Higher-Order Additive Runge–Kutta Schemes for Ordinary Differential Equations. Applied Numerical Mathematics, 136; 183–205.
Kopecz, S. and A. Meister (2018). On Order Conditions for Modified Patankar–Runge–Kutta Schemes. Applied Numerical Mathematics, 123; 159–179.
Mathews, J.H. and K.D. Fink (2004). Numerical Methods Using MATLAB. Prentice-Hall, Inc., 3rd edition.
Mechee, M., N. Senu, F. Ismail, B. Nikouravan, and Z. Siri (2013). 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, 2013; 1–9.
Monovasilis, T., Z. Kalogiratou, and T.E. Simos (2016). Construction of Exponentially Fitted Symplectic Runge–Kutta–Nyström Methods from Partitioned Runge–Kutta Methods. Mediterranean Journal of Mathematics, 13(4); 2271–2285.
Monovasilis, T., Z. Kalogiratou, and T.E. Simos (2018). Trigonometrical Fitting Conditions for Two Derivative Runge-Kutta Methods. Numerical Algorithms, 79(3); 787–800.
Parhusip, H.A., S. Trihandaru, B. Aryo, A. Wicaksono, D. Indrajaya, Y. Sardjono, and O.P. Vyas (2022). Susceptible Vaccine Infected Removed (SVIR) Model for COVID-19 Cases in Indonesia. Science and Technology Indonesia, 7(3); 400–408.
Rabiei, F. and F. Ismail (2012). Fifth-Order Improved Runge-Kutta Method With Reduced Number of Function Evaluations. Australian Journal of Basic and Applied Sciences, 6(3); 97–105.
Ram, M. (2020). Recent Advances in Mathematics for Engineering. CRC Press, 1st edition.
Sukron, M., U. Habibah, and N. Hidayat (2021). Numerical Solution of Saint-Venant Equation Using Runge-Kutta Fourth-Order Method. Journal of Physics: Conference Series, 1872(1); 012036.
Suryaningrat, W., R. Ashgi, and S. Purwani (2020). Order Runge-Kutta with Extended Formulation for Solving Ordinary Differential Equations. International Journal of Global Operations Research, 1(4); 160–167.
Authors
Habibah, U., Medrano, F. F. ., Permana, A. C. ., Ardiana, D. ., & Trisilowati. (2025). An Improved Fifth-Order Runge-Kutta Method with Higher Accuracy and Efficiency for Solving Initial Value Problems. Science and Technology Indonesia, 10(3), 802–816. https://doi.org/10.26554/sti.2025.10.3.802-816
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