Revolutionizing Numerical Approximations: A Novel Higher-Order Implicit Method vs. Runge-Kutta for Initial Value Problems
DOI:
https://doi.org/10.18196/jrc.v6i2.24511Keywords:
Obreschkoff Method, Rung-Kutta Method, ODE, Darboux’s Formula, ApproximationsAbstract
This work is dedicated to advancing the approximation of initial value problems through the introduction of an innovative and superior method inspired by Taylor’s approach. Specifically, we present an enhanced variant achieved by accelerating the expansion of the Obreschkoff formula. This results in a higher-order implicit corrected method that outperforms Rung– Kutta’s (RK) method in terms of accuracy. We derive an error bound for the Obreschkoff higher-order method, showcasing its stability, convergence, and greater efficiency than the conventional RK method. To substantiate our claims, numerical experiments are provided, highlighting the exceptional efficacy of our proposed method over the traditional RK method.
References
N. R. Anakira, A. K. Alomari, A. F. Jameel, and I. Hashim, “Multistage optimal homotopy asymptotic method for solving initial-value problems,” Journal of Nonlinear Science and Applications, vol. 9, no. 4, pp. 1826– 1843, 2016, doi: 10.22436/jnsa.009.04.37.
A. Jameel, N. R. Anakira, A. K. Alomari, I. Hashim, and M. A. Shakhatreh, “Numerical solution of n’th order fuzzy initial value problems by six stages,” Journal of Nonlinear Science and Applications, vol. 9, no. 2, pp. 627–640, 2016, doi: 10.22436/jnsa.009.02.26.
N. R. Anakira, A. K. Alomari, and I. Hashim, “Numerical Scheme for Solving Singular Two-Point Boundary Value Problems,” Journal of Applied Mathematics, vol. 2013, no. 1, 2013, doi: 10.1155/2013/468909.
N. R. Anakira, A. Almalki, D. Katatbeh, G. B. Hani, A. F. Jameel, K. S. Al Kalbani, and M. Abu-Dawas, “An algorithm for solving linear and non-linear Volterra integro-differential equations,” International Journal of Advances in Soft Computing and Its Applications, vol. 15, no. 3, pp. 77–83, 2023, doi: 10.15849/IJASCA.231130.05.
G. Farraj, B. Maayah, R. Khalil, and W. Beghami, “An algorithm for solving fractional differential equations using conformable optimized decomposition method,” International Journal of Advances in Soft Computing and Its Applications, vol. 15, no. 1, pp. 187–196, 2023, doi: 10.15849/IJASCA.230320.13.
M. Berir, “Analysis of the effect of white noise on the Halvorsen system of variable-order fractional derivatives using a novel numerical method,” International Journal of Advances in Soft Computing and Its Applications, vol. 16, no. 3, pp. 294–306, 2024, doi: 10.15849/IJASCA.241130.16.
I.M. Batiha, M.W. Alomari, N. Anakira, S. Meqdad, I.H. Jebril, and S. Momani, “Numerical advancements: A duel between EulerMaclaurin and Runge-Kutta for initial value problem,” International Journal of Neutrosophic Science, vol. 25, no. 3, pp. 76–91, 2025, doi: 10.54216/IJNS.250308.
N. Allouch, I. Batiha, I.H. Jebril, A. Al-Khateeb, and S. Hamani, “A new fractional approach for the higher-order q-Taylor method,” Image Analysis and Stereology, vol. 43, no. 3, pp.249–257, 2024, doi: 10.5566/ias.3286.
A. Zraiqat, I.M. Batiha, and S. Alshorm, “Numerical comparisons between some recent modifications of fractional Euler methods,” WSEAS Transactions on Mathematics, vol. 23, no. 1, pp. 529–535, 2024, doi: 10.37394/23206.2024.23.55.
I. M. Batiha, I.H. Jebril, N. Anakira, A.A. Al-Nana, R. Batyha, and S. Momani, “Two-dimensional fractional wave equation via a new numerical approach,” International Journal of Innovative Computing, Information & Control, vol. 20, no. 4, pp. 1045–1059, 2024, doi: 10.24507/ijicic.20.04.1045.
I. M. Batiha, R. Saadeh, I. H. Jebril, A. Qazza, A. A. Al-Nana, and S. Momani, “Composite fractional trapezoidal rule with Romberg integration,” Computer Modeling in Engineering & Sciences, vol. 140, no. 3, pp. 2729–2745, 2024, doi: 10.32604/cmes.2024.051588.
I. M. Batiha, S. Alshorm, and M. Almuzini, “Solving fractional-order monkeypox model by new numerical methods,” in Mathematical Analysis and Numerical Methods, pp. 551–561, 2024, doi: 10.1007/978-981-97- 4876-1 38.
I. Jebril, S. Alshorm, and I.M. Batiha, “Numerical solution for fractionalorder glioblastoma multiforme model,” in The International Arab Conference on Mathematics and Computations, pp. 599–607, 2024, doi: 10.1007/978-981-97-4876-1 42.
M. W. Alomari, I. M. Batiha, and S. Momani, “New higher-order implicit method for approximating solutions of the initial value problems,” Journal of Applied Mathematics and Computing, vol. 70, pp. 3369–3393, 2024, doi: 10.1007/s12190-024-02087-3.
I. M. Batiha, I. H. Jebril, A. Abdelnebi, Z. Dahmani, S. Alkhazaleh, and N. Anakira, “A new fractional representation of the higher order Taylor scheme,” Computational and Mathematical Methods, vol. 2024, no. 1, 2024, doi: 10.1155/2024/2849717.
I. M. Batiha, I. H. Jebril, S. Alshorm, M. Aljazzazi, and S. Alkhazaleh, “Numerical approach for solving incommensurate higher-order fractional differential equations,” Nonlinear Dynamics and Systems Theory, vol. 24, no. 2, pp. 123–134, 2024.
A. A. Al-Nana, I. M. Batiha, and S. Momani, “A numerical approach for dealing with fractional boundary value problems,” Mathematics, vol. 11, no. 19, 2023, doi: 10.3390/math11194082.
I. M. Batiha, A. A. Abubaker, I. H. Jebril, S. B. Al-Shaikh, and K. Matarneh, “New algorithms for dealing with fractional initial value problems,” Axioms, vol. 12, no. 5, 2023, doi: 10.3390/axioms12050488.
I. M. Batiha, S. Alshorm, A. Al-Husban, R. Saadeh, G. Gharib, and S. Momani, “The n-point composite fractional formula for approximating Riemann–Liouville integrator,” Symmetry, vol. 15, no. 4, 2023, doi: 10.3390/sym15040938.
I. M. Batiha, A. A. Abubaker, I. H. Jebril, S. B. Al-Shaikh, and K. Matarneh, “A numerical approach of handling fractional stochastic differential equations,” Axioms, vol. 12, no. 4, 2023, doi: 10.3390/axioms12040388.
I. M. Batiha, S. Momani, S. Alshorm and A. Ouannas, “Numerical Solutions of Stochastic Differential Equation Using Modified ThreePoint Fractional Formula,” 2023 International Conference on Fractional Differentiation and Its Applications (ICFDA), pp. 1-5, 2023, doi: 10.1109/ICFDA58234.2023.10153192.
I. M. Batiha, A. Bataihah, A. A. Al-Nana, S. Alshorm, I. H. Jebril, and A. Zraiqat, “A numerical scheme for dealing with fractional initial value problem,” International Journal of Innovative Computing, Information & Control, vol. 19, no. 3, pp. 763–774, 2023, doi: 10.24507/ijicic.19.03.763.
I. M. Batiha, S. Alshorm, I. Jebril, A. Zraiqat, Z. Momani, and S. Momani, “Modified 5-point fractional formula with Richardson extrapolation,” AIMS Mathematics, vol. 8, no. 4, pp. 9520–9534, 2023, doi: 10.3934/math.2023480.
I. M. Batiha, S. Alshorm, A. Ouannas, S. Momani, O. Y. Ababneh, and M. Albdareen, “Modified three-point fractional formulas with Richardson extrapolation,” Mathematics, vol. 10, no. 19, 2022, doi: 10.3390/math10193489.
R. B. Albadarneh, I. M. Batiha, A. Adwai, N. Tahat, and A. K. Alomari, “Numerical approach of Riemann-Liouville fractional derivative operator,” International Journal of Electrical and Computer Engineering, vol. 11, no. 6, pp. 5367–5378, 2021, doi: 10.11591/ijece.v11i6.pp5367-5378.
R. B. Albadarneh, I. M. Batiha, A. K. Alomari, and N. Tahat, “Numerical approach for approximating the Caputo fractional-order derivative operator,” AIMS Mathematics, vol. 6, no. 11, pp. 12743–12756, 2021, doi: 10.3934/math.2021735.
R. B. Albadarneh, I. M. Batiha, and M. Zurigat, “Numerical solutions for linear fractional differential equations of order 1 < α < 2 using finite difference method (FFDM),” International Journal of Mathematics and Computer Science, vol. 16, no. 1, pp. 103–111, 2016.
R.B. Albadarneh, M. Zerqat, and I.M. Batiha, “Numerical solutions for linear and non-linear fractional differential equations,” International Journal of Pure and Applied Mathematics, vol. 106, no. 3, pp. 859–871, 2016, doi: 10.12732/ijpam.v106i3.12.
I. M. Batiha, O. Ogilat, I. Bendib, A. Ouannas, I. H. Jebril, and N. Anakira, “Finite-time dynamics of the fractional-order epidemic model: Stability, synchronization, and simulations,” Chaos, Solitons & Fractals: X, vol. 13, 2024, doi: 10.1016/j.csfx.2024.100118.
I. M. Batiha, I. Bendib, A. Ouannas, I. H. Jebril, S. Alkhazaleh, and S. Momani, “On new results of stability and synchronization in finite-time for FitzHugh-Nagumo model using Gronwall inequality and Lyapunov ¨ function,” Journal of Robotics and Control (JRC), vol. 5, no. 6, pp. 1897– 1909, 2024, doi: 10.18196/jrc.v5i6.23211.
O. A. Almatroud, A. Hioual, A. Ouannas, and I. M. Batiha, “Asymptotic stability results of generalized discrete time reaction diffusion system applied to Lengyel-Epstein and Dagn Harrison models,” Computers & Mathematics with Applications, vol. 170, pp. 25–32, 2024, doi: 10.1016/j.camwa.2024.06.028.
S. Momani and I. M. Batiha, “Tuning of the fractional-order PID controller for some real-life industrial processes using particle swarm optimization,” Progress in Fractional Differentiation and Applications, vol. 8, no. 3, pp. 1–15, 2020, doi: 10.18576/pfda/PFDA-60-20.
T. Hamadneh, A. Zraiqat, H. Al-Zoubi, and M. Elbes, “Sufficient conditions and bounding properties for control functions using Bernstein expansion,” Applied Mathematics and Information Sciences, vol. 14, no. 6, pp. 1–9, 2020, doi: 10.18576/amis/paper.
A. Abdulsalam, N. Senu, Z. A. Majid, and N. M. A. N. Long, “Adaptive multi-step Runge–Kutta–Nystrom methods for general second-order ¨ ordinary differential equations,” Journal of Computational and Applied Mathematics, vol. 421, 2023, doi: 10.1016/j.cam.2022.114874.
A. Borri, F. Carravetta, and P. Palumbo, “Quadratized Taylor series methods for ODE numerical integration,” Applied Mathematics and Computation, vol. 458, 2023, doi: 10.1016/j.amc.2023.128237.
P. Amodio, F. Iavernaro, F. Mazzia, M.S. Mukhametzhanov, and Y. D. Sergeyev, “A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic,” Mathematics and Computers in Simulation, vol. 141, pp. 24–39, 2017, doi: 10.1016/j.matcom.2016.03.007.
M. A. Arefin, B. Gain, and R. Karim, “Accuracy analysis on solution of initial value problems of ordinary differential equations for some numerical methods with different step sizes,” International Annals of Science, vol. 10, no. 1, pp. 118–133, 2021, doi: 10.21467/ias.10.1.118-133.
A. Baeza, S. Boscarino, P. Mulet, G. Russo, and D. Zor´ıo, “Approximate Taylor methods for ODEs,” Computers & Fluids, vol. 159, pp. 156–166, 2017, doi: 10.1016/j.compfluid.2017.10.001.
A. Baeza, S. Boscarino, P. Mulet, G. Russo, and D. Zor´ıo, “Reprint of: Approximate Taylor methods for ODEs,” Computers & Fluids, vol. 169, pp. 87–97, 2018, doi: 10.1016/j.compfluid.2018.03.058.
J. C. Butcher, “A history of Runge-Kutta methods,” Applied Numerical Mathematics, vol. 20, no. 3, pp. 247–260, 1996, doi: 10.1016/0168- 9274(95)00108-5.
H. Carrillo, E. Macca, C. Pares, G. Russo, and D. Zor ´ ´ıo, “An orderadaptive compact approximation Taylor method for systems of conservation laws,” Journal of Computational Physics, vol. 438, 2021, doi: 10.1016/j.jcp.2021.110358.
G. Corliss, “Solving Ordinary Differential Equations Using Taylor Series,” ACM Transactions on Mathematical Software, vol. 8, no. 2, pp. 114–144, 1982, doi: 10.1145/355993.355995.
G. D. Hu and Z. Wang, “A modified Runge–Kutta method for increasing stability properties,” Journal of Computational and Applied Mathematics, vol. 441, 2024, doi: 10.1016/j.cam.2023.115698.
R. J. Spiteri and S. Wei, “Fractional-step Runge–Kutta methods: Representation and linear stability analysis,” Journal of Computational Physics, vol. 476, 2023, doi: 10.1016/j.jcp.2022.111900.
A. Dababneh, A. Zraiqat, A. Farah, H. Al-Zoubi, and M. Abu Hammad, “Numerical methods for finding periodic solutions of ordinary differential equations with strong nonlinearity,” Journal of Mathematics and Computer Science, vol. 11, pp. 6910–6922, 2021.
G. Gadisa and H. Garoma, “Comparison of higher order Taylor’s method and Runge-Kutta methods for solving first order ordinary differential equations,” Journal of Computer and Mathematical Sciences, vol. 8, no. 1, pp. 12–23, 2017.
A. Jorba and M. Zou, “A software package for the numerical in- ´ tegration of ODEs by means of high-order Taylor methods,” Experimental Mathematics, vol. 14, no. 1, pp. 99–117, 2005, doi: 10.1080/10586458.2005.10128904.
C. A. Kennedy and M. H. Carpenter, “Higher-order additive RungeKutta schemes for ordinary differential equations,” Applied Numerical Mathematics, vol. 136, pp. 183–205, Jan. 2017, doi: 10.1016/j.apnum.2018.10.007.
R. Kruse and Y. Wu, “Error analysis of randomized Runge–Kutta methods for differential equations with time-irregular coefficients,” Computational Methods in Applied Mathematics, vol. 17, no. 3, pp. 479–498, 2017, doi: 10.1515/cmam-2016-0048.
P. Kumar, V. S. Erturk, M. M. Arcila, and C. Harley, “Generalized forms of fractional Euler and Runge–Kutta methods using non-uniform grid,” Journal of Nonlinear Sciences and Numerical Simulation, vol. 24, no. 6, pp. 2089–2111, 2023, doi: 10.1515/ijnsns-2021-0278.
A. Moisa, “A family of two-step second-order Runge–Kutta–Chebyshev methods,” Journal of Computational and Applied Mathematics, vol. 446, p. 115868, Jan. 2024.
A. Moisa and B. Faleichik, “Second-order stabilized two-step Runge–Kutta methods,” Journal of Computational and Applied Mathematics, vol. 437, 2023, doi: 10.1016/j.cam.2024.115868.
J. Ndam, “Comparison of the solution of the Van der Pol equation using the modified Adomian decomposition method and truncated Taylor series method,” Journal of the Nigerian Society of Physical Sciences, vol. 2, no. 2, pp. 106–114, 2020, doi: 10.46481/jnsps.2020.44.
K. L. Narayanan, J. Kavitha, R. U. Rani, M. E. Lyons, and L. Rajendran, “Mathematical modelling of Amperometric Glucose biosensor based on immobilized enzymes: New approach of Taylor’s series method,” International Journal of Electrochemical Science, vol. 17, no. 10, 2022, doi: 10.20964/2022.10.47.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, London: Cambridge University Press, 1940.
G. Darboux, “Sur les developpements en s ´ erie des fonctions d’une seule ´ variable,” Journal de Mathematiques Pures et Appliqu ´ ees ´ , vol. 2, pp. 291– 312, 1876.
F. Qi, Q.-M. Luo, and B.-N. Guo, “Darboux’s formula with integral remainder of functions with two independent variables,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 691–703, 2008, doi: 10.1016/j.amc.2007.10.028.
P. Appell, “Sur une classe de polynomes,” ˆ Annales Scientifiques de l’Ecole ´ Normale Superieure ´ , vol. 9, pp. 119–144, 1880.
M. Matic, J. Pe ´ cari ˇ c, and N. Ujevi ´ c, “On new estimation of the remainder ´ in generalized Taylor’s formula,” Mathematical Inequalities and Applications, vol. 2, no. 3, pp. 343–361, 1999.
P. W. Hummel and C. L. Seebeck, “A generalization of Taylor’s expansion,” The American Mathematical Monthly, vol. 56, no. 4, pp. 243–247, 1949, doi: 10.1080/00029890.1949.11999368.
N. Obreschkoff, “Neue Quadraturtormeln,” Abhandlungen der Preußischen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse, vol. 4, pp. 1–20, 1940.
R. P. Boas Jr., “Generalized Taylor series, quadrature formulas, and a formula by Kronecker,” SIAM Review, vol. 12, no. 1, pp. 116–119, 1970, doi: 10.1137/1012008.
R. L. Burden and J.D. Faires, Numerical Analysis, Boston, USA: Brooks/Cole, Cengage Learning, 2011.
H. Robbins, “A remark on Stirling’s formula,” The American Mathematical Monthly, vol. 62, no. 1, pp. 26–29, 1955, doi: 10.2307/2308012.
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