Adams-Bashforth methods on bigeometric calculus

Mehmet Çağrı Yilmazer; Sertac Goktas; Emrah Yilmaz; Mikail Et; Mehmet Çağrı Yilmazer; Sertac Goktas; Emrah Yilmaz; Mikail Et

doi:10.3934/math.2025832

AIMS Mathematics

2025, Volume 10, Issue 8: 18627-18640. doi: 10.3934/math.2025832

Previous Article Next Article

Research article

Adams-Bashforth methods on bigeometric calculus

1.
Department of Mathematics, Faculty of Science, Firat University, Elazığ, Türkiye
2.
Department of Mathematics, Faculty of Science, Mersin University, Mersin, Türkiye

Received: 15 January 2025 Revised: 01 July 2025 Accepted: 28 July 2025 Published: 18 August 2025
MSC : 11A05, 65L05, 65L70

In this study, bigeometric Adams-Bashforth methods were defined as a strong alternative to classical numerical methods. After methods were established theoretically, error estimation formulas for each method were developed. Finally, a classical differential equation was converted to a bigeometric multiplicative differential equation, and it was solved by using bigeometric Adams-Bashforth methods. Then, error estimations were given.
- multiplicative calculus,
- bigeometric Adams-Bashforth methods,
- error estimation
Citation: Mehmet Çağrı Yilmazer, Sertac Goktas, Emrah Yilmaz, Mikail Et. Adams-Bashforth methods on bigeometric calculus[J]. AIMS Mathematics, 2025, 10(8): 18627-18640. doi: 10.3934/math.2025832

Related Papers:

Abstract

In this study, bigeometric Adams-Bashforth methods were defined as a strong alternative to classical numerical methods. After methods were established theoretically, error estimation formulas for each method were developed. Finally, a classical differential equation was converted to a bigeometric multiplicative differential equation, and it was solved by using bigeometric Adams-Bashforth methods. Then, error estimations were given.

References

[1]	V. Volterra, B. Hostinsky, Operations infinitesimales lineares, 1938.
[2]	M. Grossman, R. Katz, Non-Newtonian calculus, 1972.
[3]	M. Grossman, Bigeometric Calculus: A system with a scale-free derivative, 1983.
[4]	M. Grossman, The first nonlinear system of differential and integral calculus, 1979.
[5]	K. Boruah, B. Hazarika, G-calculus, TWMS J. Appl. Eng. Math., 8 (2018), 94–105.
[6]	K. Boruah, B. Hazarika, Bigeometric integral calculus, TWMS J. Appl. Eng. Math., 8 (2018), 374–385.
[7]	B. Eminağa, Bigeometric Taylor theorem and its application to the numerical solution of bigeometric differential equations, 2015.
[8]	S. Kaymak, N. Yalçın, On bigeometric Laplace integral transform, J. Inst. Sci. Technol., 13 (2023), 2042–2056. https://doi.org/10.21597/jist.1283580 doi: 10.21597/jist.1283580
[9]	S. Georgiev, K. Zennir, Multiplicative differential calculus, Chapman and Hall/CRC, 2022. https://doi.org/10.1201/9781003299080
[10]	S. Georgiev, K. Zennir, Multiplicative differential equations: Volume II, Chapman and Hall/CRC, 2023. https://doi.org/10.1201/9781003394549
[11]	S. G. Georgiev, Multiplicative differential geometry, Chapman and Hall/CRC, 2022. https://doi.org/10.1201/9781003299844
[12]	D. Aniszewska, Multiplicative Runge-Kutta methods, Nonlinear Dyn., 50 (2007), 265–272. https://doi.org/10.1007/s11071-006-9156-3 doi: 10.1007/s11071-006-9156-3
[13]	U. Kadak, M. Özlük, Generalized Runge‐Kutta method with respect to the Non‐Newtonian calculus, Abstr. Appl. Anal., 2015 (2015), 594685. https://doi.org/10.1155/2015/594685 doi: 10.1155/2015/594685
[14]	N. Yalçın, E. Celik, A new approach for the bigeometric newton method, MANAS J. Eng., 11 (2023), 235–240. https://doi.org/10.51354/mjen.1372839 doi: 10.51354/mjen.1372839
[15]	M. Riza, H. Aktöre, The Runge-Kutta method in geometric multiplicative calculus, LMS J. Comput. Math., 18 (2015), 539–554. https://doi.org/10.1112/S1461157015000145 doi: 10.1112/S1461157015000145
[16]	M. Riza, A. Özyapici, E. Misirli, Multiplicative finite difference methods, Quart. Appl. Math., 67 (2009), 745–754. https://doi.org/10.1090/s0033-569x-09-01158-2 doi: 10.1090/s0033-569x-09-01158-2
[17]	E. Misirli, Y. Gurefe, Multiplicative Adams Bashforth-Moulton methods, Numer. Algor., 57 (2011), 425–439. https://doi.org/10.1007/s11075-010-9437-2 doi: 10.1007/s11075-010-9437-2
[18]	N. Yalçın, M. Dedeturk, Solutions of multiplicative ordinary differential equations via the multiplicative differential transform method, AIMS Math., 6 (2021), 3393–3409. https://doi.org/10.3934/math.2021203 doi: 10.3934/math.2021203
[19]	G. Singh, S. Bhalla, Two step Newton's method with multiplicative calculus to solve the non-linear equations, J. Comput. Anal. Appl., 31 (2023), 171–179.
[20]	R. Dehghan, Numerical approximation of multiplicative integral based on exponential interpolation, Int. J. Appl. Comput. Math., 8 (2022), 2. https://doi.org/10.1007/s40819-021-01201-4 doi: 10.1007/s40819-021-01201-4
[21]	N. Yalçın, M. Dedetürk, Solutions of multiplicative linear differential equations via the multiplicative power series method, Sigma J. Eng. Nat. Sci., 41 (2023), 837–847. https://doi.org/10.14744/sigma.2023.00091 doi: 10.14744/sigma.2023.00091
[22]	E. Süli, D. F. Mayers, An introduction to numerical analysis, Cambridge university press, 2003.
[23]	J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley & Sons, 2016.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)