A new approach for the bigeometric newton method

Year 2023, , 235 - 240, 25.12.2023

Numan Yalçın , Ercan Celık

https://doi.org/10.51354/mjen.1372839

Abstract

In this study, quadratic convergent new bigeometric Newton's method (nBGNM) was developed. For this, the basic definitions and theorems of bigeometric analysis, which is one of the non-Newtonian analysis, were used. Using the bigeometric Taylor expansion, a convergence analysis of this new method was given. Also, the new bigeometric Newton method (nBGNM) was compared in detail with the geometric (multiplicative) Newton method (GNM) and the classical Newton method (NM).

Keywords

Bigeometric analysis, Bigeometric Newton method, Quadratic convergence, Bigeometric Taylor expansion

References

• [1]. Mcbride, A. (1982). V. Hutson and J. S. Pym Applications of Functional Analysis and Operator Theory (Mathematics in Science and Engineering, Volume 146, Academic Press, London, 1980), xii 390 pp. Proceedings of the Edinburgh Mathematical Society, 25(3), 280-281.
• [2]. Kreyszig, E. (1991). Introductory functional analysis with applications (Vol. 17). John Wiley & Sons.
• [3]. Weerakoon, S., & Fernando, T. (2000). A variant of Newton's method with accelerated third-order convergence. Applied mathematics letters, 13(8), 87-93.
• [4]. Wait, R. (1979). The numerical solution of algebraic equations. John Wiley & Sons.
• [5]. Grossman M, Katz R. Non-Newtonian Calculus. Pigeon Cove, MA, USA: Lee Press, 1972.
• [6]. Bashirov, A. E., Kurpınar, E. M., & Özyapıcı, A. (2008). Multiplicative calculus and its applications. Journal of mathematical analysis and applications, 337(1), 36-48.
• [7]. Aniszewska, D. (2007). Multiplicative runge–kutta methods. Nonlinear Dynamics, 50(1-2), 265-272.
• [8]. Bashirov, A. E., & Bashirova, G. (2011). Dynamics of literary texts and diffusion.
• [9]. Bashirov, A. E., Mısırlı, E., Tandoğdu, Y., & Özyapıcı, A. (2011). On modeling with multiplicative differential equations. Applied Mathematics-A Journal of Chinese Universities, 26, 425-438.
• [10]. Bashirov, A. E., & Mustafa, R. İ. Z. A. (2011). On complex multiplicative differentiation. TWMS Journal of applied and engineering mathematics, 1(1), 75-85
• [11]. Bashirov, A. E., & Riza, M. (2013). On complex multiplicative integration. arXiv preprint arXiv:1307.8293.
• [12]. Córdova-Lepe, F. (2006). The multiplicative derivative as a measure of elasticity in economics. TEMAT-Theaeteto Atheniensi Mathematica, 2(3).
• [13]. Filip, D. A., & Piatecki, C. (2014). A non-Newtonian examination of the theory of exogenous economic growth.
• [14]. Florack, L., & van Assen, H. (2012). Multiplicative calculus in biomedical image analysis. Journal of Mathematical Imaging and Vision, 42, 64-75.
• [15]. Mısırlı, E., & Özyapici, A. (2009). Exponential approximations on multiplicative calculus. In Proc. Jangjeon Math. Soc (Vol. 12, No. 2, pp. 227-236).
• [16]. Stanley, D. (1999). A multiplicative calculus. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 9(4), 310-326.
• [17]. Uzer, A. (2010). Multiplicative type complex calculus as an alternative to the classical calculus. Computers & Mathematics with Applications, 60(10), 2725-2737.
• [18]. Yalcin, N., Celik, E., & Gökdoğan, A. (2016). Multiplicative Laplace transform and its applications. Optik, 127(20), 9984-9995.
• [19]. Yalçın, N., & Çelik, E. (2018). The solution of multiplicative non-homogeneous linear differential equations. J. Appl. Math. Comput, 2(1), 27-36.
• [20]. Yalcin, N., & Celik, E. (2018). Solution of multiplicative homogeneous linear differential equations with constant exponentials. New Trends in Mathematical Sciences, 6(2), 58-67.
• [21]. Yalçın, N., & Çelik, E. Çarpımsal Cauchy-Euler ve Legendre Diferansiyel Denklemi. Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 9(3), 373-382.
• [22]. Yalçın, N. (2021). The solutions of multiplicative Hermite differential equation and multiplicative Hermite polynomials. Rendiconti del Circolo Matematico di Palermo Series 2, 70, 9-21.
• [23]. Yalçın, N., & Dedeturk, M. (2021). Solutions of multiplicative ordinary differential equations via the multiplicative differential transform method. Aims Mathematics, 6(4), 3393-3409.
• [24]. Yalçın, N., & Dedeturk, M., Solutions of Multiplicative Linear Differential Equations via the Multiplicative Power Series Method. Sigma Journal of Engineering and Natural Sciences. (In press)
• [25]. Bal, A., Yalçın, N., & Dedetürk, M. Solutions of Multiplicative İntegral Equations via The Multiplicative Power Series Method. Politeknik Dergisi, 1-1.
• [26]. Yalcin, N. (2022). Multiplicative Chebyshev differential equations and multiplicative Chebyshev polynomials. Thermal Science, 26(Spec. issue 2), 785-799.
• [27]. Bilgehan, B. (2015). Efficient approximation for linear and non‐linear signal representation. IET Signal Processing, 9(3), 260-266.
• [28]. Özyapıcı, A., Riza, M., Bilgehan, B., & Bashirov, A. E. (2014). On multiplicative and Volterra minimization methods. Numerical Algorithms, 67, 623-636.
• [29]. Ozyapici, A., & Bilgehan, B. (2016). Finite product representation via multiplicative calculus and its applications to exponential signal processing. Numerical Algorithms, 71, 475-489.
• [30].Misirli, E., & Gurefe, Y. (2011). Multiplicative adams bashforth–moulton methods. Numerical Algorithms, 57, 425-439.
• [31]. Özyapıcı, A., Sensoy, Z. B., & Karanfiller, T. (2016). Effective root-finding methods for nonlinear equations based on multiplicative calculi. Journal of Mathematics, 2016.
• [32]. Riza, M., Özyapici, A., & Mısırlı, E. (2009). Multiplicative finite difference methods. Quarterly of Applied Mathematics, 67(4), 745-754.
• [33].Waseem, M., Noor, M. A., Shah, F. A., & Noor, K. I. (2018). An efficient technique to solve nonlinear equations using multiplicative calculus. Turkish Journal of Mathematics, 42(2), 679-691.
• [34]. Unal, E., Cumhur, I., & Gokdogan, A. Multiplicative Newton’s Methods with Cubic Convergence. New Trends in Mathematical Sciences, 5(3), 299-307.
• [35]. Grossman, M. (1983). Bigeometric calculus: a system with a scale-free derivative. Archimedes Foundation.
• [36]. Riza, M., & EminaĞA, B. (2014). Bigeometric Calculus and Runge Kutta Method. arXiv preprint arXiv:1402.2877.
• [37]. Riza, M., & Eminagaı, B. (2014). Bigeometric calculus—a modelling tool. Preprint.
• [38]. Boruah, K., & Hazarika, B. (2016). Bigeometric Calculus and its applications. arXiv preprint arXiv:1608.08088.
• [39]. Boruah, K., & Hazarika, B. (2018). Bigeometric integral calculus. TWMS Journal of Applied and Engineering Mathematics, 8(2), 374-385.
• [40]. Boruah, K., Hazarika, B., & Bashirov, A. E. (2021). Solvability of bigeometric differential equations by numerical methods. Boletim da Sociedade Paranaense de Matematica, 39(2), 203-222.
• [41]. Boruah, K., & Hazarika, B. (2021). Some basic properties of bigeometric calculus and its applications in numerical analysis. Afrika Matematika, 32(1-2), 211-227.
• [42]. Eminağa, B. (2015). Bigeometric Taylor Theorem and its Application to the Numerical Solution of Bigeometric Differential Equations.