Solving Bigeometric Volterra Integral Equations by Using Successive Approximations Method

Year 2021, Volume: 25 Issue: 1, 150 - 162, 01.02.2021

Nihan Güngör

https://doi.org/10.16984/saufenbilder.779325

Cited By: 1

Abstract

In this study, the successive approximations method has been applied to investigate the solution for the linear bigeometric Volterra integral equations of the second kind in the sense of bigeometric calculus. The conditions to be taken into consideration for the bigeometric continuity and the uniqueness of the solution of linear bigeometric Volterra integral equations of the second kind are researched. Finally, some numerical examples are presented to illustrate successive approximations method.

Keywords

bigeometric calculus, bigeometric Volterra integral equations, successive approximations method

References

• Referans1 D. Aniszewska and M. Rybaczuk, “Analysis of the Multiplicative Lorenz System,” Chaos Solitons Fractals, vol. 25, pp. 79–90, 2005.
• Referans2 K. Boruah and B. Hazarika, “ -Calculus,” TWMS J. Pure Appl. Math., vol. 8, no. 1, pp. 94-105, 2018.
• Referans3 K. Boruah and B. Hazarika, “Bigeometric Integral Calculus,” TWMS J. Pure Appl. Math., vol. 8, no. 2, pp. 374-385, 2018.
• Referans4 K. Boruah, B. Hazarika and A. E. Bashirov, “Solvability of Bigeometric Diferrential Equations by Numerical Methods,” Bol. Soc. Parana. Mat., doi: 10.5269/bspm.39444, 2018.
• Referans5 F. Córdova-Lepe, “The Multiplicative Derivative as a Measure of Elasticity in Economics,” TEMAT-Theaeteto Antheniensi Mathematica, vol. 2, no.3, 2015.
• Referans6 A.F. Çakmak and F. Başar, “On Line and Double Integrals in the Non-Newtonian Sense,” AIP Conference Proceedings, 1611, pp. 415-423, 2014.
• Referans7 A.F. Çakmak and F. Başar, “Certain Spaces of Functions over the Field of Non-Newtonian Complex Numbers,” Abstr. Appl. Anal., Article ID 236124, 12 pages, doi:10.1155/2014/236124, 2014.
• Referans8 C. Duyar and O. Oğur, “A Note on Topology of Non-Newtonian Real Numbers,” IOSR Journal Of Mathematics, vol. 13, no. 6, pp. 11-14, 2017.
• Referans9 C. Duyar and B. Sağır, “Non-Newtonian Comment of Lebesgue Measure in Real Numbers,” J. Math, Article ID 6507013, 2017.
• Referans10 M. Erdoğan and C. Duyar, “Non-Newtonian Improper Integrals,” Journal of Science and Arts, vol. 1, no. 42, pp. 49-74, 2018.
• Referans11 N. Güngör, “Some Geometric of The Non-Newtonian Sequence Spaces ,” Math. Slovaca, vol. 70, no. 3, pp. 689-696, 2020.
• Referans12 N. Güngör, “ -Volterra Integral Equations and Relationship with -Differential Equations,” GÜFBED, vol.10, no.3, pp. 814-829, 2020.
• Referans13 M.Grosmann and R. Katz “Non-Newtonian Calculus,” Lee Press, Pigeon Cove Massachussets, 1972.
• Referans14 M. Grosmann, “An Introduction to Non-Newtonian Calculus,” International Journal of Mathematical Education in Science and Technology, vol. 10, no. 4, pp. 525-528, 1979.
• Referans15 M. Grosmann, “Bigeometric Calculus: A system with a Scale Free Derivative,” 1st ed., Archimedes Foundation, Rockport Massachussets, 1983.
• Referans16 U. Kadak and M. Özlük, “Generalized Runge-Kutta Methods with Respect to Non-Newtonian Calculus,” Abstr. Appl. Anal., Article ID 594685, 2014.
• Referans17 M. Krasnov, K. Kiselev and G. Makarenko, “Problems and Exercises in Integral Equation,” Mır Publishers, Moscow, 1971.
• Referans18W. V. Lovitt, “Linear Integral Equations,” Dover Publications Inc., New York, 1950.
• Referans19 M. Rahman, “Integral Equations and Their Applications”(WIT press, Boston, 2007).
• Referans20 R. K. Saeed and K. A. Berdawood, "Solving Two-dimensional Linear Volterra-Fredholm Integral Equations of the Second Kind by Using Succesive Approximation Method and Method of Succesive Substitutions," ZANCO Journal of Pure and Applied Sciences, vol. 28, no.2, pp. 35-46, 2016.
• Referans21 B. Sağır and F. Erdoğan, “On the Function Sequences and Series in the Non-Newtonian Calculus,” Journal of Science and Arts, vol. 4, no. 49, pp. 915-936, 2019.
• Referans22 C. Türkmen and F. Başar, “Some Results on the Sets of Sequences with Geometric Calculus,” Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., vol. 61, no. 2, pp. 17-34, 2012.
• Referans23 V. Volterra and B. Hostinsky, “Opérations Infinitésimales linéares,” Herman, Paris, 1938.
• Referans24 A. M. Wazwaz, “Linear and Nonlinear Integral Equations Methods and Applications”, Springer Verlag Berlin Heidelberg, 2011.