Solving Bigeometric Volterra Integral Equations by Using Successive Approximations Method

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.

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[1] D. Aniszewska and M. Rybaczuk, “Analysis of the Multiplicative Lorenz System,” Chaos Solitons Fractals, vol. 25, pp. 79–90, 2005.

[2] K. Boruah and B. Hazarika, “G -Calculus,” TWMS J. Pure Appl. Math., vol. 8, no. 1, pp. 94-105, 2018.

[3] K. Boruah and B. Hazarika, “Bigeometric Integral Calculus,” TWMS J. Pure Appl. Math., vol. 8, no. 2, pp. 374-385, 2018.

[4] K. Boruah, B. Hazarika and A. E. Bashirov, “Solvability of Bigeometric Differential Equations by Numerical Methods,” Bol. Soc. Parana. Mat., doi: 10.5269/bspm.39444, 2018.

[5] F. Córdova-Lepe, “The Multiplicative Derivative as a Measure of Elasticity in Economics,” TEMAT-Theaeteto Antheniensi Mathematica, vol. 2, no.3, 2015.

[6] A.F. Çakmak and F. Başar, “On Line and Double Integrals in the Non-Newtonian Sense,” AIP Conference Proceedings, 1611, pp. 415-423, 2014.

[7] A.F. Çakmak and F. Başar, “Certain Spaces of Functions over the Field of NonNewtonian Complex Numbers,” Abstr. Appl. Anal., Article ID 236124, 12 pages, doi:10.1155/2014/236124, 2014.

[8] 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.

[9] C. Duyar and B. Sağır, “Non-Newtonian Comment of Lebesgue Measure in Real Numbers,” J. Math, Article ID 6507013, 2017.

[10] M. Erdoğan and C. Duyar, “Non-Newtonian Improper Integrals,” Journal of Science and Arts, vol. 1, no. 42, pp. 49-74, 2018.

[11] N. Güngör, “Some Geometric of The NonNewtonian Sequence Spaces l N p  ,” Math. Slovaca, vol. 70, no. 3, pp. 689-696, 2020.

[12] N. Güngör, “ BG -Volterra Integral Equations and Relationship with BG -Differential Equations,” GÜFBED, vol.10, no.3, pp. 814-829, 2020.

[13] M.Grosmann and R. Katz “Non-Newtonian Calculus,” Lee Press, Pigeon Cove Massachussets, 1972.

[14] M. Grosmann, “An Introduction to NonNewtonian Calculus,” International Journal of Mathematical Education in Science and Technology, vol. 10, no. 4, pp. 525-528, 1979.

[15] M. Grosmann, “Bigeometric Calculus: A system with a Scale Free Derivative,” 1st ed., Archimedes Foundation, Rockport Massachusetts, 1983.

[16] U. Kadak and M. Özlük, “Generalized Runge-Kutta Methods with Respect to NonNewtonian Calculus,” Abstr. Appl. Anal., Article ID 594685, 2014.

[17] M. Krasnov, K. Kiselev and G. Makarenko, “Problems and Exercises in Integral Equation,” Mır Publishers, Moscow, 1971.

[18] W. V. Lovitt, “Linear Integral Equations,” Dover Publications Inc., New York, 1950. [19] M. Rahman, “Integral Equations and Their Applications”(WIT press, Boston, 2007).

[20] R. K. Saeed and K. A. Berdawood, "Solving Two-dimensional Linear VolterraFredholm Integral Equations of the Second Kind by Using Successive Approximation Method and Method of Successive Substitutions," ZANCO Journal of Pure and Applied Sciences, vol. 28, no.2, pp. 35-46, 2016.

[21] B. Sağır and F. Erdoğan, “On the Function Sequences and Series in the NonNewtonian Calculus,” Journal of Science and Arts, vol. 4, no. 49, pp. 915-936, 2019.

[22] 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.

[23] V. Volterra and B. Hostinsky, “Opérations Infinitésimales linéares,” Herman, Paris, 1938.

[24] A. M. Wazwaz, “Linear and Nonlinear Integral Equations Methods and Applications”, Springer Verlag Berlin Heidelberg, 2011.