On Bigeometric Laplace Integral Transform

Abstract

The purpose of this study is to mention the Laplace integral transform in bigeometric analysis, which is one of the non-Newtonian analysis by using the fundamental definitions and theorems of the Laplace integral transform, which is one of the integral transform methods of classical analysis. First of all, the concept of exponential arithmetic, which forms the basis of non Newtonian analysis, is given. As in classical analysis, definitions of the concepts of bigeometric limit, bigeometric continuity, bigeometric derivative and bigeometric integral are given in bigeometric analysis. Here, the definition of the bigeometric Laplace integral transform in bigeometric analysis is given. Then, some basic concepts and theorems of the bigeometric Laplace integral transform are given. For this purpose, the definitions of the concepts of bigeometric derivative and bigeometric indefinite integral and bigeometric definite integral in bigeometric analysis and the properties of these concepts are used. In addition, the properties of the bigeometric Laplace integral transform are investigated. Finally, solutions of bigeometric linear differential equations are investigated with the help of the bigeometric Laplace integral transform.

Keywords

• Bigeometric analysis
• Exponential arithmetic
• Bigeometric derivative
• Bigeometric integral
• Bigeometric Laplace integral transform

Bigeometric Laplace İntegral Dönüşümü Üzerine

Abstract

Bu çalışmanın amacı, klasik analizin integral dönüşüm metotlarından biri olan Laplace integral dönüşümünün temel tanım ve teoremleri kullanılarak, Newtonyen olmayan analizlerden biri olan bigeometrik analizde Laplace integral dönüşümünü tanımlamaktır. Öncelikli olarak Newtonyen olmayan analizlerin temelini oluşturan üstel aritmetik kavramı verilmiştir. Klasik analizde olduğu gibi bigeometrik analizde de bigeometrik limit, bigeometrik süreklilik, bigeometrik türev ve bigeometrik integral kavramlarının tanımları verilmiştir. Ardından, bigeometrik Laplace integral dönüşümünün tanımı yapılmıştır. Sonra, bigeometrik Laplace integral dönüşümünün bazı temel kavramları ve teoremleri verilmiştir. Bunun için bigeometrik analizde yer alan bigeometrik türev, bigeometrik belirsiz integral ve bigeometrik belirli integral kavramlarının tanımları ve bu kavramların özellikleri kullanılmıştır. Ayrıca, bigeometrik Laplace integral dönüşümünün özellikleri incelenmiştir. Son olarak bigeometrik Laplace integral dönüşümü yardımıyla bigeometrik lineer diferansiyel denklemlerin çözümleri araştırılmıştır.

Keywords

• Bigeometrik analiz
• Üstel aritmetik
• Bigeometrik türev
• Bigeometrik integral
• Bigeometrik Laplace integral dönüşümü

References

1. Bal, A., Yalcin, N. & Dedeturk, M. (2023). Solutions of Multiplicative İntegral Equations via The Multiplicative Power Series Method. Journal of Polytechnic, 26 (1) , 311-320.
2. Bashirov, A.E., Mısırlı, E., & Özyapıcı, A., 2008. Multiplicative calculus and its applications, J. Math. Anal. Appl., 337: 36-48.
3. Bashirov, A. E., & Bashirova, G. (2011). Dynamics of literary texts and diffusion.
4. Boruah, K., & Hazarika, B. (2018a). G-calculus. TWMS Journal of Applied and Engineering Mathematics, 8(1), 94-105.
5. Boruah, K., & Hazarika, B. (2018b). Bigeometric integral calculus. TWMS Journal of Applied and Engineering Mathematics, 8(2), 374-385.
6. Boruah, K., & Hazarika, B. (2021a). Some basic properties of bigeometric calculus and its applications in numerical analysis. Afrika Matematika, 32(1-2), 211-227.
7. Boruah, K., Hazarika, B., & Bashirov, A. E. (2021b). Solvability of bigeometric differential equations by numerical methods. Bol. Soc. Parana. Mat, 39, 203-222.
8. Boruah, K., & Hazarika, B. (2022). Topology on Geometric Sequence Spaces. In Approximation Theory, Sequence Spaces and Applications (pp. 1-19). Singapore: Springer Nature Singapore.

9. Córdova-Lepe, F. (2006). The multiplicative derivative as a measure of elasticity in economics. TEMAT-Theaeteto Atheniensi Mathematica, 2(3).
10. Çakmak, A. F., & Başar, F. (2012). Some new results on sequence spaces with respect to non-Newtonian calculus. Journal of Inequalities and Applications, 2012(1), 1-17.
11. Erdoğan, M. & Duyar, C., 2018. Non-Newtonian Improper Integrals. Journal of Science and Arts, 1(42), 49-74.
12. Filip, D., & Piatecki, C. (2014a). An overview on the non-newtonian calculus and its potential applications to economic.
13. Filip, D., & Piatecki, C. (2014b). 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(1), 64-75.
15. Grossman, M., & Katz, R. (1972). Non-Newtonian Calculus: A Self-contained, Elementary Exposition of the Authors' Investigations... Non-Newtonian Calculus.
16. Grossman, M. (1979). An introduction to non‐Newtonian calculus. International Journal of Mathematical Educational in Science and Technology, 10(4), 525-528.
17. Grossman, M. (1983). Bigeometric calculus: a system with a scale-free derivative. Archimedes Foundation.
18. Güngör, N. (2020). BG-Volterra integral equations and relationship with BG-differential equations. Gümüşhane University Journal of Science and Technology, 10(3), 814-829.
19. Güngör, N. (2021). Solving Bigeometric Volterra Integral Equations by Using Successive Approximations Method. Sakarya University Journal of Science, 25(1), 150-162.
20. Kaymak, S. (2023). Bigeometrik Analizde Laplace İntegral Dönüşümü ve Uygulamaları (Unpublished master's thesis). Gumushane University, Gümüşhane.
21. Riza, M., & Eminağa, B. (2014). Bigeometric calculus–a modelling tool. arXiv preprint arXiv:1402.2877. Rybaczuk, M. and Stoppel, P. (2000). The fractal growth of fatigue defects in materials. International Journal of Fracture, 103(1), 71-94.
22. Rybaczuk, M., & Cetera, A. (2001). Non-homogeneous Fractal Growth of Fatigue Defects in Materials (No. 2001-01-4057). SAE Technical Paper.
23. Turkmen, C., & Basar, F. (2012). Some basic results on the sets of sequences with geometric calculus, Commun. Fac. Fci. Univ. Ank. Series A, 1, 17-34.
24. Yalcin, N., Celik, E., & Gokdogan, A. (2016). Multiplicative Laplace transform and its applications. Optik, 127(20), 9984-9995.
25. Yalcin, N., & Celik, E. (2018). The solution of multiplicative non-homogeneous linear differential equations. J. Appl. Math. Comput, 2(1), 27-36.
26. Yalcin, N., & Celik, E. (2018). Solution of multiplicative homogeneous linear differential equations with constant exponentials. New Trends in Mathematical Sciences, 6(2), 58-67.
27. Yalcin, N., & Celik, E. Çarpımsal Cauchy-Euler ve Legendre Diferansiyel Denklemi. Gümüşhane University Journal of Science and Technology, 9(3), 373-382.
28. Yalcin, N. (2021). The solutions of multiplicative Hermite differential equation and multiplicative Hermite polynomials. Rendiconti del Circolo Matematico di Palermo Series 2, 70(1), 9-21.
29. Yalcin, N., & Dedeturk, M. (2021). Solutions of multiplicative ordinary differential equations via the multiplicative differential transform method. Aims Mathematics, 6(4), 3393-3409.
30. Yalcin, N. (2022). Multiplicative Chebyshev differential equations and multiplicative Chebyshev polynomials. Thermal Science, 26(Spec. issue 2), 785-799.