A STUDY ON NON-HOMOGENEOUS MULTIPLICATIVE BOUNDARY VALUE PROBLEMS

Abstract

The paper deals with the non-homogeneous boundary value problems established in the multiplicative calculus. For this problem, the multiplicative meanings of concepts such as adjoint operator, self-adjoint operator, Lagrange identity, Green's formula are obtained. Then, a criterion is given for the existence of solutions to non-homogeneous multiplicative boundary value problems.

Keywords

• Adjoint Operator
• Boundary Value Problems
• Green’s Formula
• Multiplicative Calculus
• Orthogonality

References

1. M. E. Aydin, A. Has, B. Yilmaz, A Non-Newtonian Approach İn Differential Geometry Of Curves: Multiplicative Rectifying Curves, Bulletin of the Korean Mathematical Society, Vol.61, No.3, pp.849-866 (2024).
2. A. Bal, N. Yalcin, M. Dedeturk, Solutions of Multiplicative Integral Equations Via The Multiplicative Power Series Method, Politeknik Dergisi, Vol.26, No.1, pp.311-320 (2023).
3. J. S. Bardi, The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time. New York: Thunder's Mouth Press, İSBN 1-56025-706-7 (2006).
4. A. Bashirov, E. M. Kurpinar, A. Ozyapici, Multiplicative Calculus and İts Applications, Journal of Mathematical Analysis and Applications, Vol.337, No.1, pp.36-48 (2008).
5. A. Bashirov, G. Bashirova, Dynamics of Literary Texts and Diffusion, Online Journal of Communication and Media Technologies, Vol.1, No.3, pp.60-82 (2011).
6. A. Bashirov, E. Misirli, Y. Tandogdu, A. Ozyapici, On Modeling with Multiplicative Differential Equations, Applied Mathematics-A Journal of Chinese Universities, Vol.26, No.4, pp.425-438 (2011).
7. B. Bilgehan, A. Ozyapici, Z. Hammouch, Y. Gurefe, Predicting the Spread of COVİD-19 with a Machine Learning Technique and Multiplicative Calculus, Soft Computing, Vol.26, No.16, pp.8017-8024 (2022).
8. C. B. Boyer, The History of the Calculus and its Conceptual Development, New York: Dover. OCLC 643872 (1959).

9. D. Campbell, Multiplicative Calculus and Student Projects, Projects, Problems, Resources and İssues in Mathematics Undergraduate Studies, Vol.9, No.4, pp.327-332 (1999).
10. F. Cordova-Lepe, The Multiplicative Derivative As a Measure of Elasticity in Economics, Theaeteto Atheniensi Mathematica, Vol.2 (2006).
11. D. Filip, C. Piatecki, A Non-Newtonian Examination of The Theory of Exogenous Economic Growth, Mathematica Aeterna, Vol.10 (2014).
12. D. Filip, C. Piatecki, In defense of a non-newtonian economic analysis. (2014).
13. L. Florack, H. Van Assen, Multiplicative Calculus in Biomedical İmage Analysis, Journal of Mathematical İmaging and Vision, Vol.42, pp.64-75 (2012).
14. S. Goktas, A New Type of Sturm‐Liouville Equation in the Non Newtonian Calculus, Journal of Function Spaces, Vol.2021, No.1, pp.1-8 (2021).
15. M. Grossman, R. Katz, Non-Newtonian Calculus. Pigeon Cove, Massachusetts. (1972)
16. M. Grossman, An introduction to non Newtonian calculus, International Journal of Mathematical Educational in Science and Technology, Vol.10, No.4, pp.525-528 (1979).
17. A. Has, B. Yilmaz, A non-Newtonian Conics in Multiplicative Analytic Geometry, Turkish Journal of Mathematics, Vol.48, No.5, pp.976-994 (2024).
18. A. Has, B. Yilmaz, H. Yildirim, A Non-Newtonian Magnetic Curves in Multiplicative Riemann Manifolds, Physica Scripta, Vol.99, No.4, 045239 (2024).
19. L. D. Hoffmann, G. L. Bradley , Calculus for Business, Economics, and the Social and Life Sciences (8th ed.). Boston: McGraw Hill (2004).
20. U. Kadak, H. Efe , The Construction of Hilbert Spaces over the Non-Newtonian Field, İnternational Journal of Analysis, pp.1-10. (2014).
21. E. Misirli, Y. Gurefe , Multiplicative Adams Bashforth-Moulton Methods, Numerical Algorithms, Vol.57, pp.425-439 (2011).
22. E. Misirli, A. Ozyapici , Exponential Approximations On Multiplicative Calculus, İn Proc. Jangjeon Math. Soc, Vol.12, No.2, pp.227-236 (2009).
23. G. B. Oznur, G. G. Ozbey, Y. Aygar Kucukevcilioglu, R. Aktas Karaman, A Study Of The Scattering Analysis of the Multiplicative Sturm-Liouville Problem, Turkish Journal of Mathematics, Vol.48, No.3, pp.608-622 (2024).
24. D. Stanley, A Multiplicative Calculus, Problems, Resources and İssues in Mathematics Undergraduate Studies, Vol.9, pp.310-326 (1999)
25. N. Yalcin, The Solutions Of Multiplicative Hermite Differential Equation And Multiplicative Hermite Polynomials, Rendiconti del Circolo Matematico di Palermo Series 2, Vol.70, No.1, pp.9-21 (2021).
26. Multiplicative Chebyshev Differential Equations and Multiplicative Chebyshev Polynomials, Thermal Science, Vol.26, No.Spec. issue 2, pp.785-799 (2022).
27. N. Yalcin, E. Celik, A. Gokdogan , Multiplicative Laplace Transform and its Applications, Optik, Elsevier GmbH. 127, pp.9984-9995 (2016).
28. N. Yalcin, E. Celik, Solution of Multiplicative Homogeneous Linear Differential Equations with Constant Exponentials, New Trends in Mathematical Sciences, Vol.6, No.2, pp.58-67 (2018).