${q}_{k}-$Laplace Transform on Quantum Integral

Abstract

In this study, we present $q_{k}-$Laplace transform by $q_{k}-$integral on quantum analogue. We give some properties of $q_{k}-$Laplace transform. The $q_{k}-$Laplace transforms of some common functions are calculated.

Keywords

• Laplace transformation
• $q-$calculus
• ${q}_{k}-$calculus

References

1. Abdi, W.H., Certain inversion and representation formulae for q−Laplace transforms, Mathematische Zeitschrift, 83(1964), 238–249.
2. Abdulazeez, S.T., Modanli, M. Analytic solution of fractional order Pseudo-Hyperbolic Telegraph equation using modified double Laplace transform method, International Journal of Mathematics and Computer in Engineering, 1(2023), 105–114.
3. Adams, C.R., On the linear ordinary q−difference equation, Annals of mathematics, (1928), 195–205.
4. Ahmad, B., Boundary-value problems for nonlinear third-order q−difference equations, Electronic Journal of Differential Equations, 94(2011), 1–7.
5. Ahmad, B., Alsaedi, A., Ntouyas, S.K., A study of second-order q−difference equations with boundary conditions, Advances in difference equations, 35(2012), 1–10.
6. Ahmad, B., Alsaedi, A., Ntouyas, S.K., Tariboon, J., Alzahrani, F., Nonlocal boundary value problems for impulsive fractional qk-difference equations, Advances in Difference Equations, 124(2016), 1–16.
7. Ahmad, B., Ntouyas, S.K., Tariboon, J., Quantum Calculus: New Concepts, Impulsive IVPs and BVPs, Inequalities, New York, World Scientific, 2016.
8. Alp, N., Sarıkaya, M.Z., q−Laplace transform on quantum integral, Kragujevac Journal of Mathematics, 47(2023) 153–164.

9. Alp, N., Sarıkaya, M. Z., A new definition and properties of quantum integral which calls q−integral, Konuralp Journal of Mathematics, 5(2017), 146–159.
10. Bangerezako, G., Variational q−calculus, Journal of Mathematical Analysis and Applications, 289(2004), 650–665.
11. Bohner, M., Guseinov, G.Sh., The h−Laplace and q−Laplace transforms, Journal of Mathematical Analysis and Applications, 365(2010), 75–92.
12. Bohner, M, Hudson, T., Euler-type boundary value problems in quantum calculus, International Journal of Applied Mathematics and Statistics, 9(2007), 19–23.
13. Bohner, M, Peterson, A, Dynamic Equations on Time Scales: An Introduction with Applications. Birkh¨auser, Boston, 2001.
14. Carmichael, R.D., The general theory of linear q−difference equations, American journal of mathematics, 34(1912), 147–168.
15. Cieslinski, J.L., Improved q−exponential and q−trigonometric functions, Applied Mathematics Letters, 24(2011), 2110–2114.
16. Chung, W. S., Kim, T., Kwon, H.I., On the q−analog of the Laplace transform, Russian Journal of Mathematical Physics, 21(2014), 156–168.
17. Dennis, G.Z., Advanced Engineering Mathematics, Jones & Bartlett Publishers, 2020.
18. Diaz, R., Teruel, C., qk-Generalized Gamma and Beta Functions, Journal of Nonlinear Mathematical Physics, 12(2005), 118–134.
19. Jackson, F., q−Form of Taylor’s theorem, Messenger of Mathematics, 39(1909), 62–64.
20. Jackson, F.H., On q−definite integrals, Quarterly Journal of Pure and Applied Mathematics, 41(1910), 193–203.
21. Kac, V., Cheung, P., Quantum Calculus, New York, Springer, 2002.
22. Rajkovic, P.M., Stankovic, M.S., Marinkovic, S. D. The zeros of polynomials orthogonal with respect to q-integral on several intervals in the complex plane, In Proceedings of The Fifth International Conference on Geometry, Integrability and Quantization Vol. 5(2004), 178–189.
23. Sudsutad, W., Ntouyas, S.K., Tariboon, J., Quantum integral inequalities for convex functions, Journal of Mathematical Inequalities, 9(2015), 781–793.
24. Tariboon, J., Ntouyas, S.K., Quantum calculus on finite intervals and applications to impulsive difference equations, Advances in Difference Equations, 282(2013), 1–19.
25. Tariboon, J.i Ntouyas, S.K., Quantum integral inequalities on finite intervals, Journal of Inequalities and Applications, 121(2014).
26. Trjitzinsky, W.J., Analytic theory of linear q− difference equations, Acta mathematica, 61(1933), 1–38.
27. Uçar, F., Albayrak, D., On q−Laplace type integral operators and their applications, Journal of Difference Equations and Applications, 18(2012), 1001–1014.
28. Yilmaz, E., Goktas, S., On the solution of a Sturm-Liouville problem by using Laplace transform on time scales, Cumhuriyet Science Journal, 42(2021), 132–140.
29. Yu, C, Wang, J., Existence of solutions for nonlinear second-order q−difference equations with first-order q−derivatives, Advances in Difference Equations, 124(2013), 1–11.