On the Solutions of Some Equations in $(p,q)$-Calculus

Year 2024, Volume: 12 Issue: 1, 21 - 27, 30.04.2024

Nihan Turan , Metin Basarır

Abstract

In this paper, we introduce the Laplace equation in $(p, q)$-calculus and give the solutions of the equation using the separation method into its variables. We also give the $(p,q)$-calculus version of the equation of motion, which expresses the displacement of a falling field in a resistant environment. Finally, we obtain the solution of the Bernoulli’s equation in $(p,q)$-calculus.

Keywords

Falling Field Problem , Laplace Equation , Bernoulli

References

• [1] H. Akc¸a, J. Benbourenane H. Eleuch, The q -derivative and differential equation, J. Phys. Conf. Ser., (2019), 1411:012002.
• [2] A. M. Alanazi, A. Ebaid, W. M. Alhawiti and G. Muhiuddin, The falling body problem in quantum calculus, Front. Phys., 8(43)(2020).
• [3] S. Y. Arbab and S. H. Altoum, q-Bernoulli’s Equation, Am. J. Appl. Sci., 17(1)(2020), 214-217.
• [4] N. Bettaibi and K. Mezlini, On the use Of the q-Mellin transform to solve some q-heat and q-wave equations, Int. J. Math. Arch., 3(2)(2012), 446-455.
• [5] R. Chakrabarti and R. Jagannathan, A (p;q)-oscillator realization of two parameter quantum algebras, J. Phys. A: Math. Gen., 24(13)(1991), 5683-5701.
• [6] H. Cheng, Canonical Quantization of Yang-Mills Theories, Perspectives in Mathematical Physics, International Press, Somerville, MA, USA, 1996.
• [7] U. Duran, S. Aracı, and M. Ac¸ıkg¨oz, A Study on Some New Results Arising from (p;q)-Calculus, TWMS J. Pure Appl. Math., 11(1)(2020), 57-71.
• [8] A. Ebaid, B. Masaedeh, E. El-Zahar, A new fractional model for the falling body problem, Chin. Phys. Lett., 34(2)(2017), 020201.
• [9] R. Floreanini and L. Vinet, q-gamma and q-beta functions in quantum algebra representation theory, J. Comput. Appl. Math., 68(1-2)(1996), 57–68.
• [10] J.J.R. Garcia, M.G. Calderon, J.M. Ortiz , D. Baleanu, Motion of a particle in a resisting medium using fractional calculus approach, Proc. Roman Acad Ser A., 14(1)(2013), 42-47.
• [11] ˙I. Genc¸t¨urk , Boundary value problems for a second-order (p;q)-difference equation with integral conditions, Turk. J. Math., 46(2)(2022), 499-515.
• [12] G.H. Hardy and E.M.Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1960.
• [13] F.H. Jackson, q-Diference Equations, American J. Math., 32(4)(1910), 305-314.
• [14] R. Jagannathan, and K.S. Rao, Two-parameter quantum algebras, twin basic numbers and associated generalized hypergeometric series, In Proceeding of the International Conference on Number Theory and Mathematical Physics, Srinivasa Ramanujan Centre, Kumbakonam, India (2005).
• [15] V. Kac and C. Pokman, Quantum Calculus, Springer, 2002.
• [16] N. Kamsrisuk, C. Promsakon, S.K. Ntouyas, J. Tariboon, Nonlocal boundary value problems for (p;q)-difference equations, Differ. Equ. Appl., 10(2)(2018), 183-195.
• [17] G. Kaniadakis, A. Lavagno, and P. Quarati, Kinetic model for q-deformed bosons and fermions, Phys. Lett. A, 227(3)(1997), 227-231.
• [18] G.V. Milovanovic, V. Gupta, N. Malik, (p;q)-Beta functions and applications in approximation, Bol. Soc. Mat. Mex., 24(1)(2018), 219-237.
• [19] M. Mursaleen, M. Nasiruzzaman, A. Khan, K.J. Ansari, Some approximation results on Bleimann Butzer Hahn operators defined by (p;q)-integers, Filomat, 30(3)(2016), 639-648.
• [20] P.N. Sadjang, On the fundamental theorem of (p;q)-calculus and some (p;q)-Taylor formulas, Result. Math., 73(1)(2018), 1-21.
• [21] W. J. Trjitzinsky, Analytic theory of linear q-difference equations, Act. Math., 62(1)(1933), 167–226.
• [22] N. Turan and M. Bas¸arır, The Solutions of Motion, Laplace And Bernoulli’s Equations in q-Calculus, 9th Internatıonal Zeugma Conference On Scıentıfıc Research, February 19-21, 2023 / Gaziantep, T¨urkiye, full text book, page 144-151, ISBN:978-625-6404-76-2.