On Certain Properties of the Laplace-type Integral Transform Via Post Quantum Calculus
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Abstract
In this paper, we apply the notion of (p,q)-calculus or post quantum calculus to establish theoretical results of (p,q)-analogues of Laplace-type integral transform of the first and second kind, which is a symmetric relation between (p,q)-analogues of the Laplace-type integral and Laplace transforms. Additionally, we discuss (p,q)-analogues of Laplace-type integral transform on various classes of some (p,q)-special functions, (p,q)-exponential function, (p,q)-trigono-metric types, (p,q)-differential operator, and (p,q)-convolution theorem. Finally, we establish results related to (p,q)-Aleph function.
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References
Jackson, F. H. (1910). On a q-Definite Integrals. The Quarterly Journal of Pure and Applied Mathematics. Vol. 41, No. 2, pp. 193-203
Jackson, F. H. (1910). q-Difference Equations. American Journal of Mathematics. Vol. 32, No. 4, pp. 305-314. DOI: 10.2307/2370183
Ernst, T. (2012). A Comprehensive Treatment of q-Calculus. Springer: Basel
Aral, A., Gupta, V., and Agarwal, R. P. (2013). Applications of q-Calculus in OperatorTheory. Springer Science + Business Media: New York
Sole, A. D. and Kac, V. (2003). On Integral Representations of q-Gamma and q-Beta Functions. Access (20 January 2023). Available (https://arxiv.org/abs/math/0302032)
Tariboon, J. and Ntouyas, S. K. (2013). Quantum Calculus on Finite Intervals and Applications to Impulsive Difference Equations. Advances in Difference Equations. Vol. 2013, pp. 1-19. DOI: 10.1186/1687-1847-2013-282
Tariboon, J. and Ntouyas, S. K. (2014). Quantum Integral Inequalities on Finite Intervals. Journal of Inequalities and Applications. Vol. 2014, pp, 1-13. DOI: 10.1186/1029-242X-2014-121
Jhanthanam, S., Tariboon, J., Ntouyas, S. K., and Nonlaopon, K. (2019). On q-Hermite-Hadamard Ibequalities for Differentiable Convex Function. Mathematics. Vol. 7, Issue 7, pp. 1-9. DOI: 10.3390/math7070632
Prabseang, J., Nonlaopon, K., and Tariboon, J. (2019). Quantum Hermite-Hadamard Inequalities for Double Integral and q-Differentiable Convex Functions. Journal of Mathematical Inequalities. Vol. 13, No. 3, pp. 675-686. DOI: 10.7153/jmi-2019-13-45
Ahmad, B., Alsaedi, A., and Ntouyas, S. K. (2012). A Study of Second-Order q-Difference Equations with Boundary Conditions. Advances in Difference Equations. Vol. 2012, pp. 1-10. DOI: 10.1186/1687-1847-2012-35
Ahmad, A., Jain, R., and Jain, D. K. (2017). q-Analogue of Aleph-Function and Itstransformation Formula with q-Derivative. Journal of Statistics Applications & Probability. Vol. 6, No. 3, pp. 1-9. DOI: 10.18576/jsap/060312
Ahmad, B., Ntouyas, S. K., and Purnaras, I. K. (2012). Existence Results for Nonlinear q-Difference Equations with Nonlocal Boundary Conditions. Communications in Nonlinear Analysis. Vol. 19, pp. 59-72
Kac, V. and Cheung, P. (2002). Quantum Calculus. Springer-Verlag: New York
Ganie, J. A. and Jain, R. (2020). On a System of q-Laplace Transform of Two Variables with Applications. Journal of Computational and Applied Mathematics. Vol. 366, pp. 1-12. DOI: 10.1016/j.cam.2019.112407
Albayrak, D., Purohit, S. D., and Uçar, F. (2013). On q-Analogues of Sumudu Transform. Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica. Vol. 21, No. 1, pp. 239-260. DOI: 10.2478/auom-2013-0016
Chung, W. S., Kim, T., and Kwon, H. I. (2014). On the q-Analog of the Laplace Transform. Russian Journal of Mathematical Physics. Vol. 21, No. 2, pp. 156-168. DOI: 10.1134/S1061920814020034
Al-Omari, S. K. Q. (2018). Certain Results Related to the N-Transform of a Certain Class of Functions and Differential Operators. Advances in Difference Equations. Vol. 2018, pp. 1-14. DOI: 10.1186/s13662-017-1457-y
Al-Omari, S. K. Q. (2020). q-Analogues and Properties of the Laplace-Type Integral Operator in the Quantum Calculus Theory. Journal of Inequalities and Applications. Vol. 2020, pp. 1-14. DOI: 10.1186/s13660-020-02471-0
Al-Omari, S. K. Q. (2021). Estimates and Properties of Certain q-Mellin Transform on Generalized q-Calculus Theory. Advances in Difference Equations. Vol. 2021, pp. 1-15. DOI: 10.1186/s13662-021-03391-z
Uçar, F. and Albayrak, D. (2011). On q-Laplace Type Integral Operators and Their Applications. Journal of Difference Equations and Applications. Vol. 18, Issue 6, pp. 1001-1014. DOI: 10.1080/10236198.2010.540572
Alp, N. and Sarikaya, M. Z. (2023). q-Laplace Transform on Quantum Integral. Kragujevac Journal of Mathematics. Vol. 47, No. 1, pp. 153-164
Al-Khairy, R. T. (2020). q-Laplace Type Transforms of q 2 - Analogues of Bessel Functions. Journal of King Saud University-Science. Vol. 32, No. 1, pp. 563-566. DOI: 10.1016/j.jksus.2018.08.012
Chakrabarti, R. and Jagannathan, R. (1991). A (p,q)-Oscillator Realization of Two-Parameter Quantum Algebras. Journal of Physics A: Mathematical and General. Vol. 24, pp. 711-718. DOI: 10.1088/0305-4470/24/13/002
Sadjang, P. N. (2018). On the Fundamental Theorem of (p,q)-Calculus and Some (p,q)-Taylor Formulas. Results in Mathematics. Vol. 73, No. 39, pp. 1-21. DOI: 10.1007/s00025-018-0783-z
Milovanovic, G. V., Gupta, V., and Malik, N. (2018). (p,q)-Beta Functions and Applications in Approximation. Boletín de la Sociedad Matemática Mexicana. Vol. 24, pp. 219-237. DOI: 10.1007/s40590-016-0139-1
Prabseang, J., Nonlaopon, K., Tariboon, J., and Ntouyas, S. K. (2021). Refinements of Hermite-Hadamard Inequalities for Continuous Convex Functions Via (p,q)-Calculus. Mathematics. Vol. 9, Issue 4, pp. 1-12. DOI: 10.3390/math9040446
Jain, A., Bhat, A. A., Jain, R., and Jain, D. K. (2021). Certain Results of (p,q)-Analogue of Aleph-Function with (p,q)-Derivative. Journal of Statistics Applications & Probability Letters. Vol. 10, Issue 1, pp. 45-52. DOI: 10.18576/jsap/100105
Ahmad, A., Jain, R., and Jain, D. K. (2017). Certain Results of (p,q)-Analogue of I-Function with (p,q)-Derivative. In Proceedings of the MSSCID-2017. 20th Annual Conference of VPI. pp. 31-44. India: Jaipur
Sadjang, P. N. (2017). On Two (p,q)-Analogues of the Laplace Transform. Journal of Difference Equations and Applications. Vol. 23, No. 9, pp. 1562-1583. DOI: 10.1080/10236198.2017.1340469
Sadjang, P. N. (2019). On (p,q)-Analogues of the Sumudu Transform. Access (20 January 2023). Available (10.13140/RG.2.2.34688.48645)
Jirakulchaiwong, S., Nonlaopon, K., Tariboon, J., Ntouyas S. K., and Kim, H. (2021). On (p,q)-Analogues of Laplace-Typed Integral Transforms and Applications. Symmetry. Vol. 13, Issue 4, pp. 1-22. DOI: 10.3390/sym13040631
Schiff, J. L. (1999). The Laplace Transform Theory and Application. Springer: New York
Yurekli, O. and Sadek, I. (1991). A Parseval-Goldstein Type Theorem on the Widder Potential Transform and Its Applications. International Journal of Mathematics and Mathematical Sciences. Vol. 14, No. 3, pp. 517-524. DOI: 10.1155/S0161171291000704
Sudland, N., Baumann, B., and Nonnenmacher, T. F. (1998). Open Problem: Who knows about the Aleph-function?. Fractional Calculus and Applied Analysis. Vol. 1, pp. 401-402
Saxena, V. P. (2008). The I-function; Anamaya publishers. New Delhi: India
Bhat, A. A., Singh, G., Jain, R., and Jain, D. K. (2019). q-Sumudu and q-Laplace Transforms of the Basic Analogue of Aleph-Function. Far East Journal of Mathematical Sciences. Vol. 118, No. 2, pp. 189-211. DOI: 10.17654/MS118020189
Tassaddiq, A., Bhat, A. A., Jain, D. K., and Ali, F. (2020). On (p,q)-Sumudu and (p,q)-Laplace Transforms of the Basic Analogue of Aleph-Function. Symmetry. Vol. 12,pp. 1-17
Mathai, A. M., Saxena, R. K., and Haubold, H. J. (2010). The H-Function: Theory and Applications. Springer: New York
Pathak, S. and Jain, R. (2017). Some Identities Involving (p,q)-Analogue of Meijer’s G-Function. Indian Academy Of Mathematics in Navlakha, Indore. Vol. 39, pp. 137-146