On the solutions of the q-analogue of the telegraph differential equation

Year 2022, Volume: 71 Issue: 3, 846 - 855, 30.09.2022

Döne Karahan

https://doi.org/10.31801/cfsuasmas.1009068

Abstract

In this work, q-analogue of the telegraph differential equation is investigated. The approximation solution of q-analogue of the telegraph differential equation is founded by using the Laplace transform collocation method (LTCM). Then, the exact solution is compared with the approximation solution for q-analogue of the telegraph differential equation. The results showed that the method is useful and effective for q-analogue of the telegraph differential equation.

Keywords

q-analogue of the telegraph differential equation, Laplace transform collocation method, exact solution, approximation solution.

References

• Euler, L., Introduction in Analysin Infinitorum, vol. 1, Lausanne, Switzerland, Bousquet, 1748.
• Ernst, T., The History of q-Calculus and a New Method, U.U.D.M. Report 2000, 16, Uppsala, Department of Mathematics, Uppsala University, 2000.
• Kac, V., Cheung, P., Quantum Calculus, Universitext, Springer, New York, 2002.
• Annaby, M.H., Mansour, Z.S., q-Fractional Calculus and Equations, Lecture Notes in Mathematics, vol. 2056, Springer, Heidelberg, 2012. https://doi.org/10.1007/978-3-642- 30898-7
• Karahan, D., Mamedov, Kh.R., Sampling theory associated with q-Sturm-Liouville operator with discontinuity conditions, Journal of Contemporary Applied Mathematics, 10(2) (2020), 1-9.
• Liu, Z.G., On a System of q-Partial Differential Equations with Applications to q-Series, In:Andrews G., Garvan F., Analytic Number Theory, Modular Forms and q-Hypergeometric Series, ALLADI60 2016, Springer Proceedings in Mathematics and Statistics, Vol. 221, Springer, 2017.
• Liu, Z.G., On the q-Partial Differential Equations and q-Series. In: The Legacy of Srinivasa Ramanujan, 213-250, Ramanujan Mathematical Society Lecture Notes Series, Vol. 20, Ramanujan Mathematical Society, Mysore, 2013.
• Cao, J., Homogeneous q-partial difference equations and some applications, Advances in Applied Mathematics, 84 (2017), 47-72. https://doi.org/10.1016/j.aam.2016.11.001
• Adewumi, A.O., Akindeinde, S.O., Aderogba, A.A., Ogundare, B.S., Laplace transform collocation method for solving hyperbolic telegraph equation, International Journal of Engineering Mathematics, (2017). https://doi.org/10.1155/2017/3504962
• Modanli, M., Laplace transform collocation and Daftar-Gejii-Jafaris method for fractional order time varying linear dynamical systems, Physica Scripta, 96(9) (2021), 094003. https://doi.org/10.1088/1402-4896/ac00e7
• Hahn, W., Beitrage zur Theorie der Heineschen Reihen (German), Math. Nachr., 2 (1949), 340-379. https://doi.org/10.1002/mana.19490020604
• Daiz, R., Ternel C., q, k-generalized gamma and beta functions, J. Nonlinear Math. Phys., 12(1) (2005), 118-131.
• De Sole, A., Kac, V.G., On integral representations of q-gamma and q-beta functions, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, 16(1) (2005), 11-29.