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On the Heat and Wave Equations with the Sturm‐Liouville Operator in Quantum Calculus

Author

Listed:
  • Serikbol Shaimardan
  • Lars-Erik Persson
  • Nariman Tokmagambetov

Abstract

In this paper, we explore a generalised solution of the Cauchy problems for the q‐heat and q‐wave equations which are generated by Jackson’s and the q‐Sturm‐Liouville operators with respect to t and x, respectively. For this, we use a new method, where a crucial tool is used to represent functions in the Fourier series expansions in a Hilbert space on quantum calculus. We show that these solutions can be represented by explicit formulas generated by the q‐Mittag‐Leffler function. Moreover, we prove the unique existence and stability of the weak solutions.

Suggested Citation

  • Serikbol Shaimardan & Lars-Erik Persson & Nariman Tokmagambetov, 2023. "On the Heat and Wave Equations with the Sturm‐Liouville Operator in Quantum Calculus," Abstract and Applied Analysis, John Wiley & Sons, vol. 2023(1).
    • Handle: RePEc:wly:jnlaaa:v:2023:y:2023:i:1:n:2488165
    DOI: 10.1155/2023/2488165
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    References listed on IDEAS

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    1. Herrmann, Richard, 2010. "Common aspects of q-deformed Lie algebras and fractional calculus," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(21), pages 4613-4622.
    2. Thomas Ernst, 2012. "A Comprehensive Treatment of q-Calculus," Springer Books, Springer, edition 127, number 978-3-0348-0431-8, October.
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