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

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  • Serikbol Shaimardan
  • Lars-Erik Persson
  • Nariman Tokmagambetov
  • Douglas R. Anderson

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 & Douglas R. Anderson, 2022. "On the Heat and Wave Equations with the Sturm-Liouville Operator in Quantum Calculus," Abstract and Applied Analysis, Hindawi, vol. 2022, pages 1-8, December.
    • Handle: RePEc:hin:jnlaaa:2488165
    DOI: 10.1155/2022/2488165
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