In this paper, we obtain exact solutions of a new modification of the Schrödinger equation related to the Bessel q-operator. The theorem is proved on the existence of this solution in the Sobolev-type space W^2_q(R^+_q) in the q-calculus. The results on correctness in the corresponding spaces of the Sobolev-type are obtained. For simplicity, we give results involving fractional q-difference equations of real order a > 0 and given real numbers in q-calculus. Numerical treatment of fractional q-difference equations is also investigated. The obtained results can be used in this field and be supplement for studies in this field.
This paper is devoted to explicit and numerical solutions to linear fractional q-difference equations and the Cauchy type problem associated with the Riemann-Liouville fractional q-derivative in q-calculus. The approaches based on the reduction to Volterra q-integral equations, on compositional relations, and on operational calculus are presented to give explicit solutions to linear q-difference equations. For simplicity, we give results involving fractional q-difference equations of real order a > 0 and given real numbers in q-calculus. Numerical treatment of fractional q-difference equations is also investigated. Finally, some examples are provided to illustrate our main results in each subsection.
In this paper we derive a sufficient condition for the existence of a unique
solution of a Cauchy type q-fractional problem (involving the fractional
q-derivative of Riemann-Liouville type) for some nonlinear differential
equations. The key technique is to first prove that this Cauchy type
q-fractional problem is equivalent to a corresponding Volterra q-integral
equation. Moreover, we define the q-analogue of the Hilfer fractional
derivative or composite fractional derivative operator and prove some
similar new equivalence, existence and uniqueness results as above. Finally,
some examples are presented to illustrate our main results in cases where we
can even give concrete formulas for these unique solutions.
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.
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