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

. 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-Le ﬄ er function. Moreover, we prove the unique existence and stability of the weak solutions.


Introduction
In the last decade, the theory of quantum groups and q -deformed algebras have been the subject of intense investigation. Many physical applications have been investigated on the basis of the q-deformation of the Heisenberg algebra (see [1,2]). For instance, the q-deformed Schrödinger equations have been proposed in [3,4], and applications to the study of q-deformed version of the hydrogen atom and of the quantum harmonic oscillator have been presented (see [5]). Fractional calculus and the q-deformed Lie algebras are closely related. A new class of fractional q-deformed Lie algebras is proposed, which for the first time allows a smooth transition between the different Lie algebras (see [6]).
The origin of the q-difference calculus can be traced back to the works by Jackson (see [7,8]) and Carmichael (see [9]) from the beginning of the twentieth century, while basic definitions and properties can be found, e.g., in the monographs [10,11] and the PhD thesis [12]. Recently, the fractional q-difference calculus has been proposed by Al-salam (see [13]) and Agarwal (see [14]). We can also mention papers [15,16], where the authors investigated the explicit solutions to linear fractional q-differential equations with the q-fractional derivative, and in [17], the q-analogue nonhomogeneous wave equations were studied.
A motivation behind this work is to state some new results about the q-heat and q-wave equations associated to the q-Sturm-Liouville operator (see (10)). We attempt to extend the heat representation theory studied in some cases (see [18][19][20], etc.). We define a generalised solution of the Cauchy problem for these equations generated by the q-Mittag-Leffler function and the q-associated functions of a biorthogonal system (see (13)). We investigate the well-posedness of the Cauchy problem for the q-heat and q-wave equations for operators with a discrete nonnegative spectrum acting on L 2 q ½0, 1. In particular, we prove both unique existence and stability of the corresponding the generalised solution.
The paper is organized as follows: the main results are presented and proved in Section 3 and Section 4. In order to not disturb these presentations, we include in Section 2 some necessary Preliminaries.

Preliminaries
In this section, we recall some notations and basic facts in q-calculus. We will always assume that 0 < q < 1. The q-real number ½α q is defined by The q-shifted factorial is defined by ( Moreover, their natural expansions to the reals are The Jackson's q-difference operator D q f ðxÞ is (see, [8,12] The q-derivative D q of a product of the functions f and g as defined by As given in [10], two q-analogues of the exponential functions are defined by Moreover, we have that Due to the various types of q-differences introduced in quantum calculus, trigonometric functions have various q-analogues (see, [21] Section 2 [10], Section 10 and [12], Section 2.12). The following definition of cosine and sine will be useful in this investigation (see [20]): where the q-analogue of the binomial coefficients ½n q ! is defined by The q-integral (or Jackson's integral) is defined by (see [8]) and a more general form is given by The q-version of integration by parts reads and if f ≡ 1, then we get that The q -Sturm-Liouville Problem. Let L 2 q ½0, 1 be the space of all real-valued functions defined on ½0, 1 such that The space L 2 q ½0, 1 is a separable Hilbert space with the inner product: Now, we shortly describe the study introduced by Annaby and Mansour in of a basic q-Sturm-Liouville eigenvalue problem in a Hilbert space (see [21], Chapter 3). In particular, they investigated the basic q-Sturm-Liouville equation: where vð·Þ is defined on ½0, 1 and continuous at zero. Let C 2 q,0 ½0, 1 denotes the space of all functions yð·Þ such that y 2 Abstract and Applied Analysis and D q y are continuous at zero. If v ≡ 0, then we get the operator L in the following form: for 0 ≤ x ≤ 1 and λ ∈ ℝ. The operator L is self adjoint on C 2 q,0 ½0, 1 ∩ L 2 q ½0, 1 (see [21], Theorem 3.4.). A fundamental set of solutions of (10) are cos ð ffiffiffi λ p ; q 2 Þ and sin ð ffiffiffi Additionally, the corresponding set of eigenfunctions is an orthogonal basis in L 2 q ð0, 1Þ. Thus, we can identify f ∈ L 2 q ½0, 1 with its Fourier series: The Sobolev Space Associated with L. The next step is to recall the essential elements of the Fourier analysis presented in [22][23][24], as well as its applications to the spectral properties of L.
For g ∈ C ∞ L ½0, 1, we introduce the Fréchet topology of C ∞ L ½0, 1 by the family of norms: Thus, for s ∈ ℝ, we can also define the Sobolev spaces W s q,L associated to L in the following form: where the q-partial differential operator D q,t uðt, xÞ with respect to t has the following form: Notation: the symbol M≾K means that there exists γ > 0 such that M ≤ γK, where γ is a constant.

The q-Heat Equation
We start with a study of the following Cauchy problem: with the initial condition We say a generalised solution of the problem (14)- (15) is a function uðt, xÞ such that they satisfy equation (14) and condition (15).
q,L ½0, 1 and f ∈ Cð½0, T ; W 2 q,L ½0, 1Þ. Then, there exists the generalised solution of u to problem (14)- (15), and Moreover, this solution can be written in the following explicit form Proof. Existence. Since the system of eigenfunctions fϕ k g ∞ k=1 is a basis in L 2 q ½0, 1 (see (11)), we seek for a function uðt, xÞ in the form 3 Abstract and Applied Analysis for each fixed 0 < t < T < ∞. The coefficients will then be given by the Fourier coefficients formula u k ðtÞ = huðt, ·Þϕ k i.
We can similarly expand the source function, From (11) and (18), we have that Hence, and Substituting (20) and (21) into the equation (14), we find that But then, due to the completeness, which are ODEs for the coefficients u k ðtÞ of the series (18). Using the integrating factor E λ k qt q and (2) and (3), we can rewrite the ODE as Form (3), (5), and (24), we get that which, in its turn, implies that and we conclude that But the initial conditions (16) and (22) imply that u k ð0Þ = φ k . Thus, Therefore, the solution uðt, xÞ can be written in the series form as so, also (17) is proved. Convergence. From (1), (4), and (5), we have that for x ∈ ½0, 1. Hence, using for 0 < t < T < ∞, (5), (23), and (25), we get that and Hence, Since φ ∈ W 2 q,L , f ∈ Cð½0, 1 ; W 2 q,L Þ, and, hence, by using the Plancherel identity and (27) and (28), we can conclude 4 Abstract and Applied Analysis Uniqueness. It only remains to prove the uniqueness of the solution. We assume the opposite; namely, that there exist functions uðt, xÞ and vðt, xÞ, which are two different solutions of problem (14)- (15). Let 0 < t < T < ∞. Then, we have that ( We define Wðt, xÞ = uðt, xÞ − vðt, xÞ. Then, the function Wðt, xÞ is a solution of the following problem ( From (18), it follows that Wðt, xÞ ≡ 0, that is, uðx, tÞ ≡ vðx, tÞ, and this contradiction to our assumption proves the uniqueness of the solution. The proof is complete.
Uniqueness. This part can be proved completely similar as the proof of Theorem 1.. So we omit the details.

Data Availability
Data supporting this manuscript are available from Scopus, Web of Science, and Google Scholar.

Conflicts of Interest
The authors declare that they have no conflicts of interest.