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

Abstract

In this paper, we explore a generalised solution of the Cauchy problems for the -heat and -wave equations which are generated by Jackson’s and the -Sturm-Liouville operators with respect to and , 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 -Mittag-Leffler function. Moreover, we prove the unique existence and stability of the weak solutions.

1. Introduction

In the last decade, the theory of quantum groups and -deformed algebras have been the subject of intense investigation. Many physical applications have been investigated on the basis of the -deformation of the Heisenberg algebra (see [1, 2]). For instance, the -deformed Schrödinger equations have been proposed in [3, 4], and applications to the study of -deformed version of the hydrogen atom and of the quantum harmonic oscillator have been presented (see [5]). Fractional calculus and the -deformed Lie algebras are closely related. A new class of fractional -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 -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 -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 -differential equations with the -fractional derivative, and in [17], the -analogue nonhomogeneous wave equations were studied.

A motivation behind this work is to state some new results about the -heat and -wave equations associated to the -Sturm-Liouville operator (see (10)). We attempt to extend the heat representation theory studied in some cases (see [18–20], etc.). We define a generalised solution of the Cauchy problem for these equations generated by the -Mittag-Leffler function and the -associated functions of a biorthogonal system (see (13)). We investigate the well-posedness of the Cauchy problem for the -heat and -wave equations for operators with a discrete nonnegative spectrum acting on . 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.

2. Preliminaries

In this section, we recall some notations and basic facts in -calculus. We will always assume that . The -real number is defined by

The -shifted factorial is defined by

Moreover, their natural expansions to the reals are

The Jackson’s -difference operator is (see, [8, 12] Section 2.1])

The -derivative of a product of the functions and as defined by

As given in [10], two -analogues of the exponential functions are defined by

Moreover, we have that

Due to the various types of -differences introduced in quantum calculus, trigonometric functions have various -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 -analogue of the binomial coefficients is defined by

The -integral (or Jackson’s integral) is defined by (see [8]) and a more general form is given by for .

The -version of integration by parts reads and if , then we get that

The-Sturm-Liouville Problem. Let be the space of all real-valued functions defined on such that

The space is a separable Hilbert space with the inner product:

Now, we shortly describe the study introduced by Annaby and Mansour in of a basic -Sturm-Liouville eigenvalue problem in a Hilbert space (see [21], Chapter 3). In particular, they investigated the basic -Sturm-Liouville equation: where is defined on and continuous at zero. Let denotes the space of all functions such that and are continuous at zero. If , then we get the operator in the following form: for and . The operator is self adjoint on (see [21], Theorem 3.4.). A fundamental set of solutions of (10) are and . Moreover, the eigenvalues are the zeros of , where and , , and

Additionally, the corresponding set of eigenfunctions is an orthogonal basis in . Thus, we can identify with its Fourier series: where

The Sobolev Space Associated with. The next step is to recall the essential elements of the Fourier analysis presented in [22–24], as well as its applications to the spectral properties of . The space is called the space of test functions for , where

For , we introduce the Fréchet topology of by the family of norms:

The space of -distributions is the space of all linear continuous functionals on .

Thus, for , we can also define the Sobolev spaces associated to in the following form: with the norm .

For , we introduce the space defined by the norms where the -partial differential operator with respect to has the following form:

Notation: the symbol means that there exists such that , where is a constant.

3. The -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 such that they satisfy equation (14) and condition (15).

Theorem 1. We assume that . Let and . Then, there exists the generalised solution of to problem (14)-(15), and Moreover, this solution can be written in the following explicit form Proof. Existence. Since the system of eigenfunctions is a basis in (see (11)), we seek for a function in the form for each fixed . The coefficients will then be given by the Fourier coefficients formula .
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 of the series (18). Using the integrating factor and (2) and (3), we can rewrite the ODE as Form (3), (5), and (24), we get that so that which, in its turn, implies that and we conclude that But the initial conditions (16) and (22) imply that . Thus, Therefore, the solution can be written in the series form as so, also (17) is proved.
Convergence. From (1), (4), and (5), we have that for . Hence, using for , (5), (23), and (25), we get that and Hence, Since , , and, hence, by using the Plancherel identity and (27) and (28), we can conclude that and and which mean that .
Uniqueness. It only remains to prove the uniqueness of the solution. We assume the opposite; namely, that there exist functions and , which are two different solutions of problem (14)-(15). Let . Then, we have that We define . Then, the function is a solution of the following problem From (18), it follows that , that is, , and this contradiction to our assumption proves the uniqueness of the solution. The proof is complete.

4. The -Wave Equation

In this section, we will seek for a generalised function , which satisfies the following -wave equation for with the initial conditions

Theorem 2. We assume that . Let and . Then, there exists the generalised solution of problem (29)-(30): Moreover, this solution can be written in the following explicit form: where the -Mittag-Leffler function is given by (see [25] and [26], Section 7): for and , where the gamma function is defined by Proof. Existence. By repeating the arguments in the proof of Theorem 1., we have the Cauchy type problem: with the initial conditions where , and .
Then, the solution to this Cauchy type in problem (29)-(30) is given (see [25], Example 6) By using (2) and we find that By applying (8) and using (36) and (37), we get that Since (see [21], Theorem 7.12]), by using (18) and (38), it follows that solution exists and can be written as i.e., on the explicit form (34).
Convergence. Firstly, using the results in [27], Lemma 6 and in [17], Lemma 1 for the -trigonometric functions in (6), we see that and are also bounded with . Then, forms (4), (12), and (32) follow that and where are any constant which only depends on .
Next, by using (38), (39), and (40), we obtain that where .
Therefore, by using (7), (8), (30), (34), (33), and (41), we have that and Thus, and and which is means that .
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.

Acknowledgments

SS and NT were supported in parts by the MESRK (Ministry of Education and Science of the Republic of Kazakhstan) grant AP08052208.