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

Authors: Serikbol Shaimardan; Lars-Erik Persson; Nariman Tokmagambetov
Journal: Abstract and Applied Analysis
 (2023)

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## Abstract

In this paper, we explore a generalised solution of the Cauchy problems for theq-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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## Body

## 1. Introduction

In the last decade, the theory of quantum groups andq-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 theq-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 theq-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 [18–20], 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 Lq20,1. 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 Section3 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 inq-calculus. We will always assume that 0<q<1. The q-real number αq is defined by
αq=1−qα1−q.Theq-shifted factorial is defined by
a;qn=1,n=0,1−a1−aq⋯1−aqn−1,n∈ℕ.Moreover, their natural expansions to the reals are(1)a−bqα=aαb/a;q∞qαb/a;q∞,a;qα=a;q∞aqα;q∞,a;q∞=∏i=0∞1−aqi.The Jackson’sq-difference operator Dqfx is (see, [8, 12] Section 2.1])
(2)Dqfx=fx−fqxx1−q.Theq-derivative Dq of a product of the functions f and g as defined by
(3)Dqfgx=fqxDqgx+Dqfxgx.As given in [10], two q-analogues of the exponential functions are defined by
(4)eqx=11−qx;q∞,Eqx=−1−qx;q∞.Moreover, we have that(5)Dqeqx=eqx,DqEq−x=Eq−qx,eqxEq−x=1.Due to the various types ofq-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]):
(6)cosz;q2=∑k=0∞−1kqk2z2k2kq!,sinz;q2=∑k=0∞−1kqkk+1z2k+12k+1q!,where the q-analogue of the binomial coefficients nq! is defined by
nq!=1,ifn=0,1q×2q×⋯×nq,ifn∈N.Theq-integral (or Jackson’s integral) is defined by (see [8])
(7)∫0xftdqt=1−qx∑m=0∞qmfxqm,and a more general form is given by
∫abfxdqx=∫0bfxdqx−∫0afxdqx,for 0<a<b.Theq-version of integration by parts reads
(8)∫abfxDqgxdqx=fgab−∫abgqxDqfxdqx,and if f≡1, then we get that
(9)∫abDqgxdqx=gb−ga.Theq-Sturm-Liouville Problem. Let Lq20,1 be the space of all real-valued functions defined on 0,1 such that
fLq20,1≔∫01fx2dqx1/2<∞.The spaceLq20,1 is a separable Hilbert space with the inner product:
f,g≔∫01fxgxdqx,f,g∈Lq20,1.Now, we shortly describe the study introduced by Annaby and Mansour in of a basicq-Sturm-Liouville eigenvalue problem in a Hilbert space (see [21], Chapter 3). In particular, they investigated the basic q-Sturm-Liouville equation:
−1qDq−1Dqyx+vxyx=λyx,0≤x≤1;λ∈ℂ,where v· is defined on 0,1 and continuous at zero. Let Cq,020,1 denotes the space of all functions y· such that y and Dqy are continuous at zero. If v≡0, then we get the operator L in the following form:
(10)L≔−1qDq−1Dqyx=λyx,y0=y1=0,for 0≤x≤1 and λ∈ℝ. The operator L is self adjoint on Cq,020,1∩Lq20,1 (see [21], Theorem 3.4.). A fundamental set of solutions of (10) are cosλ;q2 and sinλ;q2/λ. Moreover, the eigenvalues λkk=1∞ are the zeros of sinλk;q2, where
(11)λk=1−q−2q−2k+2μk−1/2,k=0,1,⋯,and ∑k=1∞μk<∞, 0≤μk≤1, and
(12)λ0≔1−q−2q≤λk,k=1,2,3,⋯.Additionally, the corresponding set of eigenfunctionssinλk;q2/λkk=1∞ is an orthogonal basis in Lq20,1. Thus, we can identify f∈Lq20,1 with its Fourier series:
fx≔∑k=1∞f,ϕkϕkx,where
(13)ϕkx=sinλkx;q2λk.The Sobolev Space Associated withL. 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 L. The space CL∞0,1≔⋂m=1∞DomLm is called the space of test functions for L, where
DomLm≔f∈Lq20,1:Ljf∈DomL,j=0,1,2,⋯,m−1.Forg∈CL∞0,1, we introduce the Fréchet topology of CL∞0,1 by the family of norms:
gCLm0,1≔maxi≤mLigLq20,1.The space ofL-distributions DL′0,1≔LCL∞0,1,ℝ is the space of all linear continuous functionals on CL∞0,1.Thus, fors∈ℝ, we can also define the Sobolev spaces Wq,Ls associated to L in the following form:
Wq,Ls≔f∈D′L0,1:Ls/2f∈Lq20,1,with the norm fWq,Ls≔Ls/2fLq20,1.Form∈ℕ0, we introduce the space Cqm0,1;Wq,Ls0,1 defined by the norms
uCqm0,T;Wq,Ls0,1≔∑n=0mmax0≤t≤TDq,tnut,.Wq,Ls0,1,0<T<∞,where the q-partial differential operator Dq,tut,x with respect to t has the following form:
Dq,tut,x=ut,x−uqt,x1−qt.Notation: the symbolM≾K means that there exists γ>0 such that M≤γK, where γ is a constant.


## 3. Theq-Heat Equation

We start with a study of the following Cauchy problem:(14)Dq,tut,x+Lut,x=ft,x,x∈0,1,t>0,with the initial condition
(15)u0,x=φx,x∈0,1.We say a generalised solution of the problem (14)-(15) is a function ut,x such that they satisfy equation (14) and condition (15).Theorem 1.
We assume that0<T<∞. Let φ∈Wq,L20,1 and f∈C0,T;Wq,L20,1. Then, there exists the generalised solution of u to problem (14)-(15), and
(16)u∈Cq10,T;Lq20,1∩C0,T;Wq,L20,1.
Moreover, this solution can be written in the following explicit form(17)ut,x=∑k∈ℕeq−λktφk+eq−λkt∫0tEqλkqsfks,·dqsϕkx.
Proof. Existence. Since the system of eigenfunctions ϕkk=1∞ is a basis in Lq20,1 (see (11)), we seek for a function ut,x in the form
(18)ut,x=∑k∈ℕuktϕkx,for each fixed 0<t<T<∞. The coefficients will then be given by the Fourier coefficients formula ukt=ut,·ϕk.
We can similarly expand the source function,(19)ft,x=∑k∈ℕfktϕkx,fkt=ft,·,ϕk.
From (11) and (18), we have that
Lϕkx=λkϕkx,k∈ℕ.
Hence,(20)Lut,x=∑k∈ℕ0uktλkϕkx,and
(21)Dq,tut,x=∑k∈ℕDquktϕkx.
Substituting (20) and (21) into the equation (14), we find that
(22)∑k∈ℕDqukt+λkuktϕkx=∑k∈ℕfktϕkx.
But then, due to the completeness,(23)Dqukt+λkukt=fkt,k∈ℕ,which are ODEs for the coefficients ukt of the series (18). Using the integrating factor Eqλkqt and (2) and (3), we can rewrite the ODE as
(24)Eqλkqtfkt=EqλkqtDqukt+Eqλkqtλkukt=EqλkqtDqukt+DqEqλktukt=DqEqλktukt.
Form (3), (5), and (24), we get that
∫0tDqEqλktuktdqs=∫0tEqλkqsfktfksdqs,so that
Eqλktukt=uk0+∫0tEqλkqtfksdqs,which, in its turn, implies that
ukt=uk0Eqλkt+1Eqλkt∫0tEqλkqsfksdqs,and we conclude that
ukt=eq−λktuk0+eq−λkt∫0tEqλkqsfksdqs.
But the initial conditions (16) and (22) imply that uk0=φk. Thus,
(25)ukt=eq−λktφk+eq−λkt∫0tEqλkqsfks,·dqs.
Therefore, the solutionut,x can be written in the series form as
ut,x=∑k∈ℕeq−λktφk+eq−λkt∫0tEqλkqsfks,·dqsϕkx,so, also (17) is proved.
Convergence. From (1), (4), and (5), we have that
eq−x=1−1−qx;q∞≤11+1−qx≤1,Eqqx≤Eqx,for x∈0,1. Hence, using for 0<t<T<∞, (5), (23), and (25), we get that
(26)ukt≤25eq−λktφk+∫0tEqλkqseqλktfksdqs≤φ,ϕk+∫0tfs,·,ϕkdqs≤φ,ϕk+Tmax0≤s≤Tfs,·,ϕk≤max1,Tφ,ϕk+max0≤s≤Tfs,·,ϕk≾φ,ϕk+max0≤s≤Tfs,·,ϕk,and
(27)Dqukt≤23λkukt+fkt≾26λkφ,ϕk+λkfkt,·,ϕk+λk−1λkfkt,·,ϕk≤λkφ,ϕk+1+λ0λkfkt,·,ϕk≾Lφ,ϕk+max0≤t≤TLfkt,·,ϕk.
Hence,(28)Lut,·=λkukt,ϕk≾26λkφ,ϕk+max0≤s≤Tλkfs,·,ϕk=Lφ,ϕk+max0≤s≤TLfs,·,ϕk.
Sinceφ∈Wq,L2, f∈C0,1;Wq,L2, and, hence, by using the Plancherel identity and (27) and (28), we can conclude that
ut,·Lq20,12=∑k∈ℕukt2≾3.13∑k∈ℕφ,ϕk2+max0≤s≤T∑k∈ℕfs,·,ϕk2=φWq,L20,12+fC0,T;Wq,L20,12<∞,and
Dqut,·Lq20,12=∑k∈ℕDqukt,·2≾27∑k∈ℕLφ,ϕk2+max0≤s≤T∑k∈ℕLfs,·,ϕk2=φWq,L20,12+fC0,T;Wq,L20,12<∞,and
Lut,·Lq20,12≾28φWq,L20,12+fC0,T;Wq,L20,12<∞,which mean that u∈Cq10,T;Lq20,1∩C0,T;Wq,L20,1.
Uniqueness. It only remains to prove the uniqueness of the solution. We assume the opposite; namely, that there exist functions ut,x and vt,x, which are two different solutions of problem (14)-(15). Let 0<t<T<∞. Then, we have that
Dq,tut,x+Lut,x=ft,x,0<x<1,u0,x=φx,0≤x≤1,Dq,tvt,x+Lvt,x=ft,x,0<x<1,v0,x=φx,0≤x≤1.
We defineWt,x=ut,x−vt,x. Then, the function Wt,x is a solution of the following problem
Dq,twt,x+Lwt,x=0,0<x<1,w0,x=0,0≤x≤1.
From (18), it follows that Wt,x≡0, that is, ux,t≡vx,t, and this contradiction to our assumption proves the uniqueness of the solution. The proof is complete.


## 4. Theq-Wave Equation

In this section, we will seek for a generalised functionut,x, which satisfies the following q-wave equation
(29)Dq,t2ut,x+Lut,x=ft,x,0<x<1,for 0<t<T<∞ with the initial conditions
(30)u0,x=ψx,Dq,tu0,x=ηx,0<x≤1.Theorem 2.
We assume that0<T<∞. Let ψ,η∈Wq,L20,1 and f∈Cq10,T;Wq,L20,1. Then, there exists the generalised solution of problem (29)-(30):
u∈Cq20,1;Lq20,T∩C0,T;Wq,L20,1.
Moreover, this solution can be written in the following explicit form:(31)ut,x=∑k∈ℕ0ψke2,1−λkt2;q+tηke2,2λkt2;q−1λkfk0e2,1−λkt2;q−1λk∫0te2,1λkt−q3sq2;qDq,sfksdqsϕk,where the q-Mittag-Leffler function eα,βλkt−q2sqα;q is given by (see [25] and [26], Section 7):
(32)eα,βλkt−qsqα;q=∑m=0∞λkmt−qsqmαΓqmα+β,for α,β∈ℝ and 0<s≤t<∞, where the gamma function Γqx is defined by
(33)Γqx=q,qq∞qx,qq∞1−q1−x,Γqn+1=nq!,n∈ℕ.
Proof. Existence. By repeating the arguments in the proof of Theorem 1., we have the Cauchy type problem:
(34)Dq2ukt+λkukt=fkt,k∈ℕ0,with the initial conditions
(35)uk0=ψk,Dquk0=ηk,k∈ℕ0,where fkt=ft,·ϕk, ψk=ψ·ϕk and ηk=η·ϕk.
Then, the solution to this Cauchy type in problem (29)-(30) is given (see [25], Example 6)
(36)ukt=ψke2,1−λkt2;q+tηke2,2−λkt2;q+∫0tt−qse2,2−λkt−q2sq2;qfksdqs.
By using (2) and we find that
(37)Dq,se2,1−λkt−q2sq2;q=∑k∈ℕ−λkmΓq2m+1Dq,st−q2sq2m=−∑k∈ℕ−λkmΓq2m+12mqt−q3sq2m−1=λkt−qs∑k∈ℕ−λkm−1Γq2mt−q2sq2m−2=λkt−qse2,2λkt−q2sq2;q.
By applying (8) and using (36) and (37), we get that
(38)ukt=ψke2,1−λkt2;q+tηke2,2−λkt2;q+1λk∫0tDq,se2,1−λkt−q2sq2;qfksdqs=ψke2,1−λkt2;q+tηke2,2−λkt2;q+1λkfkte2,1−λkt−q2q2;q−1λkfk0e2,1−λkt2;q−1λk∫0te2,1−λkt−q3sq2;qDq,sfksdqs.
Sincee2,1−λkt2q2;q2≡0 (see [21], Theorem 7.12]), by using (18) and (38), it follows that solution exists and can be written as
ut,x=∑k∈ℕψke2,1−λkt2;q+tηke2,2λkt2;q−1λkfk0e2,1−λkt2;q−1λk∫0te2,1−λkt−q3sq2;qDq,sfksdqsϕk,i.e., on the explicit form (34).
Convergence. Firstly, using the results in [27], Lemma 6 and in [17], Lemma 1 for the q-trigonometric functions in (6), we see that e2,2−λkt2;q and e2,1−λkt2;q are also bounded with t>0. Then, forms (4), (12), and (32) follow that
(39)e2,2−λkt2;q=∑m=0∞−λkmt2m2m+1q!≤∑m=0∞λkmT2m2m+1q!=sinλkT;q22Tλk≤C1,q2Tλk,and
(40)e2,1−λkt2;q=∑m=0∞−1mTλk2m2mq!≤cosλkT;q2≤C2,q,where C1,q,C2,q are any constant which only depends on q.
Next, by using (38), (39), and (40), we obtain that
(41)ukt≾40ψ,ϕk+η,ϕk+f0,·,ϕkλk+1λk∫0tDqfs,·,ϕkdqs≤ψ,ϕk+η,ϕk+f0,·,ϕkλk+Tλkmax0≤s≤TDqfs,·,ϕk≤Cλ0ψ,ϕk+η,ϕk+∑m=01max0≤t≤TDqmft,·,ϕk,where Cλ0≔max1,1/λ0,T/λ0.
Therefore, by using (7), (8), (30), (34), (33), and (41), we have that
(42)Dqukt=−Dquk0+∫0tDq2uksdqs≤3034η,ϕk+∫0tfs,·,ϕkdqs+λk∫0tuksdqs≾411λkλkη,ϕk+Tλkmax0≤s≤Tλkfs,·,ϕk+TCλ0λkψ,ϕk+λkη,ϕk+∑m=01max0≤t≤TλkDqmft,·,ϕk≾Lψ,ϕk+Lη,ϕk+∑m=01max0≤t≤TLDqmft,·,ϕk,and
(43)Da2ukt≾34λkukt+fkt≾41λkψ,ϕk+λkη,ϕk+∑m=01max0≤t≤TλkDqmft,·,ϕk=Lψ,ϕk+Lη,ϕk+∑m=01max0≤t≤TLDqmft,·,ϕk.
Thus,utLq20,12=∑k∈ℕukt2≾41∑k∈ℕψ,ϕk2+∑k∈ℕη,ϕk2+∑m=01max0≤t≤T∑k∈ℕDqmft,·,ϕk2=ηWq,L20,12+ψWq,L20,12+fCq10,T;Wq,L20,12<∞,DqutLq20,12=∑k∈ℕDqukt2≾∑k∈ℕLψ,ϕk2+∑k∈ℕLη,ϕk2+∑m=01max0≤t≤T∑k∈ℕLDqmft,·,ϕk2≤ηWq,L20,12+ψWq,L20,12+fCq10,T;Wq,L20,12<∞,and
Dq2utLq20,12=∑k∈ℕDq2ukt2≾ηWq,L20,12+ψWq,L20,12+fCq10,T;Wq,L20,12<∞,and
LutH2=∑k∈ILut,ϕkH2=∑k∈Iλkukt2≾ηWq,L20,12+ψWq,L20,12+fCq10,T;Wq,L20,12,which is means that u∈Cq20,1;Lq20,T∩Cq0,T;Wq,L20,1.
Uniqueness. This part can be proved completely similar as the proof of Theorem 1.. So we omit the details.



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