Research article

Theory of discrete fractional Sturm–Liouville equations and visual results

  • Received: 11 March 2019 Accepted: 09 May 2019 Published: 03 June 2019
  • MSC : Primary: 34A08, 39A70, 34B24, 28A75; Secondary: 44A10

  • In this article, we study discrete fractional Sturm-Liouville (DFSL) operators within Riemann-Liouville and Grünwald-Letnikov fractional operators with both delta and nabla operators. Self-adjointness of the DFSL operator is analyzed and fundamental spectral properties are proved. Besides, we get sum representation of solutions for DFSL problem by means of Laplace transform for nabla fractional difference equations and find the analytical solutions of the problem. Moreover, the results for DFSL problem, discrete Sturm-Liouville (DSL) problem with the second order, and fractional Sturm-Liouville (FSL) problem are compared with the second order classical Sturm-Liouville (CSL) problem. We display the results comparatively by tables and figures.

    Citation: Erdal Bas, Ramazan Ozarslan. Theory of discrete fractional Sturm–Liouville equations and visual results[J]. AIMS Mathematics, 2019, 4(3): 593-612. doi: 10.3934/math.2019.3.593

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  • In this article, we study discrete fractional Sturm-Liouville (DFSL) operators within Riemann-Liouville and Grünwald-Letnikov fractional operators with both delta and nabla operators. Self-adjointness of the DFSL operator is analyzed and fundamental spectral properties are proved. Besides, we get sum representation of solutions for DFSL problem by means of Laplace transform for nabla fractional difference equations and find the analytical solutions of the problem. Moreover, the results for DFSL problem, discrete Sturm-Liouville (DSL) problem with the second order, and fractional Sturm-Liouville (FSL) problem are compared with the second order classical Sturm-Liouville (CSL) problem. We display the results comparatively by tables and figures.


    Fractional calculus is a notably attractive subject owing to having wide-ranging application areas of theoretical and applied sciences. Despite the fact that there are a large number of worthwhile mathematical works on the fractional differential calculus, there is no noteworthy parallel improvement of fractional difference calculus up to lately. This statement has shown that discrete fractional calculus has certain unforeseen hardship.

    Fractional sums and differences were obtained firstly in Diaz-Osler [1], Miller-Ross [2] and Gray and Zhang [3] and they found discrete types of fractional integrals and derivatives. Later, several authors began to touch upon discrete fractional calculus; Goodrich-Peterson [4], Baleanu et al. [5], Ahrendt et al. [6]. Nevertheless, discrete fractional calculus is a rather novel area. The first studies have been done by Atıcı et al. [7,8,9,10,11], Abdeljawad et al. [12,13,14], Mozyrska et al. [15,16,17], Anastassiou [18,19], Hein et al. [20] and Cheng et al. [21] and so forth [22,23,24,25,26].

    Self-adjoint operators have an important place in differential operators. Levitan and Sargsian [27] studied self-adjoint Sturm-Liouville differential operators and they obtained spectral properties based on self-adjointness. Also, they found representation of solutions and hence they obtained asymptotic formulas of eigenfunctions and eigenvalues. Similarly, Dehghan and Mingarelli [28,29] obtained for the first time representation of solution of fractional Sturm-Liouville problem and they obtained asymptotic formulas of eigenfunctions and eigenvalues of the problem. In this study, firstly we obtain self-adjointness of DFSL operator within nabla fractional Riemann-Liouville and delta fractional Grünwald-Letnikov operators. From this point of view, we obtain orthogonality of distinct eigenfunctions, reality of eigenvalues. In addition, we open a new gate by obtaining representation of solution of DFSL problem for researchers study in this area.

    Self-adjointness of fractional Sturm-Liouville differential operators have been proven by Bas et al. [30,31], Klimek et al. [32,33]. Variational properties of fractional Sturm-Liouville problem has been studied in [34,35]. However, self-adjointness of conformable Sturm-Liouville and DFSL with Caputo-Fabrizio operator has been proven by [36,37]. Nowadays, several studies related to Atangana-Baleanu fractional derivative and its discrete version are done [38,39,40,41,42,43,44,45].

    In this study, we consider DFSL operators within Riemann-Liouville and Grünwald-Letnikov sense, and we prove the self-adjointness, orthogonality of distinct eigenfunctions, reality of eigenvalues of DFSL operator. However, we get sum representation of solutions for DFSL equation by means Laplace transform for nabla fractional difference equations. Finally, we compare the results for the solution of DFSL problem, discrete Sturm-Liouville (DSL) problem with the second order, fractional Sturm-Liouville (FSL) problem and classical Sturm-Liouville (CSL) problem with the second order. The aim of this paper is to contribute to the theory of DFSL operator.

    We discuss DFSL equations in three different ways with;

    i) Self-adjoint (nabla left and right) Riemann-Liouville (R-L) fractional operator,

    L1x(t)=μa(p(t)bμx(t))+q(t)x(t)=λr(t)x(t), 0<μ<1,

    ii) Self-adjoint (delta left and right) Grünwald-Letnikov (G-L) fractional operator,

    L2x(t)=Δμ(p(t)Δμ+x(t))+q(t)x(t)=λr(t)x(t), 0<μ<1,

    iii)(nabla left) DFSL operator is defined by R-L fractional operator,

    L3x(t)=μa(μax(t))+q(t)x(t)=λx(t), 0<μ<1.

    Definition 2.1. [4] Delta and nabla difference operators are defined by respectively

    Δx(t)=x(t+1)x(t),x(t)=x(t)x(t1). (1)

    Definition 2.2. [46] Falling function is defined by, αR

    tα_=Γ(α+1)Γ(α+1n), (2)

    where Γ is Euler gamma function.

    Definition 2.3. [46] Rising function is defined by, αR,

    t¯α=Γ(t+α)Γ(t). (3)

    Remark 1. Delta and nabla operators have the following properties

    Δtα_=αtα1_, (4)
    t¯α=αt¯α1.

    Definition 2.4. [2,7] Fractional sum operators are defined by,

    (i) The left defined nabla fractional sum with order μ>0 is defined by

    μax(t)=1Γ(μ)ts=a+1(tρ(s))¯μ1x(s), tNa+1, (5)

    (ii) The right defined nabla fractional sum with order μ>0 is defined by

    bμx(t)=1Γ(μ)b1s=t(sρ(t))¯μ1x(s), t b1N, (6)

    where ρ(t)=t1 is called backward jump operators, Na={a,a+1,...}, bN={b,b1,...}.

    Definition 2.5. [47] Fractional difference operators are defined by,

    (i) The nabla left fractional difference of order μ>0 is defined

    μax(t)=n(nμ)ax(t)=nΓ(nμ)ts=a+1(tρ(s))¯nμ1x(s), tNa+1, (7)

    (ii) The nabla right fractional difference of order μ>0 is defined

    bμx(t)=(1)nΔn b(nμ)x(t)=(1)nΔnΓ(nμ)b1s=t(sρ(t))¯nμ1x(s), t b1N. (8)

    Fractional differences in (78) are called the Riemann-Liouville (R-L) definition of the μ-th order nabla fractional difference.

    Definition 2.6. [1,21,48] Fractional difference operators are defined by,

    (i) The left defined delta fractional difference of order μ, 0<μ1, is defined by

    Δμx(t)=1hμts=0(1)sμ(μ1)...(μs+1)s!x(ts), t=1,...,N. (9)

    (ii) The right defined delta fractional difference of order μ, 0<μ1, is defined by

    Δμ+x(t)=1hμNts=0(1)sμ(μ1)...(μs+1)s!x(t+s), t=0,..,N1. (10)

    Fractional differences in (910) are called the Grünwald-Letnikov (G-L) definition of the μ-th order delta fractional difference.

    Theorem 2.7. [47] We define the summation by parts formula for R-L fractional nabla difference operator, u is defined on bN and v is defined on Na, then

    b1s=a+1u(s)μav(s)=b1s=a+1v(s)bμu(s). (11)

    Theorem 2.8. [26,48] We define the summation by parts formula for G-L delta fractional difference operator, u, v is defined on {0,1,...,n}, then

    ns=0u(s)Δμv(s)=ns=0v(s)Δμ+u(s). (12)

    Definition 2.9. [20] f:NaR, s, Laplace transform is defined as follows,

    La{f}(s)=k=1(1s)k1f(a+k),

    where =C{1} and is called the set of regressive (complex) functions.

    Definition 2.10. [20] Let f,g:NaR, all tNa+1, convolution property of f and g is given by

    (fg)(t)=ts=a+1f(tρ(s)+a)g(s),

    where ρ(s) is the backward jump function defined in [46] as

    ρ(s)=s1.

    Theorem 2.11. [20] f,g:NaR, convolution theorem is expressed as follows,

    La{fg}(s)=La{f}La{g}(s).

    Lemma 2.12. [20] f:NaR, the following property is valid,

    La+1{f}(s)=11sLa{f}(s)11sf(a+1).

    Theorem 2.13. [20] f:NaR, 0<μ<1, Laplace transform of nabla fractional difference

    La+1{μaf}(s)=sμLa+1{f}(s)1sμ1sf(a+1),tNa+1.

    Definition 2.14. [20] For |p|<1, α>0, βR and tNa, discrete Mittag-Leffler function is defined by

    Ep,α,β(t,a)=k=0pk(ta)¯αk+βΓ(αk+β+1),

    where t¯n={t(t+1)(t+n1),nZΓ(t+n)Γ(t),nR is rising factorial function.

    Theorem 2.15. [20] For |p|<1, α>0, βR, |1s|<1, and |s|α>p, Laplace transform of discrete Mittag-Leffler function is as follows,

    La{Ep,α,β(.,a)}(s)=sαβ1sαp.

    Definition 2.16. Laplace transform of f(t)R+, t0 is defined as follows,

    L{f}(s)=0estf(t)dt.

    Theorem 2.17. For z, θC,Re(δ)>0, Mittag-Leffler function with two parameters is defined as follows

    Eδ,θ(z)=k=0zkΓ(δk+θ).

    Theorem 2.18. Laplace transform of Mittag-Leffler function is as follows

    L{tθ1Eδ,θ(λtδ)}(s)=sδθsδλ.

    Property 2.19. [28] f:NaR, 0<μ<1, Laplace transform of fractional derivative in Caputo sense is as follows, 0<α<1,

    L{CDα0+f}(s)=sαL{f}(s)sα1f(0).

    Property 2.20. [28] f:NaR, 0<μ<1, Laplace transform of left fractional derivative in Riemann-Liouville sense is as follows, 0<α<1,

    L{Dα0+f}(s)=sαL{f}(s)I1α0+f(t)|t=0,

    here Iα0+ is left fractional integral in Riemann-Liouville sense.

    We consider discrete fractional Sturm-Liouville equations in three different ways as follows:

    First Case: Self-adjoint L1 DFSL operator is defined by (nabla right and left) R-L fractional operator,

    L1x(t)=μa(p(t)bμx(t))+q(t)x(t)=λr(t)x(t), 0<μ<1, (13)

    where p(t)>0, r(t)>0, q(t) is a real valued function on [a+1,b1] and real valued, λ is the spectral parameter, t[a+1,b1], x(t)l2[a+1,b1]. In 2(a+1,b1), the Hilbert space of sequences of complex numbers u(a+1),...,u(b1) with the inner product is given by,

    u(n),v(n)=b1n=a+1u(n)v(n),

    for every uDL1, let's define as follows

    DL1={u(n), v(n)2(a+1,b1):L1u(n), L1v(n)2(a+1,b1)}.

    Second Case: Self-adjoint L2 DFSL operator is defined by(delta left and right) G-L fractional operator,

    L2x(t)=Δμ(p(t)Δμ+x(t))+q(t)x(t)=λr(t)x(t), 0<μ<1, (14)

    where p,r,λ is as defined above, q(t) is a real valued function on [0,n], t[0,n], x(t)l2[0,n]. In 2(0,n), the Hilbert space of sequences of complex numbers u(0),...,u(n) with the inner product is given by, n is a finite integer,

    u(i),r(i)=ni=0u(i)r(i),

    for every uDL2, let's define as follows

    DL2={u(i), v(i)2(0,n):L2u(n), L2r(n)2(0,n)}.

    Third Case:L3 DFSL operator is defined by (nabla left) R-L fractional operator,

    L3x(t)=μa(μax(t))+q(t)x(t)=λx(t), 0<μ<1, (15)

    p,r,λ is as defined above, q(t) is a real valued function on [a+1,b1], t[a+1,b1].

    Firstly, we consider the first case and give the following theorems and proofs;

    Theorem 3.1. DFSL operator L1 is self-adjoint.

    Proof.

    u(t)L1v(t)=u(t)μa(p(t)bμv(t))+u(t)q(t)v(t), (16)
    v(t)L1u(t)=v(t)μa(p(t)bμu(t))+v(t)q(t)u(t). (17)

    If (1617) is subtracted from each other

    u(t)L1v(t)v(t)L1u(t)=u(t)μa(p(t)bμv(t))v(t)μa(p(t)bμu(t))

    and sum operator from a+1 to b1 to both side of the last equality is applied, we get

    b1s=a+1(u(s)L1v(s)v(s)L1u(s))=b1s=a+1u(s)μa(p(s)bμv(s)) (18)
    b1s=a+1v(s)μa(p(s)bμu(s)).

    If we apply the summation by parts formula in (11) to right hand side of (18), we have

    b1s=a+1(u(s)L1v(s)v(s)L1u(s))=b1s=a+1p(s)bμv(s)bμu(s)b1s=a+1p(s)bμu(s)bμv(s)=0,
    L1u,v=u,L1v.

    Hence, the proof completes.

    Theorem 3.2. Two eigenfunctions, u(t,λα) and v(t,λβ), of the equation (13) are orthogonal as λαλβ.

    Proof. Let λα and λβ are two different eigenvalues corresponds to eigenfunctions u(t) and v(t) respectively for the the equation (13),

    μa(p(t)bμu(t))+q(t)u(t)λαr(t)u(t)=0,μa(p(t)bμv(t))+q(t)v(t)λβr(t)v(t)=0.

    If we multiply last two equations by v(t) and u(t) respectively, subtract from each other and apply definite sum operator, owing to the self-adjointness of the operator L1, we have

    (λαλβ)b1s=a+1r(s)u(s)v(s)=0,

    since λαλβ,

    b1s=a+1r(s)u(s)v(s)=0,u(t),v(t)=0.

    Hence, the proof completes.

    Theorem 3.3. All eigenvalues of the equation (13) are real.

    Proof. Let λ=α+iβ, owing to the self-adjointness of the operator L1, we can write

    L1u(t),u(t)=u(t),L1u(t),λru(t),u(t)=u(t),λr(t)u(t),
    (λ¯λ)u(t),u(t)r=0.

    Since u(t),u(t)r0,

    λ=¯λ

    and hence β=0. The proof completes.

    Secondly, we consider the second case and give the following theorems and proofs;

    Theorem 3.4. DFSL operator L2 is self-adjoint.

    Proof.

    u(t)L2v(t)=u(t)Δμ(p(t)Δμ+v(t))+u(t)q(t)v(t), (19)
    v(t)L2u(t)=v(t)Δμ(p(t)Δμ+u(t))+v(t)q(t)u(t). (20)

    If (1920) is subtracted from each other

    u(t)L2v(t)v(t)L2u(t)=u(t)Δμ(p(t)Δμ+v(t))v(t)Δμ(p(t)Δμ+u(t))

    and definite sum operator from 0 to t to both side of the last equality is applied, we have

    ts=0(u(s)L1v(s)v(s)L2u(s))=ts=0u(s)Δμ(p(s)Δμ+v(s))ts=0v(s)Δμ(p(s)Δμ+u(s)). (21)

    If we apply the summation by parts formula in (12) to r.h.s. of (21), we get

    ts=0(u(s)L2v(s)v(s)L2u(s))=ts=0p(s)Δμ+v(s)Δμ+u(s)ts=0p(s)Δμ+u(s)Δμ+v(s)=0,
    L2u,v=u,L2v.

    Hence, the proof completes.

    Theorem 3.5. Two eigenfunctions, u(t,λα) and v(t,λβ), of the equation (14) are orthogonal as λαλβ. orthogonal.

    Proof. Let λα and λβ are two different eigenvalues corresponds to eigenfunctions u(t) and v(t) respectively for the the equation (14),

    Δμ(p(t)Δμ+u(t))+q(t)u(t)λαr(t)u(t)=0,Δμ(p(t)Δμ+v(t))+q(t)v(t)λβr(t)v(t)=0.

    If we multiply last two equations to v(t) and u(t) respectively, subtract from each other and apply definite sum operator, owing to the self-adjointness of the operator L2, we get

    (λαλβ)ts=0r(s)u(s)v(s)=0,

    since λαλβ,

    ts=0r(s)u(s)v(s)=0u(t),v(t)=0.

    So, the eigenfunctions are orthogonal. The proof completes.

    Theorem 3.6. All eigenvalues of the equation (14) are real.

    Proof. Let λ=α+iβ, owing to the self-adjointness of the operator L2

    L2u(t),u(t)=u(t),L2u(t),λr(t)u(t),u(t)=u(t),λr(t)u(t),
    (λ¯λ)u,ur=0.

    Since u,ur0,

    λ=¯λ,

    and hence β=0. The proof completes.

    Now, we consider the third case and give the following theorem and proof;

    Theorem 3.7.

    L3x(t)=μa(μax(t))+q(t)x(t)=λx(t),0<μ<1, (22)
    x(a+1)=c1,μax(a+1)=c2, (23)

    where p(t)>0, r(t)>0, q(t) is defined and real valued, λ is the spectral parameter. The sum representation of solution of the problem (22)(23) is found as follows,

    x(t)=c1[(1+q(a+1))Eλ,2μ,μ1(t,a)λEλ,2μ,2μ1(t,a)] (24)
    +c2[Eλ,2μ,2μ1(t,a)Eλ,2μ,μ1(t,a)]ts=a+1Eλ,2μ,2μ1(tρ(s)+a)q(s)x(s),

    where |λ|<1, |1s|<1, and |s|α>λ from Theorem 2.15.

    Proof. Let's use the Laplace transform of both side of the equation (22) by Theorem 2.13, and let q(t)x(t)=g(t),

    La+1{μa(μax)}(s)+La+1{g}(s)=λLa+1{x}(s),=sμLa+1{μax}(s)1sμ1sc2=λLa+1{x}(s)La+1{g}(s),=sμ(sμLa+1{x}(s)1sμ1sc1)1sμ1sc2=λLa+1{x}(s)La+1{g}(s),
    =La+1{x}(s)=1sμ1s1s2μλ(sμc1+c2)1s2μλLa+1{g}(s),

    from Lemma 2.12, we get

    La{x}(s)=c1(sμλs2μλ)1ss2μλ(11sLa{g}(s)11sg(a+1))+c2(1sμs2μλ). (25)

    Applying inverse Laplace transform to the equation (25), then we get representation of solution of the problem (22)(23),

    x(t)=c1((1+q(a+1))Eλ,2μ,μ1(t,a)λEλ,2μ,2μ1(t,a))+c2(Eλ,2μ,2μ1(t,a)Eλ,2μ,μ1(t,a))ts=a+1Eλ,2μ,2μ1(tρ(s)+a)q(s)x(s).

    Now, let us consider comparatively discrete fractional Sturm-Liouville (DFSL) problem, discrete Sturm-Liouville (DSL) problem, fractional Sturm-Liouville (FSL) problem and classical Sturm-Liouville (CSL) problem respectively as follows by taking q(t)=0,

    DFSL problem:

    μ0(μ0x(t))=λx(t), (26)
    x(1)=1, μax(1)=0, (27)

    and its analytic solution is as follows by the help of Laplace transform in Lemma 2.12

    x(t)=Eλ,2μ,μ1(t,0)λEλ,2μ,2μ1(t,0), (28)

    DSL problem:

    2x(t)=λx(t), (29)
    x(1)=1, x(1)=0, (30)

    and its analytic solution is as follows

    x(t)=12(1λ)t[(1λ)t(1+λ)(1+λ)(1+λ)t], (31)

    FSL problem:

     CDμ0+(Dμ0+x(t))=λx(t), (32)
    I1μ0+x(t)|t=0=1, Dμ0+x(t)|t=0=0, (33)

    and its analytic solution is as follows by the help of Laplace transform in Property 2.19 and 2.20

    x(t)=tμ1E2μ,μ(λt2μ), (34)

    CSL problem:

    x(t)=λx(t), (35)
    x(0)=1, x(0)=0, (36)

    and its analytic solution is as follows

    x(t)=coshtλ, (37)

    where the domain and range of function x(t) and Mittag-Leffler functions must be well defined. Note that we may show the solution of CSL problem can be obtained by taking μ1 in the solution of FSL problem and similarly, the solution of DSL problem can be obtained by taking μ1 in the solution of DFSL problem.

    Firstly, we compare the solutions of DFSL and DSL problems and from here we show that the solutions of DFSL problem converge to the solutions of DSL problem as μ1 in Figure 1 for discrete Mittag-Leffler function Ep,α,β(t,a)=1000k=0pk(ta)¯αk+βΓ(αk+β+1); let λ=0.01,

    Figure 1.  Comparison of solutions of DFSL–DSL problems.

    Secondly, we compare the solutions of DFSL, DSL, FSL and CSL problems for discrete Mittag-Leffler function Ep,α,β(t,a)=1000k=0pk(ta)¯αk+βΓ(αk+β+1). At first view, we observe the solution of DSL and CSL problems almost coincide in any order μ, and we observe the solutions of DFSL and FSL problem almost coincide in any order μ. However, we observe that all of the solutions of DFSL, DSL, FSL and CSL problems almost coincide to each other as μ1 in Figure 2. Let λ=0.01,

    Figure 2.  Comparison of solutions of DFSL–DSL–CSL–SL problems.

    Thirdly, we compare the solutions of DFSL problem (2223) with different orders, different potential functions and different eigenvalues for discrete Mittag-Leffler function Ep,α,β(t,a)=1000k=0pk(ta)¯αk+βΓ(αk+β+1) in the Figure 3;

    Figure 3.  Analysis of solutions of DFSL problem.

    Eigenvalues of DFSL problem (2223), correspond to some specific eigenfunctions for numerical values of discrete Mittag-Leffler function Ep,α,β(t,a)=ik=0pk(ta)¯αk+βΓ(αk+β+1), is given with different orders while q(t)=0 in Table 1;

    Table 1.  Approximations to three eigenvalues of the problem (22–23).
    i λ1,i λ2,i λ3,i λ1,i λ2,i λ3,i λ1,i λ2,i λ3,i
    750 0.992 0.982 0.057 0.986 0.941 0.027 0.483 0.483 0
    1000 0.989 0.977 0.057 0.990 0.954 0.027 0.559 0.435 0
    2000 0.996 0.990 0.057 0.995 0.978 0.027 0.654 0.435 0
    x(5),μ=0.5 x(10),μ=0.9 x(2000),μ=0.1
    i λ1,i λ2,i λ3,i λ1,i λ2,i λ3,i λ1,i λ2,i λ3,i
    750 0.951 0.004 0 0.868 0.793 0.0003 0.190 3.290×106 0
    1000 0.963 0.004 0 0.898 0.828 0.0003 0.394 3.290×106 0
    2000 0.981 0.004 0 0.947 0.828 0.0003 0.548 3.290×106 0
    x(20),μ=0.5 x(100),μ=0.9 x(1000),μ=0.7
    i λ1,i λ2,i λ3,i λ1,i λ2,i λ3,i λ1,i λ2,i λ3,i
    750 0.414 9.59×107 0 0.853 0.0003 0 0.330 4.140×106 0
    1000 0.478 9.59×107 0 0.887 0.0003 0 0.375 4.140×106 0
    2000 0.544 9.59×107 0 0.940 0.0003 0 0.361 4.140×106 0
    x(1000),μ=0.3 x(100),μ=0.8 x(1000),μ=0.9
    i λ1,i λ2,i λ3,i λ1,i λ2,i λ3,i λ1,i λ2,i λ3,i
    750 0.303 3.894×106 0 0.192 0.066 0 0.985 0.955 0.026
    1000 0.335 3.894×106 0 0.197 0.066 0 0.989 0.941 0.026
    2000 0.399 3.894×106 0 0.289 0.066 0 0.994 0.918 0.026
    x(1000),μ=0.8 x(2000),μ=0.6 x(10),μ=0.83

     | Show Table
    DownLoad: CSV

    Finally, we give the solutions of DFSL problem (2223) with different orders, different potential functions and different eigenvalues for discrete Mittag-Leffler function Ep,α,β(t,a)=100k=0pk(ta)¯αk+βΓ(αk+β+1) in Tables 24;

    Table 2.  q(t)=0,λ=0.2.
    x(t) μ=0.1 μ=0.2 μ=0.5 μ=0.7 μ=0.9
    x(1) 1 1 1 1 1
    x(2) 0.125 0.25 0.625 0.875 1.125
    x(3) 0.075 0.174 0.624 1.050 1.575
    x(5) 0.045 0.128 0.830 1.968 4.000
    x(7) 0.0336 0.111 1.228 4.079 11.203
    x(9) 0.0274 0.103 1.878 8.657 31.941
    x(12) 0.022 0.098 3.622 27.05 154.56
    x(15) 0.0187 0.0962 7.045 84.75 748.56
    x(16) 0.0178 0.0961 8.800 124.04 1266.5
    x(18) 0.0164 0.0964 13.737 265.70 3625.6
    x(20) 0.0152 0.0972 21.455 569.16 10378.8

     | Show Table
    DownLoad: CSV
    Table 3.  λ=0.01,μ=0.45.
    x(t) q(t)=1 q(t)=t q(t)=t
    x(1) 1 1 1
    x(2) 0.2261 0.1505 0.1871
    x(3) 0.1138 0.0481 0.0767
    x(5) 0.0518 0.0110 0.0252
    x(7) 0.0318 0.0043 0.0123
    x(9) 0.0223 0.0021 0.0072
    x(12) 0.0150 0.0010 0.0039
    x(15) 0.0110 0.0005 0.0025
    x(16) 0.0101 0.0004 0.0022
    x(18) 0.0086 0.0003 0.0017
    x(20) 0.0075 0.0002 0.0014

     | Show Table
    DownLoad: CSV
    Table 4.  λ=0.01,μ=0.5.
    x(t) q(t)=1 q(t)=t q(t)=t
    x(1) 1 1 1
    x(2) 0.2261 0.1505 0.1871
    x(3) 0.1138 0.0481 0.0767
    x(5) 0.0518 0.0110 0.0252
    x(7) 0.0318 0.0043 0.0123
    x(9) 0.0223 0.0021 0.0072
    x(12) 0.0150 0.0010 0.0039
    x(15) 0.0110 0.0005 0.0025
    x(16) 0.0101 0.0004 0.0022
    x(18) 0.0086 0.0003 0.0017
    x(20) 0.0075 0.0002 0.0014

     | Show Table
    DownLoad: CSV

    Now, let's consider the problems together DFSL (26)(27), DSL (29)(30), FSL (32)(33) and CSL (35)(36). Eigenvalues of these problems are the roots of the following equation

    x(35)=0.

    Thus, if we apply the solutions (28), (31), (34) and (37) of these four problems to the equation above respectively, we can find the eigenvalues of these problems for the orders μ=0.9 and μ=0.99 respectively in Table 5, and Table 6,

    Table 5.  μ=0.9.
    λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8 λ9 λ10
    DFSL 0.904 0.859 0.811 0.262 0.157 0.079 0.029 0.003 0.982
    FSL 0.497 0.383 0.283 0.196 0.124 0.066 0.026 0.003 0 ...
    DSL 1.450 0.689 0.469 0.310 0.194 0.112 0.055 0.019 0.002
    CSL 0.163 0.128 0.098 0.072 0.050 0.032 0.008 0.002 0

     | Show Table
    DownLoad: CSV
    Table 6.  μ=0.99.
    λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8 λ9 λ10
    DFSL 0.866 0.813 0.200 0.115 0.057 0.020 0.002 0 0.982
    FSL 0.456 0.343 0.246 0.165 0.100 0.051 0.018 0.002 0 ...
    DSL 1.450 0.689 0.469 0.310 0.194 0.112 0.055 0.019 0.002 ...
    CSL 0.163 0.128 0.098 0.072 0.050 0.032 0.008 0.002 0

     | Show Table
    DownLoad: CSV

    In here, we observe that these four problems have real eigenvalues under different orders μ=0.9 and μ=0.99, hence we can find eigenfunctions putting these eigenvalues into the four solutions. Furthermore, as the order changes, we can see that eigenvalues change for DFSL problems.

    We consider firstly discrete fractional Sturm-Liouville (DFSL) operators with nabla Riemann-Liouville and delta Grünwald-Letnikov fractional operators and we prove self-adjointness of the DFSL operator and fundamental spectral properties. However, we analyze DFSL problem, discrete Sturm-Liouville (DSL) problem, fractional Sturm-Liouville (FSL) problem and classical Sturm-Liouville (CSL) problem by taking q(t)=0 in applications. Firstly, we compare the solutions of DFSL and DSL problems and we observe that the solutions of DFSL problem converge to the solutions of DSL problem when μ1 in Fig. 1. Secondly, we compare the solutions of DFSL, DSL, FSL and CSL problems in Fig. 2. At first view, we observe the solutions of DSL and CSL problems almost coincide with any order μ, and we observe the solutions of DFSL and FSL problem almost coincide with any order μ. However, we observe that all of solutions of DFSL, DSL, FSL and CSL problems almost coincide with each other as μ1. Thirdly, we compare the solutions of DFSL problem (2223) with different orders, different potential functions and different eigenvalues in Fig. 3.

    Eigenvalues of DFSL problem (2223) corresponded to some specific eigenfunctions is given with different orders in Table 1. We give the eigenfunctions of DFSL problem (2223) with different orders, different potential functions and different eigenvalues in Table 2, Table 3 and Table 4.

    In Section 4.1, we consider DFSL, DSL, FSL and CSL problems together and thus, we can compare the eigenvalues of these four problems in Table 5 and Table 6 for different values of μ. We observe that these four problems have real eigenvalues under different values of μ, from here we can find eigenfunctions corresponding eigenvalues. Moreover, when the order change, eigenvalues change for DFSL problems.

    Consequently, important results in spectral theory are given for discrete Sturm-Liouville problems. These results will lead to open gates for the researchers studied in this area. Especially, representation of solution will be practicable for future studies. It worths noting that visual results both will enable to be understood clearly by readers and verify the results to the integer order discrete case while the order approaches to one.

    This paper includes a part of Ph.D. thesis data of Ramazan OZARSLAN.

    The authors declare no conflict of interest.



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