
This paper is concerned with the inverse problem of determining an obstacle and the corresponding incident point sources in the Helmholtz equation from near-field scattering data. An optimization method is proposed to simultaneously recover both the obstacle and source locations. Moreover, a two-step sampling scheme with novel indicator functions is proposed to produce a good initial guess for solving the optimization problem. Theoretically, we analyze the convergence properties of the optimization method and the behaviors of the indicator functions. Several numerical examples are presented to show the effectiveness of the proposed method.
Citation: Yan Chang, Yukun Guo. Simultaneous recovery of an obstacle and its excitation sources from near-field scattering data[J]. Electronic Research Archive, 2022, 30(4): 1296-1321. doi: 10.3934/era.2022068
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This paper is concerned with the inverse problem of determining an obstacle and the corresponding incident point sources in the Helmholtz equation from near-field scattering data. An optimization method is proposed to simultaneously recover both the obstacle and source locations. Moreover, a two-step sampling scheme with novel indicator functions is proposed to produce a good initial guess for solving the optimization problem. Theoretically, we analyze the convergence properties of the optimization method and the behaviors of the indicator functions. Several numerical examples are presented to show the effectiveness of the proposed method.
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)b∇μx(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(t−1). | (1) |
Definition 2.2. [46] Falling function is defined by, α∈R
tα_=Γ(α+1)Γ(α+1−n), | (2) |
where Γ is Euler gamma function.
Definition 2.3. [46] Rising function is defined by, α∈R,
t¯α=Γ(t+α)Γ(t). | (3) |
Remark 1. Delta and nabla operators have the following properties
Δtα_=αtα−1_, | (4) |
∇t¯α=αt¯α−1. |
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)t∑s=a+1(t−ρ(s))¯μ−1x(s), t∈Na+1, | (5) |
(ii) The right defined nabla fractional sum with order μ>0 is defined by
b∇−μx(t)=1Γ(μ)b−1∑s=t(s−ρ(t))¯μ−1x(s), t∈ b−1N, | (6) |
where ρ(t)=t−1 is called backward jump operators, Na={a,a+1,...}, bN={b,b−1,...}.
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∇−(n−μ)ax(t)=∇nΓ(n−μ)t∑s=a+1(t−ρ(s))¯n−μ−1x(s), t∈Na+1, | (7) |
(ii) The nabla right fractional difference of order μ>0 is defined
b∇μx(t)=(−1)nΔn b∇−(n−μ)x(t)=(−1)nΔnΓ(n−μ)b−1∑s=t(s−ρ(t))¯n−μ−1x(s), t∈ b−1N. | (8) |
Fractional differences in (7−8) 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<μ≤1, is defined by
Δμ−x(t)=1hμt∑s=0(−1)sμ(μ−1)...(μ−s+1)s!x(t−s), t=1,...,N. | (9) |
(ii) The right defined delta fractional difference of order μ, 0<μ≤1, is defined by
Δμ+x(t)=1hμN−t∑s=0(−1)sμ(μ−1)...(μ−s+1)s!x(t+s), t=0,..,N−1. | (10) |
Fractional differences in (9−10) 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
b−1∑s=a+1u(s)∇μav(s)=b−1∑s=a+1v(s)b∇μu(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
n∑s=0u(s)Δμ−v(s)=n∑s=0v(s)Δμ+u(s). | (12) |
Definition 2.9. [20] f:Na→R, s∈ℜ, Laplace transform is defined as follows,
La{f}(s)=∞∑k=1(1−s)k−1f(a+k), |
where ℜ=C∖{1} and ℜ is called the set of regressive (complex) functions.
Definition 2.10. [20] Let f,g:Na→R, all t∈Na+1, convolution property of f and g is given by
(f∗g)(t)=t∑s=a+1f(t−ρ(s)+a)g(s), |
where ρ(s) is the backward jump function defined in [46] as
ρ(s)=s−1. |
Theorem 2.11. [20] f,g:Na→R, convolution theorem is expressed as follows,
La{f∗g}(s)=La{f}La{g}(s). |
Lemma 2.12. [20] f:Na→R, the following property is valid,
La+1{f}(s)=11−sLa{f}(s)−11−sf(a+1). |
Theorem 2.13. [20] f:Na→R, 0<μ<1, Laplace transform of nabla fractional difference
La+1{∇μaf}(s)=sμLa+1{f}(s)−1−sμ1−sf(a+1),t∈Na+1. |
Definition 2.14. [20] For |p|<1, α>0, β∈R and t∈Na, discrete Mittag-Leffler function is defined by
Ep,α,β(t,a)=∞∑k=0pk(t−a)¯αk+βΓ(αk+β+1), |
where t¯n={t(t+1)⋯(t+n−1),n∈ZΓ(t+n)Γ(t),n∈R is rising factorial function.
Theorem 2.15. [20] For |p|<1, α>0, β∈R, |1−s|<1, and |s|α>p, Laplace transform of discrete Mittag-Leffler function is as follows,
La{Ep,α,β(.,a)}(s)=sα−β−1sα−p. |
Definition 2.16. Laplace transform of f(t)∈R+, t≥0 is defined as follows,
L{f}(s)=∞∫0e−stf(t)dt. |
Theorem 2.17. For z, θ∈C,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{tθ−1Eδ,θ(λtδ)}(s)=sδ−θsδ−λ. |
Property 2.19. [28] f:Na→R, 0<μ<1, Laplace transform of fractional derivative in Caputo sense is as follows, 0<α<1,
L{CDα0+f}(s)=sαL{f}(s)−sα−1f(0). |
Property 2.20. [28] f:Na→R, 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)−I1−α0+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)b∇μx(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,b−1] and real valued, λ is the spectral parameter, t∈[a+1,b−1], x(t)∈l2[a+1,b−1]. In ℓ2(a+1,b−1), the Hilbert space of sequences of complex numbers u(a+1),...,u(b−1) with the inner product is given by,
⟨u(n),v(n)⟩=b−1∑n=a+1u(n)v(n), |
for every u∈DL1, let's define as follows
DL1={u(n), v(n)∈ℓ2(a+1,b−1):L1u(n), L1v(n)∈ℓ2(a+1,b−1)}. |
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 ℓ2(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)⟩=n∑i=0u(i)r(i), |
for every u∈DL2, let's define as follows
DL2={u(i), v(i)∈ℓ2(0,n):L2u(n), L2r(n)∈ℓ2(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,b−1], t∈[a+1,b−1].
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)b∇μv(t))+u(t)q(t)v(t), | (16) |
v(t)L1u(t)=v(t)∇μa(p(t)b∇μu(t))+v(t)q(t)u(t). | (17) |
If (16−17) is subtracted from each other
u(t)L1v(t)−v(t)L1u(t)=u(t)∇μa(p(t)b∇μv(t))−v(t)∇μa(p(t)b∇μu(t)) |
and sum operator from a+1 to b−1 to both side of the last equality is applied, we get
b−1∑s=a+1(u(s)L1v(s)−v(s)L1u(s))=b−1∑s=a+1u(s)∇μa(p(s)b∇μv(s)) | (18) |
−b−1∑s=a+1v(s)∇μa(p(s)b∇μu(s)). |
If we apply the summation by parts formula in (11) to right hand side of (18), we have
b−1∑s=a+1(u(s)L1v(s)−v(s)L1u(s))=b−1∑s=a+1p(s)b∇μv(s)b∇μu(s)−b−1∑s=a+1p(s)b∇μu(s)b∇μv(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 λα≠λβ.
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)b∇μu(t))+q(t)u(t)−λαr(t)u(t)=0,∇μa(p(t)b∇μv(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
(λα−λβ)b−1∑s=a+1r(s)u(s)v(s)=0, |
since λα≠λβ,
b−1∑s=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)⟩r≠0,
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 (19−20) 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
t∑s=0(u(s)L1v(s)−v(s)L2u(s))=t∑s=0u(s)Δμ−(p(s)Δμ+v(s))−t∑s=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
t∑s=0(u(s)L2v(s)−v(s)L2u(s))=t∑s=0p(s)Δμ+v(s)Δμ+u(s)−t∑s=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 λα≠λβ. 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
(λα−λβ)t∑s=0r(s)u(s)v(s)=0, |
since λα≠λβ,
t∑s=0r(s)u(s)v(s)=0⟨u(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,u⟩r=0. |
Since ⟨u,u⟩r≠0,
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,μ−1(t,a)−λEλ,2μ,2μ−1(t,a)] | (24) |
+c2[Eλ,2μ,2μ−1(t,a)−Eλ,2μ,μ−1(t,a)]−t∑s=a+1Eλ,2μ,2μ−1(t−ρ(s)+a)q(s)x(s), |
where |λ|<1, |1−s|<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)−1−sμ1−sc2=λLa+1{x}(s)−La+1{g}(s),=sμ(sμLa+1{x}(s)−1−sμ1−sc1)−1−sμ1−sc2=λLa+1{x}(s)−La+1{g}(s), |
=La+1{x}(s)=1−sμ1−s1s2μ−λ(sμc1+c2)−1s2μ−λLa+1{g}(s), |
from Lemma 2.12, we get
La{x}(s)=c1(sμ−λs2μ−λ)−1−ss2μ−λ(11−sLa{g}(s)−11−sg(a+1))+c2(1−sμs2μ−λ). | (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,μ−1(t,a)−λEλ,2μ,2μ−1(t,a))+c2(Eλ,2μ,2μ−1(t,a)−Eλ,2μ,μ−1(t,a))−t∑s=a+1Eλ,2μ,2μ−1(t−ρ(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,μ−1(t,0)−λEλ,2μ,2μ−1(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(1−λ)−t[(1−√λ)t(1+√λ)−(−1+√λ)(1+√λ)t], | (31) |
FSL problem:
CDμ0+(Dμ0+x(t))=λx(t), | (32) |
I1−μ0+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)=tμ−1E2μ,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)=cosht√λ, | (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 μ→1 in the solution of FSL problem and similarly, the solution of DSL problem can be obtained by taking μ→1 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 μ→1 in Figure 1 for discrete Mittag-Leffler function Ep,α,β(t,a)=1000∑k=0pk(t−a)¯αk+βΓ(αk+β+1); let λ=0.01,
Secondly, we compare the solutions of DFSL, DSL, FSL and CSL problems for discrete Mittag-Leffler function Ep,α,β(t,a)=1000∑k=0pk(t−a)¯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 μ→1 in Figure 2. Let λ=0.01,
Thirdly, we compare the solutions of DFSL problem (22−23) with different orders, different potential functions and different eigenvalues for discrete Mittag-Leffler function Ep,α,β(t,a)=1000∑k=0pk(t−a)¯αk+βΓ(αk+β+1) in the Figure 3;
Eigenvalues of DFSL problem (22−23), correspond to some specific eigenfunctions for numerical values of discrete Mittag-Leffler function Ep,α,β(t,a)=i∑k=0pk(t−a)¯αk+βΓ(αk+β+1), is given with different orders while q(t)=0 in Table 1;
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 | −3.290×10−6 | 0 |
1000 | −0.963 | −0.004 | 0 | −0.898 | −0.828 | −0.0003 | −0.394 | −3.290×10−6 | 0 |
2000 | −0.981 | −0.004 | 0 | −0.947 | −0.828 | −0.0003 | −0.548 | −3.290×10−6 | 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 | −9.59×10−7 | 0 | −0.853 | −0.0003 | 0 | −0.330 | −4.140×10−6 | 0 |
1000 | −0.478 | −9.59×10−7 | 0 | −0.887 | −0.0003 | 0 | −0.375 | −4.140×10−6 | 0 |
2000 | −0.544 | −9.59×10−7 | 0 | −0.940 | −0.0003 | 0 | −0.361 | −4.140×10−6 | 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 | −3.894×10−6 | 0 | −0.192 | −0.066 | 0 | −0.985 | −0.955 | −0.026 |
1000 | −0.335 | −3.894×10−6 | 0 | −0.197 | −0.066 | 0 | −0.989 | −0.941 | −0.026 |
2000 | −0.399 | −3.894×10−6 | 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 |
Finally, we give the solutions of DFSL problem (22−23) with different orders, different potential functions and different eigenvalues for discrete Mittag-Leffler function Ep,α,β(t,a)=100∑k=0pk(t−a)¯αk+βΓ(αk+β+1) in Tables 2–4;
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 |
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 |
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 |
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,
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 |
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 |
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 μ→1 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 μ→1. Thirdly, we compare the solutions of DFSL problem (22−23) with different orders, different potential functions and different eigenvalues in Fig. 3.
Eigenvalues of DFSL problem (22−23) corresponded to some specific eigenfunctions is given with different orders in Table 1. We give the eigenfunctions of DFSL problem (22−23) 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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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 | −3.290×10−6 | 0 |
1000 | −0.963 | −0.004 | 0 | −0.898 | −0.828 | −0.0003 | −0.394 | −3.290×10−6 | 0 |
2000 | −0.981 | −0.004 | 0 | −0.947 | −0.828 | −0.0003 | −0.548 | −3.290×10−6 | 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 | −9.59×10−7 | 0 | −0.853 | −0.0003 | 0 | −0.330 | −4.140×10−6 | 0 |
1000 | −0.478 | −9.59×10−7 | 0 | −0.887 | −0.0003 | 0 | −0.375 | −4.140×10−6 | 0 |
2000 | −0.544 | −9.59×10−7 | 0 | −0.940 | −0.0003 | 0 | −0.361 | −4.140×10−6 | 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 | −3.894×10−6 | 0 | −0.192 | −0.066 | 0 | −0.985 | −0.955 | −0.026 |
1000 | −0.335 | −3.894×10−6 | 0 | −0.197 | −0.066 | 0 | −0.989 | −0.941 | −0.026 |
2000 | −0.399 | −3.894×10−6 | 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 |
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 |
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 |
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 |
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 |
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 |
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 | −3.290×10−6 | 0 |
1000 | −0.963 | −0.004 | 0 | −0.898 | −0.828 | −0.0003 | −0.394 | −3.290×10−6 | 0 |
2000 | −0.981 | −0.004 | 0 | −0.947 | −0.828 | −0.0003 | −0.548 | −3.290×10−6 | 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 | −9.59×10−7 | 0 | −0.853 | −0.0003 | 0 | −0.330 | −4.140×10−6 | 0 |
1000 | −0.478 | −9.59×10−7 | 0 | −0.887 | −0.0003 | 0 | −0.375 | −4.140×10−6 | 0 |
2000 | −0.544 | −9.59×10−7 | 0 | −0.940 | −0.0003 | 0 | −0.361 | −4.140×10−6 | 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 | −3.894×10−6 | 0 | −0.192 | −0.066 | 0 | −0.985 | −0.955 | −0.026 |
1000 | −0.335 | −3.894×10−6 | 0 | −0.197 | −0.066 | 0 | −0.989 | −0.941 | −0.026 |
2000 | −0.399 | −3.894×10−6 | 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 |
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 |
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 |
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 |
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 |
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 |