COMPLEXITY Complexity 1099-0526 1076-2787 Hindawi 10.1155/2017/3720471 3720471 Research Article Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems http://orcid.org/0000-0003-4249-8903 Al-Refai Mohammed 1 http://orcid.org/0000-0002-8889-3768 Abdeljawad Thabet 2 Elsadany Abdelalim 1 Department of Mathematical Sciences UAE University P.O. Box 15551 Al Ain Abu Dhabi UAE uaeu.ac.ae 2 Department of Mathematics and General Sciences Prince Sultan University P.O. Box 66833 Riyadh 11586 Saudi Arabia psu.edu.sa 2017 1492017 2017 07 05 2017 06 08 2017 1492017 2017 Copyright © 2017 Mohammed Al-Refai and Thabet Abdeljawad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.

United Arab Emirates University 31S239-UPAR(1)
1. Introduction and Preliminaries

Fractional calculus is old as the Newtonian calculus . The name fractional was given to express the integration and differentiation up to arbitrary order. Traditionally, there are two approaches to define the fractional derivative. The first approach, Riemann-Liouville approach, is to iterate the integral with respect to certain weight function and replace the iterated integral by single integral through Leibniz-Cauchy formula and then replace the factorial function by the Gamma function. In this approach, the arbitrary order Riemann-Liouville results from the integrating measure dt and the Hadamard fractional integral results from the integrating measure dt/t. The second approach, Grünwald-Letinkov approach, is to iterate the limit definition of the derivative to get a quantity with certain binomial coefficient and then fractionalize by using the Gamma function instead of the factorial in the binomial coefficient. In case of the Riemann-Liouville and Caputo fractional derivatives, a singular kernel of the form (t-s)-α is generated for 0<α<1 to reflect the nonlocality and the memory in the fractional operator. Through history, hundreds of researchers did their best to develop the theory of fractional calculus and generalize it, either by obtaining more general fractional derivatives with different kernels or by defining the fractional operator on different time scales such as the discrete fractional difference operators (see  and the references therein) and q-fractional operators (see  and the references therein).

In 2014 , Khalil et al. introduced the so-called conformable fractional derivative by modifying the limit definition of the derivative by inserting the multiple t1-α,0<α<1 inside the definition. The word fractional there was used to express the derivative of arbitrary order although no memory effect exists inside the corresponding integral inverse operator. This conformable (fractional) derivative seems to be kind of local derivative without memory. An interesting application of the conformable fractional derivative in Physics was discussed in , where it has been used to formulate an Action Principle for particles under frictional forces. Despite the many nice properties the conformable derivative has, it has the drawbacks that when α tends to zero we do not obtain the original function and the conformable integrals inverse operators are free of memory and do not have a semigroup property. It is most likely to call them conformable derivatives or local derivatives of arbitrary order. In connection with this, at the end of reference , the author asked whether it is possible to fractionalize the conformable (fractional) derivative by using conformable (fractional) integrals of order 0<α1 or by iterating the conformable derivative. The first part, Riemann-Liouville approach, was answered in [12, 13], where the author iterated the (conformable) integral with weight tρ-1, ρ0 to define generalized fractional integrals and derivatives that unify Riemann-Liouville fractional integrals (ρ=1) and derivatives together with Hadamard fractional integrals and derivatives. Actually, the limiting case of that generalization is when ρ0+ leads to Hadamard type. However, the Grünwald-Letinkov approach for conformable derivatives is still open. The conformable time-scale fractional calculus of order 0<α<1 is introduced in  and has been used to develop the fractional differentiation and fractional integration. After then, many authors got interested in this type of derivatives for their many nice behaviors [10, 1518]. Motivated by the need of some new fractional derivatives with nice properties and that can be applied to more real world modeling, some authors introduced very recently new kinds of fractional derivatives whose kernel is nonsingular. For the fractional derivatives with exponential kernels we refer to . For fractional derivatives of nonsingular Mittag-Leffler functions we refer to .

Motivated, as mentioned above, with the need of new fractional derivatives with nice properties we study in this article the eigenvalue problems of Sturm-Liouville into conformable (fractional) calculus. Recently, there are several analytical studies devoted to fractional Sturm-Liouville eigenvalue problems; see . In these studies some of the well-known results of the Sturm-Liouville problems are extended to the fractional ones with left- and right-sided fractional derivatives of Riemann-Liouville and Caputo and Riesz derivatives. These results include orthogonality and completeness of eigenfunctions and countability of the real eigenvalues. Another class of fractional eigenvalue problem with Caputo fractional derivative has been studied in  using maximum principles and method of upper and lower solutions.

For a function f:(0,)R the (conformable) fractional derivative of order 0<α1 of f at t>0 was defined by(1)Daαft=limϵ0ft+ϵt-a1-α-ftϵ,and the fractional derivative at a is defined as (Daαf)(a)=limta+(Daαf)(t). The corresponding conformable (fractional) integral of order 0<α<1 and starting from a is defined by(2)Iaαfx=axftdαt=axftt-aα-1dt.It is to be noted that the author used this modified conformable integral in order to extend it to left-right concept and confirm it by the Q-operator and obtain a left-right integration by parts version. Otherwise the integral can be given by (Iaαf)(x)=axf(t)tα-1dt. It was shown in [9, 11] that (IaαDaαf)(x)=f(x)-f(a) and (DaαIaαf)(x)=f(x). For the higher order case and other details such as the product rule, chain rule, and integration by parts, we refer the reader to [9, 11].

2. Main Results

In this paper we consider the fractional extension of the Sturm-Liouville eigenvalue problem(3)DaαpxDaαy+qxy=-λwxy,12<α1,a<x<b,where p,Daαp,q and the weight functions w are continuous on (a,b), p(x)>0, and w(x)>0, on [a,b], and the fractional derivative Daα is the conformable fractional derivative. We discuss (3) with boundary conditions(4)c1ya+c2ya=0,c12+c22>0,r1yb+r2yb=0,r12+r22>0.

We say that y is 2α-continuously differentiable on [a,b], if DaαDaαy is continuous on [a,b], and yC2α[a,b], if yC1[a,b] and is 2α-continuously differentiable on [a,b].

Let (5)Ly,α=DaαpxDaαy+qxy;then the fractional Sturm-Liouville eigenvalue problem (3) can be written as (6)Ly,α=-λwxy.The following is a generalized result of the well-known Lagrange identity.

Theorem 1 (fractional Lagrange identity).

Letting y1,y2 be 2α-continuously differentiable on [a,b], then the following holds true:(7)aby2Ly1,α-y1Ly2,αdαx=pxy2Daαy1-y1Daαy2ab.

Proof.

We have(8)y2Ly1,α-y1Ly2,α=y2DaαpxDaαy1+qxy1y2-y1DaαpxDaαy2-qxy1y2=y2DaαpxDaαy1-y1DaαpxDaαy2.Using the integration by parts formula of the conformable fractional derivative , we have(9)aby2DaαpxDaαy1-y1DaαpxDaαy2dαx=pxy2Daαy1ab-abpxDaαy1Daαy2dαx-pxy1Daαy2ab+abpxDaαy1Daαy2dαx=pxy2Daαy1-y1Daαy2ab,which proves the result.

Proposition 2.

If yC1[0,1] and y(x0)=0, for some x0[a,b], then (Daαy)(x0)=0.

Proof.

Since yC1[0,1], then (Daαy)(x)=(x-a)1-αy(x), and the result follows for a<x0b. If x0=a, we have (Daαy)(a)=limxa+(x-a)1-αy(x)=0.

Proposition 3.

Let y1 and y2 in C1[a,b], which satisfy the boundary conditions (4); then it holds that (10)pxy2Daαy1-y1Daαy2ab=0.

Proof.

Since y1C1[a,b], then Daαy1=(x-a)1-αy1(x). Similarly, Daαy2=(x-a)1-αy2(x). We have(11)pxy2Daαy1-y1Daαy2ab=pby2bDaαy1b-y1bDaαy2b-pay2aDaαy1a-y1aDaαy2a.Since c12+c22>0, and r12+r22>0, we first assume that, without loss of generality, c10 and r10, and the proof of other cases will be obtained analogously. We have (12)ya=-c2c1ya,yb=-r2r1yb.Thus,(13)y2bDaαy1b-y1bDaαy2b=-r2r1y2bDaαy1b+r2r1y1bDaαy2b=-r2r1y2bb-a1-αy1b-y1bb-a1-αy2b=0.Analogously, (14)y2aDaαy1a-y1aDaαy2a=0,which proves the result.

Definition 4.

We say that f and g are α-orthogonal with respect to the weight function μ(x)0, if (15)abμxfxgxdαx=0.

Theorem 5.

The eigenfunctions of the fractional eigenvalue problem (3)-(4) corresponding to distinct eigenvalues are α-orthogonal with respect to the weight function w(x).

Proof.

Let λ1 and λ2 be two distinct eigenvalues and y1 and y2 are the corresponding eigenfunctions. We have(16)Ly1,α=-λ1wxy1,(17)Ly2,α=-λ2wxy2.Multiplying (16) by y2 and (17) by y1 and subtracting the two equations yield(18)y2Ly1,α-y1Ly2,α=-λ1-λ2wxy1y2.Performing the fractional integral Iaα and using the fractional Lagrange identity we have(19)-λ1-λ2abwxy1y2dαx=aby2Ly1,α-y1Ly2,αdαx=pxy2Daαy1-y1Daαy2ab=0,by virtue of Proposition 3. Since λ1λ2, we have abwxy1y2dαx=0, and the result is obtained.

Theorem 6.

The eigenvalues of the fractional eigenvalue problem (3)-(4) are real.

Proof.

Let y be a solution to the fractional Sturm-Liouville eigenvalue problem (3)-(4). Taking the complex conjugate of (3)-(4) and using the fact that p(x),q(x) and w(x) are real valued functions, we have(20)Ly¯,α=DaαpxDaαy¯+qxy¯=-λwxy¯,c1y¯a+c2y¯a=0,r1y¯b+r2y¯b=0.Applying analogous steps to the proofs of Theorem 5 and Proposition 3 with y1=y and y2=y¯, we have(21)-λ-λ¯abwxyx2dαx=aby¯Ly,α-yLy¯,αdxα=pxy¯Daαy-yDaαy¯ab=0,and thus λ=λ¯ which completes the proof.

Definition 7.

Let f and g be α-differentiable; the fractional Wronskian function is defined by (22)Wαf,g=fDaαg-gDaαf.

Theorem 8.

Let y1 and y2 be 2α-continuously differentiable on [a,b], and they are linearly independent solutions of (3); then (23)Wαy1,y2=Wαy1,y2apapx.

Proof.

Applying the product rule one can easily verify that(24)DaαWαy1,y2=y1DaαDaαy2-y2DaαDaαy1.Analogously, applying the product rule to (3) yields (25)DaαDaαy=-1pDaαpDaαy+q+λwy.Substituting the last equation in (24) yields(26)DaαWαy1,y2=-y1pDaαpDaαy2+q+λwy2+y2pDaαpDaαy1+q+λwy1=Daαppy2Daαy1-y1Daαy2=-DaαppWαy1,y2.One can easily verify that the solution of the above fractional differential equation is (27)Wαy1,y2=cp,where c is constant. Now, Wα(y1,y2)(a)=c/p(a), and thus c=Wα(y1,y2)(a)p(a), and hence the result.

Theorem 9.

The eigenvalues of the fractional eigenvalue problem (3)-(4) are simple.

Proof.

Let y1 and y2 be two eigenfunctions for the same eigenvalue λ. From (18) we have(28)0=y2Ly1,α-y1Ly2,α=y2DaαpxDaαy1-y1DaαpxDaαy2=y2DaαpDaαy1+pDaαDaαy1-y1DaαpDaαy2+pDaαDaαy2=py2DaαDaαy1-y1DaαDaαy2+Daαpy2Daαy1-y1Daαy2=Daαpy2Daαy1-y1Daαy2.Thus(29)py2Daαy1-y1Daαy2=c,and since y1 and y2 satisfy the same boundary conditions, we have c=0 and (30)y2Daαy1-y1Daαy2=0.Since Wα(y1,y2)=0, and y1 and y2 are both solutions to the fractional eigenvalue problem (3)-(4), then they are linearly dependent.

Theorem 10 (fractional Rayleigh Quotient).

The eigenvalues λ of problem (3) satisfy(31)λ=abpDaαy2dαx-abqy2dαx-pyDaαyababwy2dαx

Proof.

Multiplying (3) by y and integrating yields(32)abyDaαpxDaαydαx+01qxy2dαx=-λabwxy2dαx.Integrating the first integral by parts we have(33)pyDaαyab-abpDaαy2dαx+abqxy2dαx=-λabwxy2dαxwhich proves the result.

Corollary 11.

Letting yC1[a,b] and q(x)0, then the eigenvalues of (3) associated with homogeneous boundary conditions of Dirichlet or Neumann type are nonnegative.

Proof.

Since the boundary conditions are of Dirichlet or Neumann type then it holds that (34)yDaαyab=0.Then the result is directly obtained from the fractional Rayleigh Quotient as q(x)0.

Now if y is a stationary function for(35)Jaαy=abFy,Daαy,xdxα=abFy,Daαy,xx-aα-1dx,then it holds that, see ,(36)Fyy,Daαy,x-DaαFyαy,Daαy,x=0,the fractional Euler equation. We remark here that the above equation is a necessary condition for a stationary point and not sufficient. In the following we show that the fractional Sturm-Liouville eigenvalue problem (3)-(4) is equivalent to the following:

(i) Finding the stationary function y(x) of(37)Fy=abpDaαy2-qy2x-aα-1dx,subject to G[y]=1, where(38)Gy=abwy2x-aα-1dx.

To find the stationary of F[y] subject to G[y]=1, we first find the stationary value y of K[y]=F[y]-λG[y] and then eliminate λ using G[y]=1. Applying the fractional Euler Equation (36) to K[y] yields (39)-2qy-2λwy-Daα2pDaαy=0,or (40)DaαpDaαy+qy=λwy,which is the fractional Sturm-Liouville problem. Moreover, multiplying (3) by y and integrating yields (41)abyDaαpDaαyx-aα-1dx+abqy2x-aα-1dx=-λabwy2x-aα-1dx.Performing integration by parts of the first integral yields (42)pyDaαyab-abpDaαy2x-aα-1dx+abqy2x-aα-1dx=-λabwy2x-aα-1dx.Since (43)yDaαyab=0,we have (44)λabwy2x-aα-1dx=abpDaαy2-qy2x-aα-1dx.Since abwy2x-aα-1dx=1, we have (45)λ=abpDaαy2-qy2x-aα-1dx.That is, λ is determined by F[y] in (37).

The problem in (i) is equivalent to the problem of finding the stationary function of (ii) A[y]=F[y]/G[y]. Thus the eigenvalues of the fractional Sturm-Liouville eigenvalue problem are the values given by A[y]. The proof of (i) being equivalent to (ii) is well-known in the literature and we present it here for the sake of completeness.

We have (46)δA=GδF-FδGG2,and δA=0 if and only if GδF-FδG=0, or (47)δF-FGδG=δF-AG=0,which is the same as δK.

Using the above results and the fractional Rayleigh Quotient result we have the following.

Lemma 12.

For the fractional eigenvalue problem (3)-(4) it holds that(48)λ=abpDaαy2dαx-abqy2dαxabwy2dαxand the eigenfunction y is a stationary (minimum) value of the above ratio.

Remark 13.

Assuming that the eigenvalues of (3)-(4) are ordered, λ1<λ2<λn, then the above result can be used to give an upper estimate value of the first eigenvalue λ1, by choosing arbitrary function ψ that satisfies the same boundary conditions, and computing the ratio in (48) for ψ.

3. Illustrative Examples Example 1.

Consider the fractional eigenvalue problem (3)-(4) with p=1,q=0,w=1,0<x<1 and with Dirichlet boundary condition y(0)=y(1)=0. The eigenfunctions are ϕn=sin(nπxα) and the corresponding eigenvalues are λn=n2α2π2.

In the following we apply the fractional Rayleigh Quotient to obtain lower estimates of the first eigenvalue. We start with the atrial function ψ(x)=xα-x2α, which satisfies the homogenous boundary conditions ψ(0)=ψ(1)=0. We have D0αψ=α(1-2xα), and thus(49)λ101D0αψ2xα-1dx01ψ2xα-1dx=01α21-2xα2xα-1dx01xα-x2α2xα-1dx=10α2.So, we obtain an upper estimate λ1¯=10α2, which is comparable with the exact eigenvalue λ1=π2α2. However, this upper bound can be improved by choosing a trial function (50)ψx=xα1-xα+axα1-xα2,with parameter a and then choosing a to minimize the fractional Rayleigh Quotient. Direct calculations show that (51)01D0αψ2xα-1dx=α10535+2aa+7,01ψ2xα-1dx=21+aa+9630a.Thus, the fractional Rayleigh Quotient will produce (52)FRa,α=α263010535+aa+721+aa+9.The minimum value of (53)Ra=63010535+aa+721+aa+9is 9.86975 and occurs at a=1.13314. Hence, an upper estimate λ1¯=9.86975α2 is obtained which is very close to the exact one.

Example 2.

Consider the fractional eigenvalue problem (3)-(4) with p=1,q=0,w=1,0<x<1 and with boundary condition y(0)-y(0)=0, y(1)=0. The eigenfunctions are (54)ϕn=ansinλnxα+bncosλnxα.We choose an=0, so that ϕn=λnαxα-1ancos(λnxα)-λnαxα-1bnsin(λnxα) is defined at x=0. Thus, ϕn=bncos(λnxα), and applying the boundary conditions we have ϕn=0. That is, the problem possesses no eigenfunctions for 1/2<α<1.

Remark 3.

It is well-known that the regular Sturm-Liouville eigenvalue problem with integer derivative possesses an infinite number of eigenvalues. This result is not valid for the fractional one as shown in the previous example. However, the fractional Sturm-Liouville equation in (3) can be discussed with fractional boundary conditions of the type (55)c1ya+c2Daαya=0,c12+c22>0,r1yb+r2Daαyb=0,r12+r22>0.We believe that the above fractional eigenvalue problem possesses an infinite number of eigenvalues and we left it for a future work.

4. Conclusion

We have considered a regular conformable fractional Sturm-Liouville eigenvalue problem. We proved that the eigenvalues are real and simple and the eigenfunctions are orthogonal. We also established the fractional Wronskian result for any two linearly independent solutions of the problem. We obtained a fractional Rayleigh Quotient and applied a fractional variational principle to show that the minimum value of the Quotient is obtained at an eigenfunction. This result is used to estimate the first eigenvalue and the presented example illustrates the efficiency of the result. We illustrated by an example that the existence of eigenfunctions is not guaranteed unlike the result for the regular Sturm-Liouville eigenvalue problem. Most of the obtained results are analogous for the ones of regular Sturm-Liouville eigenvalue problems and they open the door for establishing other results such as the countability of eigenfunctions and completeness of eigenfunctions which are essential in solving fractional differential equations by fractional eigenfunction expansion.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author gratefully acknowledges the support of the United Arab Emirates University under the Grant 31S239-UPAR(1) 2016.