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Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems
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.
1. Introduction and Preliminaries
Fractional calculus is old as the Newtonian calculus [1–3]. 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 and the Hadamard fractional integral results from the integrating measure . 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 is generated for 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 [4–7] and the references therein) and -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 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 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 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 , to define generalized fractional integrals and derivatives that unify Riemann-Liouville fractional integrals () and derivatives together with Hadamard fractional integrals and derivatives. Actually, the limiting case of that generalization is when leads to Hadamard type. However, the Grünwald-Letinkov approach for conformable derivatives is still open. The conformable time-scale fractional calculus of order 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, 15–18]. 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 [20–22].
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 [23–27]. 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 the (conformable) fractional derivative of order of at was defined byand the fractional derivative at is defined as . The corresponding conformable (fractional) integral of order and starting from is defined byIt 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 -operator and obtain a left-right integration by parts version. Otherwise the integral can be given by . It was shown in [9, 11] that and . 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 problemwhere and the weight functions are continuous on , , and , on , and the fractional derivative is the conformable fractional derivative. We discuss (3) with boundary conditions
We say that is -continuously differentiable on , if is continuous on , and , if and is -continuously differentiable on .
Let then the fractional Sturm-Liouville eigenvalue problem (3) can be written as The following is a generalized result of the well-known Lagrange identity.
Theorem 1 (fractional Lagrange identity). Letting be -continuously differentiable on , then the following holds true:
Proof. We haveUsing the integration by parts formula of the conformable fractional derivative , we havewhich proves the result.
Proposition 2. If and , for some , then .
Proof. Since , then , and the result follows for . If , we have .
Proposition 3. Let and in , which satisfy the boundary conditions (4); then it holds that
Proof. Since , then . Similarly, . We haveSince , and , we first assume that, without loss of generality, and , and the proof of other cases will be obtained analogously. We have Thus,Analogously, which proves the result.
Definition 4. We say that and are -orthogonal with respect to the weight function , if
Proof. Let and be two distinct eigenvalues and and are the corresponding eigenfunctions. We haveMultiplying (16) by and (17) by and subtracting the two equations yieldPerforming the fractional integral and using the fractional Lagrange identity we haveby virtue of Proposition 3. Since , we have , and the result is obtained.
Proof. Let be a solution to the fractional Sturm-Liouville eigenvalue problem (3)-(4). Taking the complex conjugate of (3)-(4) and using the fact that and are real valued functions, we haveApplying analogous steps to the proofs of Theorem 5 and Proposition 3 with and , we haveand thus which completes the proof.
Definition 7. Let and be -differentiable; the fractional Wronskian function is defined by
Theorem 8. Let and be -continuously differentiable on , and they are linearly independent solutions of (3); then
Proof. Applying the product rule one can easily verify thatAnalogously, applying the product rule to (3) yields Substituting the last equation in (24) yieldsOne can easily verify that the solution of the above fractional differential equation is where is constant. Now, , and thus , and hence the result.
Proof. Let and be two eigenfunctions for the same eigenvalue . From (18) we haveThusand since and satisfy the same boundary conditions, we have and Since , and and are both solutions to the fractional eigenvalue problem (3)-(4), then they are linearly dependent.
Theorem 10 (fractional Rayleigh Quotient). The eigenvalues of problem (3) satisfy
Proof. Multiplying (3) by and integrating yieldsIntegrating the first integral by parts we havewhich proves the result.
Corollary 11. Letting and , 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 Then the result is directly obtained from the fractional Rayleigh Quotient as .
Now if is a stationary function forthen it holds that, see ,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 ofsubject to , where
To find the stationary of subject to , we first find the stationary value of and then eliminate using . Applying the fractional Euler Equation (36) to yields or which is the fractional Sturm-Liouville problem. Moreover, multiplying (3) by and integrating yields Performing integration by parts of the first integral yields Since we have Since , we have That is, is determined by in (37).
The problem in (i) is equivalent to the problem of finding the stationary function of (ii) . Thus the eigenvalues of the fractional Sturm-Liouville eigenvalue problem are the values given by . 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 and if and only if , or which is the same as .
Using the above results and the fractional Rayleigh Quotient result we have the following.
Remark 13. Assuming that the eigenvalues of (3)-(4) are ordered, , then the above result can be used to give an upper estimate value of the first eigenvalue , by choosing arbitrary function that satisfies the same boundary conditions, and computing the ratio in (48) for
3. Illustrative Examples
Example 1. Consider the fractional eigenvalue problem (3)-(4) with and with Dirichlet boundary condition . The eigenfunctions are and the corresponding eigenvalues are .
In the following we apply the fractional Rayleigh Quotient to obtain lower estimates of the first eigenvalue. We start with the atrial function , which satisfies the homogenous boundary conditions . We have , and thusSo, we obtain an upper estimate , which is comparable with the exact eigenvalue . However, this upper bound can be improved by choosing a trial function with parameter and then choosing to minimize the fractional Rayleigh Quotient. Direct calculations show that Thus, the fractional Rayleigh Quotient will produce The minimum value of is and occurs at . Hence, an upper estimate is obtained which is very close to the exact one.
Example 2. Consider the fractional eigenvalue problem (3)-(4) with and with boundary condition , . The eigenfunctions are We choose , so that is defined at . Thus, , and applying the boundary conditions we have . That is, the problem possesses no eigenfunctions for .
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 We believe that the above fractional eigenvalue problem possesses an infinite number of eigenvalues and we left it for a future work.
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.
The first author gratefully acknowledges the support of the United Arab Emirates University under the Grant 31S239-UPAR(1) 2016.
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