FRACTIONAL APPROACH FOR DIRAC OPERATOR INVOLVING M-TRUNCATED DERIVATIVE

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Introduction
Studies related to several types of differential equations are always attracted by scientists.Because the differential equations have the speciality to model more complex natural systems.Also, the main advantage of fractional derivatives is that it allows us to achieve better results in modeling.Many fractional integral and derivatives like Liouville-Caputo, Riemann-Liouville, Hilfer, Atangana-Baleanu, Caputo-Fabrizio, etc. has been introduced and studied by scientists in [4,12,13,28].In recently, Khalil et al. has described the local derivative, which is also referred to as the conformable derivative depending on the basic limit definitions of the derivative firstly in [20].The conformable derivative is very useful in applied mathematics because it shows parallel features to the ordinary derivative like quotient of two functions and the derivative of the product.Also it enables changing of order between 0 < α ≤ 1.Because of this reason, many scientists have applied this 260 A. ERCAN derivative to their studies like [1-3, 5, 6, 8, 9, 16-18, 20].Proportional α− derivative has similar features with conformable derivative but it differs in its limit definition which presented by Katugampola [19] .It was studied in [6,7].In recently, Mfractional derivative containing a Mittag-Leffler function with one parameter has been introduced by Sousa and Oliveira in [25][26][27].Benefiting from the definition of these four local derivatives as mentioned above, M-truncated derivative is introduced by Sousa and Oliveira in [25] and it represents a generalization of the other four local derivatives because of the additional parameter inside definition.All other definitions of local derivative such as Katugampola, M-truncated derivative are adaptations of conformable derivative.In these derivatives, basic formulas such as quotient of two function, derivative of the product, chain rule, Leibniz rule etc. shares similarities with conformable derivatives.Spectral analysis of M-truncated derivative for Sturm-Liouville problem and some applications containing truncated Mittag-Leffler function are studied in [24,[29][30][31].
Dirac equation has a big importance in the modern field of atomic physics.The deepest meaning of the Dirac equation was that any relative definition of a particle necessarily includes not only the wave function of a single particle, but also multiple wave functions representing the potential of other particles.Dirac equation systems have applications in many branches of science like electrical engineering, mathematics and physics.New applications of conclusions and opinions from this topic shed light on future problems such as inverse problems of spectral theory.A first-order matrix linear differential equation whose solution is a 4-component wave function (a spinor) is so important in physics and mathematics [10,14,15].
Let L be a matrix operator defined by where V (x) is a potential function, m is the mass of a particle and y (x) denote a two component vector function y (x) = y 1 (x) y 2 (x) .Then let's consider the equation where λ is a parameter and The basic analysis of the spectral structure for the Dirac operator means that finding asymptotic behaviors of the eigenvalues, the vector-valued eigenfunctions and the norming constants and showing the reality of the eigenvalues and the ortogonality of the eigen-vector-functions, etc.This type of analysis is called a direct problem.In this article, the reality of the eigenvalues, and the orthogonality of the eigenfunctions have been shown and the asymptotic formulas for the eigenvalues, eigen-vector-functions, the normalized eigen-vector-functions and the norming constants have been obtained in terms of M-truncated derivative for Dirac system having separated boundary conditions.The studies on the direct and inverse eigenvalue problems can be viewed from [11,[21][22][23].Basic spectral features of linear differential operators including conformable derivatives, which inspired our work, were studied by [2,3,5,6,25].Authors have established an existence and uniqueness theorem for a conformable fractional Dirac system in study [2].Also they have addresses the existence of a spectral function for a singular conformable Dirac system in [3].The M-truncated derivative can be employed in studies related to eigenvalue problems and spectral analysis.It is particularly beneficial in such analyses related to Dirac operators based on fractional derivatives.Our primary reason for selecting this particular local derivative is its inclusivity of other local derivatives, owing to the presence of an additional parameter associated with the Mittag-Leffler function.Differing from the literature, our results are more comprehensive compared to other local derivatives due to the presence of the parameter associated with the Mittag-Leffler function.
The layout of this research is presented in the following way: in section 2, we present to definitions and fundamental properties of MTD.In section 3, the spectral structure of Dirac system is studied.This main part of our study involves the reality of the eigenvalues, the orthogonality of the eigenvector-functions, and asymptotic formulas for the vector-valued eigen-functions, the eigenvalues, the norming constants and their normalized forms.Section 4 prerents detailed discussion about simulation anaysis by supporting with the graphs for different values of α, γ and λ.In part 5, the remarks of main results close the paper.

Preliminaries
In this part, we assign some necessary definitions, theorems and lemmas related to MTD.Definition 1. [25] The concept of the truncated Mittag-Leffler function with a single parameter is introduced through, . (1) where i E γ (.) is the truncated Mittag-Leffler function defined in (1) for γ > 0.

Definition 3. [25]
The M-integral is defined as follows dt .
In the next section, we will analyze the Dirac systems in terms of the MTD and we are able to obtain general representations of solutions that involve parameters α and γ.Additionally, using the MTD approach, we can also present asymptotic formulas for eigen-vector-functions and eigenvalues.The general results which found in main results correspond to classical Dirac systems when α = 1 and γ = 1.

Main Results
Let us consider Dirac system containing M-tuncated derivative as follows: where i T α,γ M is MTD operator, p (x) and r (x) are continuous and real-valued functions on [0, π] , y (x) is 2α−continuously differentiable on [0, π], i T α,γ M y (x) is continuous on [0, π] .Deal with the system (3) subject to boundary conditions where a and b are real constants.
Proof.Let λ 1 = a + ib be an eigenvalue with eigenfunction y (x, λ 1 ).Since p (x) and r (x) real-valued functions, λ 2 = λ1 = a − ib is also an eigenvalue with the eigenfunctions ȳ (x, λ 2 ) .By considering Theorem 2, we have and since y(x) ̸ = 0 , we have λ = λ.□ Theorem 4. The solution of the system (3) satisfying the initial conditions (6) provides the following integral equation system, Proof.By using the variation of parameters method given in [17], we express the representation of the solutions as follow: where If we benefit from the initial conditions ( 6), it can be easily seen ( 8) and (9).□ Theorem 5.As |λ| → ∞, the estimates are provided as follows: for 0 ≤ x ≤ π where ξ (x, λ) = λΓ (γ + 1) Proof.Let us introduce by φ (x, λ) the solution of the system (3) satisfying the initial conditions (6).If the problem (3), ( 6) is considered for p (x) = r (x) ≡ 0, the solution of this problem stand for ψ (x, λ) .Therby, one can easily obtain that If the solution of the problem (3), ( 6) is applied to the transformation matrix operator, we have [22] φ (x, λ) = R (x) ψ (x, λ) in here R (x) and K (x, s) are matrices of second-order that can be continuously differentiated twice, and γ (x) and β (x) can be computed as below: for κ = 1.Thereby considering by ( 17) and ( 18), we find the formulas where K ij (x, s) are the components of the matrix K (x, s) for i, j = 1, 2 from (17).
To gain the asymptotics in (10) and (11), it is enough to integrate by parts the integrals including in (19), because of the differentiability of the functions K ij (x, s) .
In a similar manner, if we differentiate (19) in terms of λ, we obtain the asymptotics in ( 12) and ( 13).
Additionally, we demonstrate the asymptotic behaviors of the eigenvalues using the MTD approach, enabling us to observe how the formulas change as α and γ vary.□ Theorem 6.The eigenvalues of the problem outlined by equations (3) to (5) in their asymptotic forms are given as follows: where Proof.The eigenvalues of the given problem overlap with the roots of the characteristic function If we put asymptotics of the eigen-vector-functions φ 1 (π, λ) and φ 2 (π, λ) from the estimates (11) into △ (λ) , we obtain After some calculation with the aid of trigonometric functions, we reach sin λΓ (γ + 1) It is clearly seen that the equation ( 21), for large |λ|, has solutions in the form Therefore the asymptotic formula for eigenvalues is obtained in (20).□ Theorem 7. The asymptotic formula for the norming constants is given by Proof.By utilizing the asymptotic formula for eigenvalues given in (20), we can reobtain the asymptotics for eigen-vector-functions as follows: where ξ (x, To reach at the asymptotic expression for the norming constants, take in consideration the following integral Hence, the proof is completed.□ 268 A. ERCAN Theorem 8. Asymptotic expression of the normalized vector-valued eigenfunctions is given in the form, Proof.The proof can be easily seen with the help of Theorem 7. □

Illustrative Results
In the current section, the representation of the solutions y 1 (x) and y 2 (x) for Dirac equation is offered by means of MTD under different orders of α, different potentials and different values of λ.If the values of α increases while the value of γ is constant, Figure 1 (a) and (b) have showed a right-sided shift for the solution curves.If the values of γ increases while the value of α is constant, Figure 2 (a) and (b) have showed a smaller right-sided shift in the solutions than Figure 1. Figure 3 demonstrates the acting for the solutions when q = 0, 1, 2, 3. Also, the roots of the characteristic function are computed detailed under different values of α in Table 1.If one pays attention to Figure 4 (a), (b) and (c), it can be easily seen that the value of α increases which is equal to 0.1, 0.3, 0.5, respectively, the frequency of the oscillation interval decreases.That is as the value of α decreases the number of eigenvalues of considered problem increases.Thereby α is changed the mobility of the solutions curves increases and it provides important advantage in applications of spectral analysis.Lastly, in Figure 5 (a) and (b), the graphs of eigenfunctions corresponding to different eigenvalues were plotted according to the changing the value of α and γ, respectively.Also Figure 5 (b) shows that eigenfunctions overlap for different values of γ.The main purpose in drawing graphs with different values is that one can observe the behavior of representations of solutions curves for Dirac equation in light of MTD.Also the approximate eigenvalues are given for different orders of α and γ in Table 1.Assume that a = 1, b = π 4 for all figures.   1 under different order of α and γ

Conclusion
In here, we analyzed spectral structure of Dirac systems which has been studied by Levitan and Sargsjan [22] for integer order case in light of MTD.For onedimensional Dirac operator in sense of MTD, its fundamental spectral theory is given systematically and behaviours of eigen-vector-functions are observed with graphics under different orders, potentials and eigenvalues.We obtain the representations of the solutions and asymptotics for the norming constants, the eigenvalues, eigen-vector-functions, and the normalized eigen-vector-functions.To gain these important results, certain calculations like variation of parameters method, Leibniz rule, and so forth are made in sense of MTD.The most important advantage of MTD is that this definition offers the features of the integer-order calculus.MTD give us the change to examine derivatives of infinite order .Also, we give comparative analysis of the solutions by graphs with different orders α and γ, different eigenvalues and different potentials.Thereby, we observe the behaviours of the mobility of the solutions.Thus, we have supplied a large amount of spectral theory for the considered problem in terms of MTD.