Delta Fractional Sturm--Liouville Problems: From Discrete to Continuous

In this study, we consider delta fractional Sturm--Liouville (DFSL) initial value problems in the sense of delta Caputo and delta Riemann-Liouville (R--L) operators. One of the properties of delta fractional difference operators which makes it different from nabla counterpart is to shift its domain. This feature makes it more complex than the nabla fractional operator. We obtain sum representation of solutions for DFSL initial value problems with the help of $\mathcal{Z}-$ transformation. Moreover, we get analytical solutions of homogeneous DFSL problem within Riemann-Liouville (R--L) and Caputo sense, discrete Sturm--Liouville (DSL) problem, continuous fractional Sturm--Liouville (FSL) problem in the sense of R--L and Caputo operators, and continuous Sturm--Liouville (SL) differential problem. From this point of view, we compare all the solutions with each other. Consequently, we show that all results for these four eigenvalue problems are compatible with each other and approach to each other while the orders tends to one, i.e. $\Delta^{\mu }\left( \Delta x\left( t-\mu \right) \right)\cong D_{0^{+}}^{\mu }\left( x^{\prime }\left( t\right) \right)\cong \Delta^2x(n-1) \cong x^{\prime \prime }\left( t\right) =\lambda x\left( t\right),\ \mu\rightarrow1 $ . We support our results comparatively by tables and simulations in detail.


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The idea of a fractional derivative was first proposed in 1695 on the question of L'Hospital about the 22 meaning of the derivative d n y dx n while n = 1 2 to Leibniz. Leibniz gave that answer, "It appears that one day 23 useful consequences will be drawn from these paradoxes.". Thereafter, Leibniz and Bernoulli began  Fractional difference operators were first given by Chapman in 1911 [6] for ∆ s a n (for s ∈ R). [20] R-L fractional derivative is given as, α > 0, α ∈ R and n − 1 ≤ α < n, n ∈ N, Definition 2.

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Theorem 2. [55] t, r ∈ R + , generalized factorial function is given as here, t and r must be selected so that the gamma function is defined.

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Now, let us give the relation between gamma function and falling factorial function:

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Proof. If we take Z−transformation of both side of the equation (21) from Theorem 5, we have Now, let us take inverse Z−transformation of both side of the equation (24). For this aim firstly let 1 us make the terms in the rhs of the equation (24) suitable respectively to apply Theorem 6 i.. For the 2 first term at the rhs of the equation (24) from Theorem 6 i.; For the second term at the rhs of the equation (24) from Theorem 6 ii.; For the third term at the rhs of the equation (24) from Theorem 6 iii.; Also, if we use convolution property in Theorem 3, then we have Finally, sum representation of solution of the problem (21) − (22) is found as follows Hence, the proof is completed.
here |t| > 1, |t − 1| α |t| 1−α > |λ| due to Theorem 6. Proof. If we take Z−transformation of both side of the equation (25) from Theorem 5, we have Now, let us take inverse Z−transformation of both side of the equation (28). For this aim firstly let us 7 make the terms in the rhs of the equation (28) suitable respectively to apply Theorem 6 i..

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For the first term at the rhs of the equation (28) from Theorem 6 i.; For the second term at the rhs of the equation (28) from Theorem 6 ii.; For the third term at the rhs of the equation (28) from Theorem 6 iii.; Also, if we use convolution property in Theorem 3, then we have Finally, representation of solution of the problem (25) − (26) Hence, the proof is completed.
Analytical Solution of Sturm-Liouville Differential Problem 7 (SL) problem: t ∈ [0, b] and its analytic solution is as follows Where the domain and range of function x (t) and M-L functions must be selected to be defined.
If we consider SL problem as follows 13 14 t ∈ [0, b], then its analytical solution 15 Eigenvalues of these problems are the roots of the following boundary condition Hence, if we apply the solutions of these four problems to the boundary condition above, we can find 16 the eigenvalues of these problems for the orders µ = 0.9 and µ = 0.99 approximately in Table 6.4.1 and 17 Table 6.4.2.

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Eigenfunctions are obtained by writing these eigenvalues into the solution of these four problems.    Following, we consider DFSL problem in R-L sense and we compare its solutions with the solu-11 tions of FSL problem in R-L sense. We observe that solutions of these two problems almost matches 12 in any order µ, however, it is observed that there exists difference between the solutions of DFSL prob-13 lem in Caputo sense and FSL in R-L sense. Besides that, we compare solutions of DFSL in Caputo 14 sense with the solutions of FSL problem in Caputo sense and we observe that solutions of these two 15 problems almost matches in any order µ. Then, we compare solutions of DFSL problem in R-L and 16 Caputo sense and we observe that solutions coincide while the order µ → 1.