On multi-step methods for singular fractional q-integro-differential equations

[ ]( ) ( ) = k t t k k k k Ω , , , , q 1 2 3 4 , under some boundary conditions where Ω is singular at some point ≤ ≤ t 0 1, on a time scale { } { } = = ∪ t t t q : 0 t n 0 0 , for ∈ n where ∈ t0 and ( ) ∈ q 0, 1 . We consider the compact map and avail the Lebesgue dominated theorem for finding solutions of the addressed problem. Besides, we prove the main results in context of completely continuous functions. Our attention is concentrated on fractional multi-step methods of both implicit and explicit type, for which sufficient existence conditions are investigated. Finally, we present some examples involving graphs, tables and algorithms to illustrate the validity of our theoretical findings.

Abstract: The objective of this paper is to investigate, by applying the standard Caputo fractional q-derivative of order α, the existence of solutions for the singular fractional q-integro-differential equation
In 2007, Atici and Eloe studied discrete fractional calculus and considered a family of finite fractional linear difference equations. They developed the theory of linear finite fractional difference equations analogously to the theory of finite difference equations. In [23], the fractional problem was studied, where < < r 0 1, < < σ 2 3, < < λ 0 2, σ is the Caputo fractional derivative and is a continuous function. In [26], the singular fractional problem where q α c denotes Caputo fractional q-derivative of order α and [ ] ( ) × → F : 0, 1 3 is a multivalued map with ( ) a class of all subsets of [24]. In 2019, Ntouyas et al. in [20], by applying definition of the fractional q-derivative of the Caputo-type and the fractional q-integral of the Riemann-Liouville-type, studied the existence and uniqueness of solutions for a multi-term nonlinear fractional q-integro-differential equations under some boundary conditions , function Ω is a L κ -Carathéodory, ( ) r k k k Ω , , , 1 2 3 may be singular and q σ c the fractional Caputo-type q-derivative. Furthermore, they discussed the existence of solutions for the fractional q-derivative inclusions such that − > σ β 1 [16]. Relevant results have been presented in other studies, for example [27][28][29][30][31].
In this paper and motivated by the aforementioned achievements, we investigate the singular fractional q-integro-differential equation of the form 4 is singular at some points of ( ) ∈ ≔ t J 0, 1 and q σ is the Caputo fractional q-derivative of order σ. Existence of solutions is studied via multi-step methods. We prove the main results in context of completely continuous functions and by the help of the Lebesgue dominated theorem. Examples are presented and MATLAB routines [32] are implemented to demonstrate the validity of the proposed results. The rest of the paper is organized as follows: Section 2 recalls some preliminary concepts and fundamental results of q-calculus. Sections 3 and 4 are devoted to the main results and examples illustrating the obtained results and some algorithms for the addressed problem, respectively.

Essential preliminaries
This section is devoted to starting some notations and essential preliminaries that are acting as necessary prerequisites for the results of the subsequent sections.
. Then If is a closed, bounded and convex subset of a Banach space and → Φ : is completely continuous, then Φ has a fixed point in .

Linear multi-step methods
As in the case of ordinary differential equations, linear multi-step methods for fractional differential equations makes use of approximations of values of ( ) on some points of a partition < <⋯< s s s n 0 1 [32,35]. We can therefore write linear multi-step methods for the solution of (1) in the form . Numerical methods (14) are requested to be consistent with the original problem (1), in the sense that, as → h 0, the discretized problem is expected to tend asymptotically to the continuous one [32]. In order to formally introduce the consistency concept and study order conditions, it is usually to introduce, associated with (14), the linear difference operator j n j n j n j n j τ j n j q τ C n j n j n j n j is a sufficiently smooth function [32]. The linear multi-step method (14) is said to be consistent if, for any initial value problem (1), with exact solution ( ( ) ( ) ( ) ( )) k t k t k t k t , , , , it holds with h and n related by = + t s hn 0 . Moreover, the method is said to be of order ℓ if as h tends to zero. Under the assumption that ( ( ) ( ) ( ) ( )) k t k t k t k t , , , and its τ-fractional q-derivative as In this way, we can write the difference operator

Main results
We employ the multi-step methods to prove the main results in this section. First, we adopt the following lemma.
Proof. Assume that k be a solution for the problem. By applying Lemma 2.1, we get . By utilizing the boundary conditions, we conclude Therefore, we have two cases.
(1) If ≤ t τ, then we can see that (2) If ≥ t τ, then we can see that  This implies that, ( ) Now, we give our main result.
Theorem 3.2. The singular problem (1) has a solution whenever the following assumptions hold.
(1) There exist the maps → f J : (2) There exist ∈ g and ∈ Θ 4 such that for each ( )∈ k k k k , , , 4 , almost all ∈ t J. Also Proof. We first define a map → T : by   On the other hand, we get and Similarly, one can check that T maps bounded sets into bounded sets. Let ∈ t t J , 1 2 with ≤ t t 1 2 . Then, we have  . Also, we have  By using a similar way, we conclude that | ( ) as → t t  Proof. For each ∈ k and ≥ i 1 define for all i, t and k k k k , , , 4 . By simple method, we conclude that ( ) ( ) ( ) → k t k t i and each Ω i is a regular function on J . A regular function at a point a is a function that is regular in some neighborhood of a. For each i, consider the regular fractional q-integro-differential equation 2 , here M 1 and M 2 are defined in equation (19), and = ε 0 At present, consider the set Thus, Hence, By applying the Lebesgue dominated theorem, we obtain for all ∈ t J. This completes the proof. □

Illustrative examples via computational results
In this section, we present two illustrative examples. For problems for which the analytical solution is not known, we will use, as reference solution, the numerical approximation obtained with a tiny step h by the implicit trapezoidal PI rule, which, as we will see, usually shows an excellent accuracy [32]. For this purpose, we need to present a simplified analysis that is able to execute the values of the q-Gamma function. We provided a pseudo-code description of the method for calculation of the q-Gamma function of order n in Algorithms 3, 4, 6 and 7; for more details see https://www.dm.uniba.it/members/garrappa/ software. All the experiments are carried out in MATLAB Ver. 8.5.0.197613 (R2015a) on a computer equipped with a CPU AMD Athlon(tm) II X2 245 at 2.90 GHz running under the operating system Windows 7.
Example 4.2. Consider the singular fractional q-integro-differential problem  . At first by using Eqs (16) and (19), we obtain  , respectively, according to Table 4.
13 , respectively, which are shown in Tables 5-7. Note that the value of r must be more than 13 , respectively, according to Table 8.  On multi-step methods for singular fractional q-integro-differential equations  1403 Table 8 shows numerical values of ( ) k t in equation (27). Furthermore, one can see that the curve of ( ) k t with respect to t in Table 8 (Algorithm 14)). We can see that Θ 1 , Θ 2 and g satisfy the conditions of Theorem 3.3. Thus, the problem (27) has a solution.

Conclusion
The q-integro-differential boundary equations and their applications represent a matter of high interest in the area of fractional q-calculus due to their various applications in areas of science and technology. Indeed, the q-integro-differential boundary value problems often occur in mathematical modeling of a variety of physical operations. In this context, we prove the existence of a solution for a new class of singular q-integro-differential equations (18) and (27) on a time scale. The results are verified by constructing two examples along with their numerical simulations that demonstrated perfect consistency with the theoretical findings. To this end, the authors investigated a complicated case by utilizing an appropriate basic theory which facilitates a particular interest in this paper.