To investigate a class of multi-singular pointwise defined fractional q –integro-di ff erential equation with applications

: In the research work, we discuss a multi-singular pointwise defined fractional q –integro-di ff erential equation under some boundary conditions via the Riemann-Liouville q –integral and Caputo fractional q –derivatives. New existence results rely on the α -admissible map and fixed point theorem for α - ψ -contraction map. At the end, we present an example with application and some algorithms to illustrate the primary e ff ects.

Using the ideas from these works, we investigate the existence of solutions for the following nonlinear pointwise defined fractional q-integro-differential equation: for some non-negative real numbers c 1 and c 2 belonging to [0, ∞) and all u 1 , u 2 ∈B, where D α q and D β q are the Caputo fractional q-derivatives of order α and β, respectively, which are defined in (2.11), and w ∈L is singular at some points t ∈J.
In fact, the non-constant real-valued function u on the interval I = [a, b] is said to be singular on I, if it is continuous, and there exists a set S ⊆ I of measure 0 such that for all t outside of S , u ′ (t) exists, and it is zero, that is, the derivative of u vanish almost everywhere. We say that, D α q u(t) + g(t) = 0 is a pointwise defined equation onJ if there exists set S ⊂J such that the measure of S c is zero, and the equation holds on S [44].
In Section 2, we recall some essential definitions of Caputo fractional q-derivative. Section 3 contains our main results of this work, while an example is presented to support the validity of our obtained results. An application with some needed algorithms for the problems are given in Section 4. In Section 5, conclusion is presented.

Basic definitions for the problem
Throughout the paper, we apply the notations of time scales calculus [12]. The Caputo fractional q-derivative is considered here on for all ℵ ∈ N, s 0 ∈ R and q ∈ J. If there is no confusion concerning s 0 , we denote T s 0 by T.
. Also, for σ ∈ R, we have: In [10], the authors proved that (v − w) (σ+ν) The q-Gamma function is given by where v ∈ R\{· · · , −2, −1, 0} [25]. In fact, by using (2.2), we have for all t ∈ T\{0}, and D q [u](0) = lim t→0 D q [u](t) [2]. Also, the higher order q-derivative of the function [2]. In fact, Remark 2.1. By using Eq (2.1), we can change Eq (2.5) into the following: (2.6) The q-integral of the function u is defined by for 0 ≤ t ≤ b, provided that the series is absolutely convergent [2]. If a is in whenever the series converges. The operator I n q is given by I 0 q [u](t) = u(t) and for n ≥ 1 and u ∈ C([0, b]) [2]. It has been proven that whenever the function u is continuous at t = 0 [2]. The fractional Riemann-Liouville type q-integral of the function u is defined by Therefore, we have: The Caputo fractional q-derivative of the function u is defined by for t ∈J and σ > 0 [23,35]. It has been proven that where σ, ν ≥ 0 [23]. Also, where σ > 0 and n ≥ 1 [23].
Thus, we have: 1 − q i+1 q m u tq k+m . (2.12) The authors in [41] presented all algorithms and MATLAB code's lines to simplify q-factorial , and some necessary equations.

Main results
Let us first prove the following essential lemma: Lemma 3.1. Suppose that α ≥ 2, q ∈ J and g ∈L. The solution of the boundary value problem: with boundary conditions is expressed as: on a time scale T t 0 where G q (t, s) is expressed as:

3)
and where c 0 , c 1 are some real numbers, and I α q is Riemann-Liouville type q-integral of order α.
Note that, the mappings G q (t, s) and ∂G q (t,s) ∂t are continuous with respect to t. Let w be a map onJ ×B 2 such that w is singular at some points ofJ. Let us define the function Θ u :B →B by for all t ∈J, where I α q is the fractional Riemann-Liouville q-integral of order α which is defined in (2.9), and D β q is the Caputo fractional q-derivative of order β which is defined in (2.11). Then, by taking the first order derivative related to t, we have: Obviously, the singular pointwise defined Eq (1.1) has a solution iff the map Θ u has a fixed point.
If u E or v E, then the last inequality holds obviously. This shows that for all u, v ∈B. Now, Lemma 2.6 implies that Θ has a fixed point that is the solution for problem (1.1). □

An illustrative example with application
The following illustrative example is given to support the validity of our main results. A computational method is provided here to test the proposed problem (1.1). Linear motion is commonly basic among all other motions. From the 1st law of Newton's motion, objects that are not experiencing any net force will continue to move in a straight line with a constant velocity until they are subjected to a net force. Example 4.1. We consider a constrained motion of a particle along a straight line restrained by two linear springs with equal spring constant (stiffness coefficient) under external force and fractional damping along the t-axis (Figure 1). L F Figure 1. A particle along a straight line restrained by two linear springs with equal spring constant.
We put: , .   ∥μ i ∥J p i ℓ γ i = M α,a,b × 2 × 0.01 × 2 1 = 0.04M α,a,b < 1. Table 2 shows numerical results for different values of q ∈ J. Figure 2 shows the curve of these results. Now, according to the obtained results, Theorem 3.3 implies that problem (4.2) has a solution.

Conclusions
The multi-singular pointwise defined fractional q-integro-differential equation has been successfully investigated in this work. The investigation of this particular equation provides us with a powerful tool in modeling most scientific phenomena without the need to remove most parameters which have an essential role in the physical interpretation of the studied phenomena. Multi-singular pointwise defined fractional q-integro-differential equation (1.1) has been studied on a time scale under some boundary conditions. An application that describes the motion of a particle in the plane has been provided in this work to support our results' validity and applicability in the fields of physics and engineering.