To investigate a class of multi-singular pointwise defined fractional $ q $–integro-differential equation with applications

Abstract

<abstract><p>In the research work, we discuss a multi-singular pointwise defined fractional $ q $–integro-differential equation under some boundary conditions via the Riemann-Liouville $ q $–integral and Caputo fractional $ q $–derivatives. New existence results rely on the $ \alpha $-admissible map and fixed point theorem for $ \alpha $-$ \mathtt{ψ} $-contraction map. At the end, we present an example with application and some algorithms to illustrate the primary effects.</p></abstract>

Well-posed conditions on a class of fractional q-differential equations by using the Schauder fixed point theorem

“…We have also studied the perturbation on boundary condition on the function exists in the right-hand side of the problem and on the fractional order. To the leading of our information, the results have never been detailed in other works [11,12,61] that consider the problems. In this manner, it is very apparent that the solution of the problem is stable under the small perturbation.…”

“…We need to present a simplified analysis that is able to execute the values of the q-gamma function. For this purpose, we provided a pseudo-code description of the method for calculation of the q-gamma function of order n [61].…”

“…, where σ , ν ≥ 0 [29]. The authors in [61] presented all algorithms and MATLAB lines to simplify q-factorial functions (vw) (n) q , (vw) (σ ) q , q (v), I q [℘](v), and some necessary equations. Now, we introduce some basic definitions, lemmas, and theorems which are used in the subsequent sections.…”

Exact Solutions for the Generalized Atangana-Baleanu-Riemann Fractional (3 + 1)-Dimensional Kadomtsev–Petviashvili Equation

“…Many scientists in the world are absorbed by this topic, and searching for the exact solutions of these FNPDES is significant in the study of nonlinear phenomena, including the nonlinear wave observed in mechanics [1], ecological and economic systems [2], two-scale thermal science [3], optical fibers [4], chaotic oscillations [5], atmospheric science [6], telegraph transmission [7], etc. [8][9][10][11]. To date, many powerful methods have been proposed for this subject, such as the Bäcklund transformation method [12], Darboux transformation [13], Hirota bilinear method [14], improved F-expansion method [15], sine-Gordon method [16], projective Riccati equations method [17], G'/G-expansion method [18], (G'/G,1/G)-expansion method [19], improved (m + G'/G)-expansion method [20], improved G'/G 2 -expansion method [21], the first integral method [22], Generalized Exp-Function Method [23], Exp(−ϕ(ξ))-Expansion Method [24], and Lie symmetry method, which are connected to the wave transformations of equations that do not change the set of solutions [25], among others [26][27][28][29][30][31][32][33][34][35][36].…”

q -fractional functional integro-differential equation with q -integral condition

“…The attention of researchers subject of q-differential equations appeared in recent years [17]. Noted recently the attention of many researchers is in the field of fractional q-calculus [3,14,18,19].…”