On the analytical and numerical study for fractional q-integrodifferential equations

Abstract

In this paper, we give some basic concepts of q-calculus that will be needed in this paper. Then, we built the q-nonlocal condition that ensures the solution existence and uniqueness of the fractional q-integrodifferential equation. Also, we introduce the continuous dependence of the solution. We find the numerical solution using the finite-difference-Trapezoidal and the cubic B-spline-Trapezoidal methods. Finally, we give three examples to illustrate the validity of our main results.

On study the existence and uniqueness of the solution of the Caputo–Fabrizio coupled system of nonlocal fractional q‐integro differential equations

“…So, numerical techniques are used to estimate the solution. The authors of [15] used the Banach fixed point theorem to study the existence of a unique solution to a first‐order integro‐differential equation. In [16], the authors proved that the second‐order integro‐differential equation has a continuous solution.…”

“…In [16], the authors proved that the second‐order integro‐differential equation has a continuous solution. Moreover, in [14–16], the authors obtained the numerical solutions to problems using a merge of Simpson's with finite difference and trapezoidal with finite difference methods. The fractional trapezoidal rule is used by authors in [17] to get the numerical solution of fractional order integro‐differential equations.…”

The nonlocal coupled system of Caputo–Fabrizio fractional q‐integro differential equation

“…There are numerous papers that deal with the fractional q ‐difference equation [20–23]. In 2020 [24], the authors discussed the numerical and analytical solutions of the integro‐differential equation with an initial condition. Then, in 2022 [25], they investigated the existence and uniqueness of solutions for a second type Volterra–Fredholm integro‐differential equation.…”

“…They also applied the method of finite‐trapezoidal to obtain the numerical solution of the problem. Moreover, in 2022 [26], they investigated the existence and uniqueness of solutions for a second type of Volterra–Fredholm integro‐differential equation. They also applied the finite‐trapezoidal method to obtain the numerical solution of the problem.…”

“…Afterwards, [27] discussed the existence, uniqueness, and continuous dependence of solutions for the fractional q ‐integro‐differential equation, including the Caputo–Fabrizio fractional derivative and the q‐integral of the Riemann–Liouville type. They solved the problems in two paper [26, 27] by using two methods: The first is the finite‐trapezoidal method, and the second is the cubic trapezoidal method. Additionally, in 2023, the authors conducted an analytical and numerical study of the fractional q ‐integro‐differential equation with nonlocal conditions.…”