The authors investigate the existence of solutions to a class of boundary value problems for fractional q-difference equations in a Banach space that involves a q-derivative of the Caputo type and nonlinear integral boundary conditions. Their result is based on Mönch’s fixed point theorem and the technique of measures of noncompactness. This approach has proved to be an interesting and useful approach to studying such problems. Some basic concepts from the fractional q-calculus are introduced, including q-derivatives and q-integrals. An example of the main result is included as well as some suggestions for future research.

The main goal of this work is to propose a new fractional approach of the higher-order q-Taylor method with Initial Value Problems (IVPs) for fractional q-difference equations which is called the Fractional Higher-Order q-Taylor Method (FHOqTM). By applying the generalised q-Taylor theorem, this would be achieved. As a consequence, we calculate the FHOqTM’s local truncation error. Finally, we present numerical applications to validate our results by comparing the exact solution and the approximate solution obtained by (FHOqTM).

The two-energy neutron diffusion model in slab reactors characterizes neutron behavior across two energy groups: fast and thermal. Fast neutrons, generated by fission, decelerate through collisions, transitioning into thermal neutrons. This model employs diffusion equations to compute neutron flux distributions and reactor parameters, thereby optimizing reactor design and safety to ensure efficient neutron utilization and stable, sustained nuclear reactions. The primary objective of this research is to explore both analytical and numerical solutions to the two-energy neutron diffusion model in slab reactors. Specifically, we will utilize the Laplace transform method for an analytical solution of the two-energy neutron diffusion model. Subsequently, employing the Caputo differentiator, we transform the original neutron diffusion model into its fractional-order equivalents, yielding the fractional-order two-energy group neutron diffusion model in slab reactors. To address the resulting fractional-order system, we develop a novel approach aimed at reducing the 2β-order system to a β-order system, where β ∈ (0, 1]. This transformed system is then solved using the Modified Fractional Euler Method (MFEM), an advanced variation of the fractional Euler method. Finally, we present numerical simulations that validate our results and demonstrate their applicability.

This work discusses the existence and uniqueness of solutions to the boundary value problem for fractional q-difference equations, which involves the Caputo’s fractional q-derivative with nonlinear integral conditions, by using some standard fixed point theorems. In addition, we study the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of its solutions. Finally, we present two examples to explain our results.

This paper, we investigat the existence and uniqueness of solutions for a initial value problem for impulsive fractional q-difference equations involving the Caputo fractional q-difference derivative. The results are obtained using the Banach contraction principle and Krasnoselskii’s fixed point theorems. Also, we discuss the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of its solutions. Finally, an example is given to illustrate the effectiveness of our results. AMS (MOS) Subject Classifications : 26A33, 39A13, 47H10.