In the last two decades, significant numbers of articles have appeared on this topic, and some of the papers deal with the existence and uniqueness of solutions for difference equation problems (see ). It has been found that the role of FDEs is very important in treating and modeling nonlinear problems with applications in mathematical analysis and various branches of science, including diffusion, plasmas, dynamic systems, nonlinear optics, and many other areas (see ). Here, we shall focus principally on the most commonly used and classical definition, which is known as the Riemann–Liouville ( RL) fractional calculus (see, for details, see also the recent survey-cum-expository review articles ).įractional difference equations ( FDEs) have become a hot research topic in the mathematical and physical sciences. There are several possible ways to define discrete fractional operators (differences and sums), leading to a diverse and rich field of study (see ). However, since the emergence of the theory of discrete fractional calculus, different types of discrete fractional operators have been developed to deal with various situations in the applied and natural sciences due to their great importance as an advanced mathematical tool for the interpretation and modeling of many biological and physical phenomena, such as various biological studies, electrical circuits, mechanical fluids, relaxation processes, and damping-law models (see ). The idea of discrete fractional calculus is to replace the natural numbers in the order of the difference by fractional orders.
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