Abstract
To model complex systems with discrete-time features and memory effects in the uncertain environment, a definition of an uncertain fractional forward difference equation with Riemann–Liouville-like forward difference is introduced. Moreover, analytic solutions to a type of special linear uncertain fractional difference equations are presented by the Picard iteration method. Then, an existence and uniqueness theorem of the solutions is proved by applying Banach contraction mapping theorem. Finally, two examples are provided to illustrate the validity of the existence and uniqueness theorem.
Highlights
Since 1965, fractional calculus has been a topic of interest and has become a useful tool for tackling problems in physics, biology, economics, and several fields in engineering [1,2,3,4]
Numerical versions of continuous fractional calculus can result in the discrete models, but this treatment can readily lead to cumulate errors and cannot accurately depict the non-locality of complex systems
Research on discrete fractional calculus provides a very new idea to model the systems with memory effects and discrete-time features
Summary
Since 1965, fractional calculus has been a topic of interest and has become a useful tool for tackling problems in physics, biology, economics, and several fields in engineering [1,2,3,4]. Research on discrete fractional calculus provides a very new idea to model the systems with memory effects and discrete-time features. To investigate systems with memory effects in the uncertain environment, the conception of uncertain fractional differential equations (UFDEs) rested on the uncertain theory was introduced in [18]. Existence of their solutions was discussed in [19]. Motivated by the works mentioned above, to develop modeling techniques with discrete fractional calculus, we will define an uncertain fractional forward difference equation (UFFDE) and present an existence and uniqueness theorem of solutions to UFFDEs. The rest is arranged as follows: In Sect.
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