Link theorem and distributions of solutions to uncertain Liouville-Caputo difference equations

<p style='text-indent:20px;'>We consider a class of initial fractional Liouville-Caputo difference equations (IFLCDEs) and its corresponding initial uncertain fractional Liouville-Caputo difference equations (IUFLCDEs). Next, we make comparisons between two unique solutions of the IFLCDEs by deriving an important theorem, namely the main theorem. Besides, we make comparisons between IUFLCDEs and their <inline-formula><tex-math id="M1">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths by deriving another important theorem, namely the link theorem, which is obtained by the help of the main theorem. We consider a special case of the IUFLCDEs and its solution involving the discrete Mittag-Leffler. Also, we present the solution of its <inline-formula><tex-math id="M2">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths via the solution of the special linear IUFLCDE. Furthermore, we derive the uniqueness of IUFLCDEs. Finally, we present some test examples of IUFLCDEs by using the uniqueness theorem and the link theorem to find a relation between the solutions for the IUFLCDEs of symmetrical uncertain variables and their <inline-formula><tex-math id="M3">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths.</p>

Optimal control problem of variable-order delay system of advertising procedure: Numerical treatment

<p style='text-indent:20px;'>This paper presents an optimal control problem of the general variable-order fractional delay model of advertising procedure. The problem describes the flow of the clients from the unaware people group to the conscious or bought band. The new formulation generalizes the model that proposed by Muller. Two control variables are considered to increase the number of customers who purchased the products. An efficient nonstandard difference approach is used to study numerically the behavior of the solution of the mentioned problem. Properties of the proposed system were introduced analytically and numerically. The proposed difference schema maintains the properties of the analytic solutions as boundedness and the positivity. Numerical examples, for testing the applicability of the utilized method and to show the simplicity, accuracy and efficiency of this approximation approach, are presented with some comprising with standard difference methods.</p>