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A pathological study in the definition of uncertain numbers is carried out, and some solutions are proposed. Fundamental theorems for uncertain discrete fractional and integer order calculus are established. The concept of maximal solution for obtaining a unique uncertain solution is introduced. The solutions of uncertain discrete relaxation equations for the integer and the fractional order are obtained. Various numerical examples are accompanied to clarify the theoretical results and study of uncertain system behavior.

A system of fractional differential equations (FDEs) with fractional derivatives of diverse orders is called an incommensurate system of FDEs. In this paper, the well-posedness of the initial value problem for incommensurate systems of FDEs is obtained on the space of continuous functions. Three different methods for this analysis are used and compared. The complexity of such analysis is reduced by new techniques. Strong existence results are obtained by weaker conditions. The uniqueness and the continuous dependency of the solution on initial values are investigated using the Gronwall inequality. [ABSTRACT FROM AUTHOR]

Uncertain numbers, in a parallel definition of fuzzy numbers, are introduced. Model uncertainty and measurement uncertainty are our motivations for this study. A class of scalar multiplication and differences is proposed. Related algebra is investigated. A necessary and suficient condition of the existence of the introduced differences is obtained. Then, the existing result for the derivative is studied. Many interestingly important results are obtained. For example, the Hukuhara derivative does not exist for any fuzzy function with the new viewpoint. Constructive conditions for the existence of the generalized Hukuhara derivative are introduced. Four possible categories for derivatives fall into two forms of the fuzzy derivative for the generalized Hukuhara derivative. Importantly, this bifurcation in the definition of the new generalized Hukuhara derivative does not happen. Finally, all definitions related to differences and derivatives of uncertain numbers are unified in one concrete form with concrete analysis. Some examples and counterexamples are provided to illustrate theories and theorems in detail. [ABSTRACT FROM AUTHOR]

The quantitative and qualitative evaluations of long-term trends and spatial distribution of heatwaves (HW) are carried out in Southern Armenia (Syunik province). For this purpose, the daily average maximum air temperature data (Tmax) from four meteorological stations (Sisian, Goris, Kapan, and Meghri) are studied for the 1936–2019 period. The HW catchment algorithm and a new concept of HW (which is called potential HWs [PHWs]) are developed. The presence of trends and their significance for different HW measures are assessed utilizing linear and exponential trends, Sen's slope, and Mann– Kendall (MK) trend test. The results show that HWs become increasingly frequent and intense. Moreover, they have extended duration, as they emerge earlier and fade later in Syunik province. Significant variation trends of HW are observed (e.g., the frequency of HWs increased up to five times), especially after the 1990s in southern and central parts of the province. Contrary to this, relatively weaker or no trends occurred in the northern parts. Our methodological approach successfully determines the HW measures' trends in the province based on the relief features and altitude differences. The outcomes highlight their notable impact on the progression of HW. Overall, this study provides significant insights into understanding HW's variations and their transformation in the mountainous areas. [ABSTRACT FROM AUTHOR]