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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]

Power series, as an important means to analyze functions in different complex settings, are employed in various applied areas to solve differential equations and nonlinear problems and provide the assessment of intervals of convergence. Accordingly, the fuzzy logistic differential equation using the Caputo operator has been studied in this paper. Accordingly, the fuzzy logistic differential equation using the Caputo operator has been studied in this paper. The generalized Hukuhara difference and the generalized Hukuhara derivative are also used, and a power series representation is proposed for the solution of the fuzzy fractional logistic equation. Afterward, power series coefficients are obtained using a recursive formula. Finally, numerical experiments and illustrated results of the computations are presented to allow for more realistic decisions reflecting high complexity and underlying uncertainty. Thus, the numerical computations in our study reveal the effectiveness and accuracy of the power series method. Therefore, it is found that the fuzzy solution converges to the deterministic solution when uncertainty decreases, and, based on the technical analyses, it has been demonstrated that the results obtained are more fundamental in preventing geometric growth in nonlinear phenomena where uncertainties emerge due to impreciseness and inexactness. [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]

Well-posedness of the terminal value problem for nonlinear systems of generalized fractional differential equations is studied. The generalized fractional operator is formulated with a classical operator and a related weighted space. The terminal value problem is transformed into weakly singular Fredholm and Volterra integral equations with delay. A lower bound for the well-posedness of the corresponding problem is introduced. A collocation method covering all problems with generalized derivatives is introduced and analyzed. Illustrative examples for validation and application of the proposed methods are supported. The effects of various fractional derivatives on the solution, wellposedness, and fitting error are studied. An application for estimating the population of diabetes cases in the past is introduced. [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]

A neural network method for solving fractional diffusion equations is presented in this letter. An adaptive gradient descent method is proposed to minimize energy functions. Due to the memory effects of the fractional calculus, the gradient of energy function becomes much more complicated, and we suggest a simplified method. Numerical examples with one-layer and two-layer neurons show the effectiveness of the method. [ABSTRACT FROM AUTHOR]

Terminal value problems of fractional nonlinear systems are studied in this paper. The existence and uniqueness are given. The regularity of the solution is obtained in the weighted spaces. Discretized piecewise polynomial collocation methods are proposed on the graded mesh. A convergence analysis and the order are presented. Numerical examples for supporting theoretical results and applications for population models are illustrated. [ABSTRACT FROM AUTHOR]

A class of tempered fractional neural networks is proposed in this paper. Stability conditions for tempered fractional neural networks are provided by using Banach fixed point theorem. Attractivity and Mittag-Leffler stability are given. In order to show the efficiency and convenience of the method used, tempered fractional neural networks with and without delay are discussed, respectively. Furthermore, short memory and variable-order tempered fractional neural networks are proposed under the global conditions. Finally, two numerical examples are used to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

A class of tempered fractional differential equations with terminal value problems are investigated in this paper. Discretized collocation methods on piecewise polynomials spaces are proposed for solving these equations. Regularity results are constructed on weighted spaces and convergence order is studied. Several examples are supported the theoretical parts and compared with other methods. [ABSTRACT FROM AUTHOR]

The visco-elastic dampers can be economically designed for response reduction of dynamical systems. Recent developments in fractional calculus have affected the modeling of visco-elastic materials. The fractional models for visco-elastic dampers, which are parsimonious, require fewer parameters in comparison with other models. In this paper, we use the visco-elastic dampers for control of structural responses. The visco-elastic damper utilizes three important parameters including damping coefficient, stiffness, and fractional order. The dynamic model of the building with visco-elastic dampers can be acquired by a system of fractional differential equations, which includes both fractional and ordinary derivatives. We apply spline collocation methods for obtaining the numerical solution of this complex system. The collocation method is implemented in a graded mesh. The discrete data of earthquake are smoothed by spline interpolation of higher degree. The introduced method is used to simulate the response of a structure with active and passive controllers. A comparison of using different dampers is considered. Also, H ∞ norm of the transform function is used for response analysis. The results of simulations for four-story and 10-story buildings show significant reductions in structural responses. Moreover, our analysis presents that the large variety of values can be used for parameters of the visco-elastic damper to provide flexibility in designing appropriate visco-elastic-dampers. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a system of linear and nonlinear integral algebraic equations (IAEs) of Hessenberg type. Convergence analysis of the discontinuous collocation methods is investigated for the large class of IAEs based on the new definitions. Finally, some numerical experiments are provided to support the theoretical results. [ABSTRACT FROM PUBLISHER]

In this paper, we deal with a system of integral algebraic equations of the Hessenberg type. Using a new index definition, the existence and uniqueness of a solution to this system are studied. The well-known piecewise continuous collocation methods are used to solve this system numerically, and the convergence properties of the perturbed piecewise continuous collocation methods are investigated to obtain the order of convergence for the given numerical methods. Finally, some numerical experiments are provided to support the theoretical results. [ABSTRACT FROM AUTHOR]