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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]

Fractional differential equation approach is frequently used to describe long-term interactions in nonlinear systems. However, it results in difficulty in inverse problems as well as the numerical treatment. Numerical analysis of intermediate value problems and the well-posedness are investigated in this study. Two high order numerical methods for solving intermediate value problems are proposed. Convergence and sensitivity analysis are provided. A comparison is provided by a well-chosen example. The estimated order of the convergence shows the sharpness of our analysis. [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]

In this paper, systems of fuzzy fractional differential equations with a lateral type of the Hukuhara derivative and the generalized Hukuhara derivative are numerically studied. Collocation method on discontinuous piecewise polynomial spaces is proposed. Convergence of the proposed method is analyzed. The superconvergent results on the graded mesh are studied. Examples are provided to support theoretical results. Finally, the effect of uncertainty in a diabetes model and its resulting complications is investigated as a practical application. [ABSTRACT FROM AUTHOR]