Abstract
The purpose of this paper is to provide sufficient conditions for the local and global existence of solutions for the general nonlinear distributed-order fractional differential equations in the time domain. Also, we provide sufficient conditions for the uniqueness of the solutions. Furthermore, we use operational matrices for the fractional integral operator of the second kind Chebyshev wavelets and shifted fractional-order Jacobi polynomials via Gauss–Legendre quadrature formula and collocation methods to reduce the proposed equations into systems of nonlinear equations. Also, error bounds and convergence of the presented methods are investigated. In addition, the presented methods are implemented for two test problems and some famous distributed-order models, such as the model that describes the motion of the oscillator, the distributed-order fractional relaxation equation, and the Bagley–Torvik equation, to demonstrate the desired efficiency and accuracy of the proposed approaches. Comparisons between the methods proposed in this paper and the existing methods are given, which show that our numerical schemes exhibit better performances than the existing ones.
Highlights
Distributed-order fractional derivatives indicate fractional derivatives that are integrated over the order of the differentiation within a given range [55]
3 Existence and uniqueness of solutions In the following theorem, by using Schauder’s fixed point theorem [57], we prove the local existence of solutions for general distributed-order fractional differential equations (DOFDEs) in a Banach space
7 Conclusion In this research paper, based on Schauder’s and Tychonoff ’s fixed point theorems, sufficient conditions for the local and global existence of solutions were provided for general DOFDEs
Summary
Distributed-order fractional derivatives indicate fractional derivatives that are integrated over the order of the differentiation within a given range [55]. The motivation of DOFDEs is the generalization of single-order and multi-term fractional differential equations [27]. 35, 43, 45, 48] and the references therein), there have been few research studies that developed numerical methods to solve general DOFDEs (see [19, 21, 36, 38, 47, 50, 53]). 3, we provide sufficient conditions for the existence and uniqueness of solutions for general DOFDEs. In Sect. 6, we solve two test problems and some famous distributed-order models, such as the model that describes the motion of the oscillator, the distributed-order fractional relaxation equation, and the Bagley–Torvik equation, to show that our approaches will increase the accuracy of the methods used for such operational matrices.
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