A family of measures of noncompactness in the Hölder space Cn,γ(R+) and its application to some fractional differential equations and numerical methods

2020 ◽  
Vol 363 ◽  
pp. 256-272
Author(s):  
Hojjatollah Amiri Kayvanloo ◽  
Mahnaz Khanehgir ◽  
Reza Allahyari
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Holder Space ◽  
Hölder Space ◽  
Measures Of Noncompactness ◽  
Fractional Differential

scholarly journals A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications

2019 ◽  
Vol 22 (1) ◽  
pp. 27-59 ◽  
Author(s):  
HongGuang Sun ◽  
Ailian Chang ◽  
Yong Zhang ◽  
Wen Chen
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Relevant Literature ◽  
Physical Models ◽  
Variable Order ◽  
Mathematical Tool ◽  
Fractional Differential ◽  
Mathematical Foundations ◽  
Selection Of

Abstract Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

Solvability for a couple system of nonlinear fractional differential equations in a Banach space

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Jitai Liang ◽  
Zhenhai Liu ◽  
Xuhuan Wang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Couple System ◽  
Measures Of Noncompactness ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
The Fixed Point Theorem

AbstractIn this paper, we study boundary value problems of nonlinear fractional differential equations in a Banach Space E of the following form: $\left\{ \begin{gathered} D_{0^ + }^p x(t) = f_1 (t,x(t),y(t)),t \in J = [0,1], \hfill \\ D_{0^ + }^q y(t) = f_2 (t,x(t),y(t)),t \in J = [0,1], \hfill \\ x(0) + \lambda _1 x(1) = g_1 (x,y), \hfill \\ y(0) + \lambda _2 y(1) = g_2 (x,y), \hfill \\ \end{gathered} \right. $ where D 0+ denotes the Caputo fractional derivative, 0 < p,q ≤ 1. Some new results on the solutions are obtained, by the concept of measures of noncompactness and the fixed point theorem of Mönch type.

scholarly journals Applications of measures of noncompactness to infinite system of fractional differential equations

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3421-3432 ◽  
Author(s):  
Mohammad Mursaleen ◽  
Bilal Bilalov ◽  
Syed Rizvi
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence Result ◽  
Measure Of Noncompactness ◽  
Infinite System ◽  
Hausdorff Measure Of Noncompactness ◽  
Measures Of Noncompactness ◽  
Banach Sequence Spaces ◽  
Fractional Differential ◽  
Banach Sequence

In this paper, we discuss few existence result for solution of an infinite system of fractional differential equations of order ?(1 < ? < 2), with three point boundary value problem in the interval [0, T]. The problem is studied in the classical Banach sequence spaces c0 and lp (1 ? p < 1), using Hausdorff measure of noncompactness and Darbo type fixed point theorem. We also illustrate our results through some concrete examples.

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