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Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives 总被引:1,自引:0,他引:1
Varsha Daftardar-Gejji 《Journal of Mathematical Analysis and Applications》2007,328(2):1026-1033
We discuss existence, uniqueness and stability of solutions of the system of nonlinear fractional differential equations
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Churong Chen Martin Bohner Baoguo Jia 《Mathematical Methods in the Applied Sciences》2019,42(18):7461-7470
We study the Ulam‐Hyers stability of linear and nonlinear nabla fractional Caputo difference equations on finite intervals. Our main tool used is a recently established generalized Gronwall inequality, which allows us to give some Ulam‐Hyers stability results of discrete fractional Caputo equations. We present two examples to illustrate our main results. 相似文献
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On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives 总被引:2,自引:0,他引:2
Local and global existence and uniqueness theorems for a functional delay fractional differential equation with bounded delay are investigated. The continuity with respect to the initial function is proved under Lipschitz and the continuity kind conditions are analyzed. 相似文献
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Zhongli Wei Changci PangYouzheng Ding 《Communications in Nonlinear Science & Numerical Simulation》2012,17(8):3148-3160
In this paper, we investigate the existence of positive solutions of singular super-linear (or sub-linear) integral boundary value problems for fractional differential equation involving Caputo fractional derivative. Necessary and sufficient conditions for the existence of C3[0, 1] positive solutions are given by means of the fixed point theorems on cones. Our nonlinearity f(t, x) may be singular at t = 0 and/or t = 1. 相似文献
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Boundary value problem for p−Laplacian Caputo fractional difference equations with fractional sum boundary conditions 下载免费PDF全文
Thanin Sitthiwirattham 《Mathematical Methods in the Applied Sciences》2016,39(6):1522-1534
In this paper, we consider a discrete fractional boundary value problem of the form where 0 < α,β≤1, 1 < α + β≤2, 0 < γ≤1, , ρ is a constant, and denote the Caputo fractional differences of order α and β, respectively, is a continuous function, and ?p is the p‐Laplacian operator. The existence of at least one solution is proved by using Banach fixed point theorem and Schaefer's fixed point theorem. Some illustrative examples are also presented. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
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This article investigates nonlinear impulsive Caputo fractional differential equations. Utilizing Lyapunov functions, Laplace transforms of fractional derivatives and boundedness of Mittag-Leffler functions, several sufficient conditions are derived to ensure the global ultimate boundedness and the exponential stability of the systems. An example is given to explain the obtained results. 相似文献
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Areeba Ikram 《Journal of Difference Equations and Applications》2019,25(6):757-775
ABSTRACTWe will establish uniqueness of solutions to boundary value problems involving the nabla Caputo fractional difference under two-point boundary conditions and give an explicit expression for the Green's functions for these problems. Using the Green's functions for specific cases of these boundary value problems, we will then develop Lyapunov inequalities for certain nabla Caputo BVPs. 相似文献
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Said R. Grace John R. Graef Ercan Tunc 《Journal of Applied Analysis & Computation》2019,9(4):1305-1318
The authors present conditions under which every positive solution $x(t)$ of the integro--differential equation $x^{\prime \prime }(t)=a(t)+\int_{c}^{t}(t-s)^{\alpha-1}[e(s)+k(t,s)f(s,x(s))]ds, \quad c>1, \ \alpha >0,$ satisfies $x(t)=O(tA(t))\textrm{ as }t\rightarrow \infty,$ i.e, $\limsup_{t\rightarrow \infty }\frac{x(t)}{tA(t)}<\infty, \textrm{where} \ A(t)=\int_{c}^{t}a(s)ds.$ From the results obtained, they derive a technique that can be applied to some related integro--differential equations that are equivalent to certain fractional differential equations of Caputo type of any order. 相似文献
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In this paper, we prove the existence and non-existence of solutions to two impulsive fractional differential equations with strong or weak Caputo derivatives in Euclidean space, respectively. 相似文献
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This paper provides a robust convergence checking method for nonlinear differential equations of fractional order with consideration of homotopy perturbation technique. The differential operators are taken in the Caputo sense. Some theorems to prove the existence and uniqueness of the series solutions are presented. Results show that the proposed theoretical analysis is accurate. 相似文献
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《Communications in Nonlinear Science & Numerical Simulation》2014,19(8):2820-2827
In this paper, by using fixed point theorems of concave operators in partial ordering Banach spaces, we establish the existence and uniqueness of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations for any given parameter. Moreover, we present some pleasant properties of positive solutions to the boundary value problem dependent on the parameter. In the end, two examples are given to illustrate our main results. 相似文献
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Kolade M. Owolabi 《Numerical Methods for Partial Differential Equations》2021,37(1):131-151
This paper is primarily concern with the formulation and analysis of a reliable numerical method based on the novel alternating direction implicit finite difference scheme for the solution of the fractional reaction–diffusion system. In the work, the integer first‐order derivative in time is replaced with the Caputo fractional derivative operator. As a case study, the dynamics of predator–prey model is considered. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction–diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve both self‐diffusion and cross‐diffusion problems in two‐dimensions. We observed in the experimental results a range of spatiotemporal and chaotic structures that are related to Turing pattern. It was also discovered in the simulations that cross‐diffusive case gives rise to spatial patterns faster than the diffusive case. Apart from chaotic spiral‐like structures obtained in this work, it should also be mentioned that Turing patterns such as stationary spots and stripes are obtainable, depending on the initial and parameters choices. 相似文献
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We study a time fractional heat equation in a non cylindrical domain. The problem is one-dimensional. We prove existence of properly defined weak solutions by means of the Galerkin approximation. 相似文献
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Numerical methods for multi-term fractional (arbitrary) orders differential equations 总被引:2,自引:0,他引:2
A. E. M. El-Mesiry A. M. A. El-Sayed H. A. A. El-Saka 《Applied mathematics and computation》2005,160(3):683-699
Our main concern here is to give a numerical scheme to solve a nonlinear multi-term fractional (arbitrary) orders differential equation. 相似文献
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Michal Fec?kan JinRong Wang 《Communications in Nonlinear Science & Numerical Simulation》2012,17(7):3050-3060
This paper is motivated from some recent papers treating the problem of the existence of a solution for impulsive differential equations with fractional derivative. We firstly show that the formula of solutions in cited papers are incorrect. Secondly, we reconsider a class of impulsive fractional differential equations and introduce a correct formula of solutions for a impulsive Cauchy problem with Caputo fractional derivative. Further, some sufficient conditions for existence of the solutions are established by applying fixed point methods. Some examples are given to illustrate the results. 相似文献