Existence and regularity of solutions for evolution equations with Riemann–Liouville fractional derivatives

Fan, Zhenbin

doi:10.1016/j.indag.2014.01.002

Cited by 19 publications

(10 citation statements)

References 11 publications

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“…For the basic fractional diffusion equation, given by the special case F = G = 0 and a = b = 0, the solution admits a series representation via separation of variables, which, in combination with the asymptotics of the Mittag-Leffler function, yields bounds on the time derivatives of u in various spatial norms [17,18]. One may also represent the solution in terms of a fractional resolvent [19,20,21]. These simple approaches no longer work in the general case, and the analysis that follows relies instead on the tools used in our study [1] of well-posedness: energy methods and a fractional Gronwall inequality.…”

Section: Introductionmentioning

confidence: 99%

Regularity theory for time-fractional advection–diffusion–reaction equations

McLean

Mustapha

Ali

et al. 2020

Computers & Mathematics with Applications

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We investigate the behavior of the time derivatives of the solution to a linear timefractional, advection-diffusion-reaction equation, allowing space-and time-dependent coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of fractional integrals.

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Section: Introductionmentioning

confidence: 99%

Regularity theory for time-fractional advection–diffusion–reaction equations

McLean

Mustapha

Ali

et al. 2020

Computers & Mathematics with Applications

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“…From this point of view, our results extend the theorems in [17,18], where the authors assumed that the operator generates 0 -semigroups. As for examples, we refer the reader to [17,18,33] since the examples therein still hold in our framework.…”

Section: Remark 15mentioning

confidence: 99%

“…In [32,33], the authors established the general theory of fractional resolvent and studied its application to the well posed problem for the evolution equation below:…”

Section: Introductionmentioning

confidence: 99%

“…Under this general condition, the difficulty on the well-posedness is how to deal with the singularity of resolvent operators and solutions at zero. We deal with this problem by utilizing the new space 1− ([0, ], ) and the theory of fractional resolvent developed in [32,33]. In the present article, we first obtain the uniqueness and existence of mild solutions for functional differential equation (2) via fractional resolvent and topological approach.…”

Section: Introductionmentioning

confidence: 99%

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Approximate Controllability for Functional Equations with Riemann-Liouville Derivative by Iterative and Approximate Method

Ibrahim

Fan

2017

Journal of Function Spaces

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We discuss the functional control systems governed by differential equations with Riemann-Liouville fractional derivative in general Banach spaces in the present paper. First we derive the uniqueness and existence of mild solutions for functional differential equations by the approach of fixed point and fractional resolvent under more general settings. Then we present new sufficient conditions for approximate controllability of functional control system by means of the iterative and approximate method. Our results unify and generalize some previous works on this topic.

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“…Some authors have discussed the solutions to fractional differential equations with Riemann-Liouville fractional derivative [18,[31][32][33]. For the mild solution, there are two different types of representation that have been given.…”

Section: Introductionmentioning

confidence: 99%

Solutions to Riemann–Liouville fractional integrodifferential equations via fractional resolvents

Yang

2019

Adv Differ Equ

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This paper is concerned with the semilinear fractional integrodifferential system with Riemann-Liouville fractional derivative. Firstly, we introduce the suitable C 1-α-solution to Riemann-Liouville fractional integrodifferential equations in the new frame of fractional resolvents. Some properties of fractional resolvents are given. Then we discuss the sufficient conditions for the existence of solutions without the Lipschitz assumptions to nonlinear item. Finally, an example on fractional partial differential equations is presented to illustrate our abstract results.

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Existence and regularity of solutions for evolution equations with Riemann–Liouville fractional derivatives

Cited by 19 publications

References 11 publications

Regularity theory for time-fractional advection–diffusion–reaction equations

Regularity theory for time-fractional advection–diffusion–reaction equations

Approximate Controllability for Functional Equations with Riemann-Liouville Derivative by Iterative and Approximate Method

Solutions to Riemann–Liouville fractional integrodifferential equations via fractional resolvents

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