On a singular nonlocal time fractional order mixed problem with a memory term

Mesloub, Said; Algahtani, Obaid

doi:10.1002/mma.4921

Cited by 3 publications

(3 citation statements)

References 19 publications

(22 reference statements)

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“…Lemma (Mesloub and Algahtani 60 ) Let a nonnegative absolutely continuous function z ( t ) satisfy the inequality

^{C} D^{α} z (t) \leq C_{1} z (t) + C_{2} (t), 0 < α < 1,

for almost all t ∈ [0, T ], where C 1 is a positive constant and C 2 ( t ) is an integrable nonnegative function on [0, T ]. Then,

z (t) \leq z (0) E_{α} (C_{1} t^{α}) + Γ (α) E_{α, α} (C_{1} t^{α}) I^{α} C_{2} (t),

where E α and E α , β are Mittag–Leffler functions.…”

Section: Preliminariesmentioning

confidence: 97%

Nonnegative solutions to time fractional Keller–Segel system

Akilandeeswari

Tyagi

2020

Math Methods in App Sciences

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We establish the existence of nonnegative weak solutions to time fractional Keller-Segel system with Dirichlet boundary condition in a bounded domain with smooth boundary. Since the considered system has a cross-diffusion term and the corresponding diffusion matrix is not positive definite, we first regularize the system. Then under suitable assumptions on the initial conditions, we establish the existence of solutions to the system by using the Galerkin approximation method. The convergence of solutions is proved by means of compactness criteria for fractional partial differential equations. The nonnegativity of solutions is proved by the standard arguments. Furthermore, the existence of the weak solution to the system with Neumann boundary condition is discussed.

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“…Lemma (Mesloub and Algahtani 60 ) Let a nonnegative absolutely continuous function z ( t ) satisfy the inequality

^{C} D^{α} z (t) \leq C_{1} z (t) + C_{2} (t), 0 < α < 1,

for almost all t ∈ [0, T ], where C 1 is a positive constant and C 2 ( t ) is an integrable nonnegative function on [0, T ]. Then,

z (t) \leq z (0) E_{α} (C_{1} t^{α}) + Γ (α) E_{α, α} (C_{1} t^{α}) I^{α} C_{2} (t),

where E α and E α , β are Mittag–Leffler functions.…”

Section: Preliminariesmentioning

confidence: 97%

Nonnegative solutions to time fractional Keller–Segel system

Akilandeeswari

Tyagi

2020

Math Methods in App Sciences

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“…Fractional order partial differential equations have become one of the most popular areas of research in mathematical analysis. Their application has been utilized in various scientific fields, such as optimal control theory, chemistry, physics, mathematics, biology, finance and engineering [1][2][3][4][5].…”

Section: Introductionmentioning

confidence: 99%

Solvability of nonlinear fractional integro-differential equation with nonlocal condition

Aicha

Merad

2021

AJMS

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PurposeThis study describes the applicability of the a priori estimate method on a nonlocal nonlinear fractional differential equation for which the weak solution's existence and uniqueness are proved. The authors divide the proof into two sections for the linear associated problem; the authors derive the a priori bound and demonstrate the operator range density that is generated. The authors solve the nonlinear problem by introducing an iterative process depending on the preceding results.Design/methodology/approachThe functional analysis method is the a priori estimate method or energy inequality method.FindingsThe results show the efficiency of a priori estimate method in the case of time-fractional order differential equations with nonlocal conditions. Our results also illustrate the existence and uniqueness of the continuous dependence of solutions on fractional order differential equations with nonlocal conditions.Research limitations/implicationsThe authors’ work can be considered a contribution to the development of the functional analysis method that is used to prove well-positioned problems with fractional order.Originality/valueThe authors confirm that this work is original and has not been published elsewhere, nor is it currently under consideration for publication elsewhere.

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“…In the literature, many researchers used the functional analysis method to investigate the well posedness of initial boundary value problems for partial differential equations with time and space integer order having nonlocal conditions-we cite, for example, the references [14][15][16][17]. For the fractional diffusion wave equations case with higher order derivatives and classical boundary conditions, there are only few papers dealing with the existence and uniqueness of solution such as [18][19][20]. In this paper, an initial boundary value problem with purely nonlocal constraints of integral type for a Caputo time fractional 2mth order diffusion wave equation is studied by applying the functional analysis method, the so-called energy inequality method based mainly on some a priori estimates and on the density of the range of the operator generated by the studied problem.…”

Section: Introductionmentioning

confidence: 99%

Even Higher Order Fractional Initial Boundary Value Problem with Nonlocal Constraints of Purely Integral Type

Mesloub

Aldosari

2019

Symmetry

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In this paper, the a priori estimate method, the so-called energy inequalities method based on some functional analysis tools is developed for a Caputo time fractional 2 m th order diffusion wave equation with purely nonlocal conditions of integral type. Existence and uniqueness of the solution are proved. The proofs of the results are based on some a priori estimates and on some density arguments.

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On a singular nonlocal time fractional order mixed problem with a memory term

Cited by 3 publications

References 19 publications

Nonnegative solutions to time fractional Keller–Segel system

Nonnegative solutions to time fractional Keller–Segel system

Solvability of nonlinear fractional integro-differential equation with nonlocal condition

Even Higher Order Fractional Initial Boundary Value Problem with Nonlocal Constraints of Purely Integral Type

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