Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

Ahmed, Idris; Abubakar, Jamilu; Borisut, Piyachat; Sitthithakerngkiet, Kanokwan

doi:10.1186/s13662-020-02887-4

Cited by 15 publications

(6 citation statements)

References 44 publications

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“…Proof. Assume that σ > 0 and M ∈ A is any solution of the relation (22). Let Y ∈ A be a unique solution of the system (12).…”

Section: Definition 11 (Generalized Uh Stability)mentioning

confidence: 99%

“…Hyers-Ulam stability and existence of solutions to the generalized Liouville-Caputo fractional differential equations have been investigated in [21]. Ahmed et al investigated the solutions for the impulsive fractional pantograph differential equation via generalized anti-periodic boundary conditions in [22]. The numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation was proposed by Hinze and his coauthors [23].…”

Section: Introductionmentioning

confidence: 99%

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Mathematical Analysis of Biodegradation Model under Nonlocal Operator in Caputo Sense

2021

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To lower the concentration of organic pollutants in the effluent stream, wastewater must be treated before being discharged into the environment. The question of whether wastewater treatment facilities can successfully reduce the concentration of micropollutants found in their influent streams is becoming increasingly pressing. The removal of micropollutants in treatment plants is investigated using a model that incorporates biodegradation and sorption as the key processes of micropollutant removal. This article provides the mathematical analysis of the wastewater model that describes the removal of micropollutant in treatment plants under a non-local operator in Caputo sense. The positivity of the solution is presented for the Caputo fractional model. The steady state’s solution of model and their stability is presented. The fixed point theorems of Leray–Schauder and Banach are used to deduce results regarding the existence of the solution of the model. Ulam–Hyers (UH) types of stabilities are presented via functional analysis. The fractional Euler method is used to find the numerical results of the proposed model. The numerical results are illustrated via graphs to show the effects of recycle ratio and the impact of fractional order on the evolution of the model.

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“…Proof. Assume that σ > 0 and M ∈ A is any solution of the relation (22). Let Y ∈ A be a unique solution of the system (12).…”

Section: Definition 11 (Generalized Uh Stability)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mathematical Analysis of Biodegradation Model under Nonlocal Operator in Caputo Sense

2021

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“…It is worth to remark that as papers in the literature closely related to the topic of the present article one may mention [16,17,18,19,20,21,22,23,24,25] etc.. The paper is organized as follows: in Section 2 we recall some preliminary facts that we need in the sequel, Section 3 is devoted to our existence theorem and in Section 4 we obtain the arcwise connectedness of the solution set.…”

Section: Introductionmentioning

confidence: 99%

On a fractional differential inclusion involving a generalized Caputo type derivative with certain fractional integral boundary conditions

Cernea

2022

J Frac Calc & Nonlinear Sys

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We study a class of fractional differential inclusions defined by Caputo-Katugampola fractional derivativeinvolving a nonconvex set-valued map in the presence of certain fractional integral boundary conditions.Using a technique developed by Filippov we establish an existence result for the problem considered underthe hypothesis that the set-valued map is Lipschitz in the state variable. Also, based on a result concerningthe arcwise connectedness of the fixed point set of a class of set-valued contractions, we prove the arcwiseconnectedness of the solution set of the problem considered. The paper is the first in literature which containssuch kind of results in the framework of the problem studied.

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“…Fractional calculus caught much attention towards mathematical worlds (see [1,2,3,4,5,6,7,8,13,14,15,22]). In fact, fractional calculus is a branch of mathematical analysis, which separate itself from normal calculus, with non-integers order of derivatives and integrals as special characteristics.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of continuous solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order

Ahmed¹,

Limpanukorn

Ibrahim

2021

J. Math. Anal. Model.

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In this paper, the authors introduced a novel definition based on Hilfer fractional derivative, which name $q$-Hilfer fractional derivative of variable order. And the uniqueness of solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order of the form \eqref{eq:varorderfrac} with $0 < \alpha(t) < 1$, $0 \leq \beta \leq 1$, and $0 < q < 1$ is studied. Moreover, an example is provided to demonstrate the result.

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Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

Cited by 15 publications

References 44 publications

Mathematical Analysis of Biodegradation Model under Nonlocal Operator in Caputo Sense

Mathematical Analysis of Biodegradation Model under Nonlocal Operator in Caputo Sense

On a fractional differential inclusion involving a generalized Caputo type derivative with certain fractional integral boundary conditions

Uniqueness of continuous solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order

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