Qualitative Analysis for Multiterm Langevin Systems with Generalized Caputo Fractional Operators of Different Orders

Ali, Saeed M.; Abdo, Mohammed S.

doi:10.1155/2022/1879152

Cited by 5 publications

(5 citation statements)

References 29 publications

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“…Having proven the existence and the uniqueness of a solution to the problem defined by Eq. ( 6), we finally summarize the analytical solutions (see [34]) for these fractional models as follows: Based on Lemma 3.1, the analytical solution for the drug concentration model via Caputo sense Eq. ( 5) is given by…”

Section: Proof Let Us Consider the Close Setmentioning

confidence: 99%

“…has the unique solution CF C(t) = C 0 + a α f (t, CF C(t)) + b α t 0 f (s, CF C(s))ds, (34) Since CF C(0) = CF N(0), then C 0 = N 0 . Hence Eq.…”

Section: Let Us Consider the Following Cf-problemmentioning

confidence: 99%

“…This result was generalized to differential equations, and it was believed that Obloza was the first to make this generalization [28], after which Alsina and Ger published a scientific paper containing what is known today as the Hyers-Ulam stability [29], Rus investigated the Hyers-Ulam stability of differential and integral equations using the Gronwall lemma [30]. Recently, the Hyers-Ulam stability problems of several differential equations were investigated by using the method of integral operators, see [31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

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Modeling Drug Concentration in Blood through Caputo-Fabrizio and Caputo Fractional Derivatives

Awadalla¹,

Abuasbeh²,

Noupoue³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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This study focuses on the dynamics of drug concentration in the blood. In general, the concentration level of a drug in the blood is evaluated by the mean of an ordinary and first-order differential equation. More precisely, it is solved through an initial value problem. We proposed a new modeling technique for studying drug concentration in blood dynamics. This technique is based on two fractional derivatives, namely, Caputo and Caputo-Fabrizio derivatives. We first provided comprehensive and detailed proof of the existence of at least one solution to the problem; we later proved the uniqueness of the existing solution. The proof was written using the Caputo-Fabrizio fractional derivative and some fixed-point techniques. Stability via the Ulam-Hyers (UH) technique was also investigated. The application of the proposed model on two real data sets revealed that the Caputo derivative was more suitable in this study. Indeed, for the first data set, the model based on the Caputo derivative yielded a Mean Squared Error (MSE) of 0.03095 with a corresponding best value of fractional order of derivative of 1.00360. Caputo-Fabrizio-basedderivative appeared to be the second-best method for the problem, with an MSE of 0.04324 for a corresponding best fractional derivative order of 0.43532. For the second experiment, Caputo derivative-based model still performed the best as it yielded an MSE of 0.04066, whereas the classical and the Caputo-Fabrizio methods were tied with the same MSE of 0.07299. Another interesting finding was that the MSE yielded by the Caputo-Fabrizio fractional derivative coincided with the MSE obtained from the classical approach.

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Section: Proof Let Us Consider the Close Setmentioning

confidence: 99%

“…has the unique solution CF C(t) = C 0 + a α f (t, CF C(t)) + b α t 0 f (s, CF C(s))ds, (34) Since CF C(0) = CF N(0), then C 0 = N 0 . Hence Eq.…”

Section: Let Us Consider the Following Cf-problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Modeling Drug Concentration in Blood through Caputo-Fabrizio and Caputo Fractional Derivatives

Awadalla¹,

Abuasbeh²,

Noupoue³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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show abstract

“…Before giving the essential result, we shall investigate formula (17) under the following assumptions:…”

Section: Solution Formulationmentioning

confidence: 99%

“…Existence theorem for psi-fractional HEs has been proven by Suwan et al [16]. Some qualitative analyses for multiterm LEs with generalized Caputo FDs and diffusion FDE with ABC operators can be found in [17,18]. The authors in [19] considered a hybrid LE involving Caputo FD and Riemann-Liouville (RL) fractional integral (FI) as follows: Motivated by the above works aforesaid and inspired by [19,21], in this paper, we deal with the existence of solutions for the following BVP to the nonlinear fractional hybrid-Sturm-Liouville-Langevin differential equation:…”

Section: Introductionmentioning

confidence: 97%

On Solutions of Hybrid–Sturm-Liouville–Langevin Equations with Generalized Versions of Caputo Fractional Derivatives

Boutiara

Wahash²,

Zahran

et al. 2022

Journal of Function Spaces

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The main intention of this research article is to introduce a new class of generalized fractional differential equations that fall into the categories of Sturm-Liouville’s, Langevin’s, and hybrid’s problems involving Y -Caputo fractional derivatives. The existence of the solutions of the proposed equations is discussed by using the technique of the measure of noncompactness related to the fixed point theorem, which is a generalization of Darbo’s fixed point theorem. Additionally, pertinent examples are provided along with the different values of the function Y to confirm the validity of the reported results.

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Some New Properties of the Mittag-Leffler Functions and Their Applications to Solvability and Stability of a Class of Fractional Langevin Differential Equations

Baghani,

Nieto

2023

Qual. Theory Dyn. Syst.

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Qualitative Analysis for Multiterm Langevin Systems with Generalized Caputo Fractional Operators of Different Orders

Cited by 5 publications

References 29 publications

Modeling Drug Concentration in Blood through Caputo-Fabrizio and Caputo Fractional Derivatives

Modeling Drug Concentration in Blood through Caputo-Fabrizio and Caputo Fractional Derivatives

On Solutions of Hybrid–Sturm-Liouville–Langevin Equations with Generalized Versions of Caputo Fractional Derivatives

Some New Properties of the Mittag-Leffler Functions and Their Applications to Solvability and Stability of a Class of Fractional Langevin Differential Equations

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