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

Baghani, Hamid; Nieto, Juan J.

doi:10.1007/s12346-023-00870-4

Cited by 5 publications

(3 citation statements)

References 26 publications

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“…where E α,1 is the Mittag-Leffler function of parameter ᾱ [29]. Hence, we obtain that x θ − x * converges to zero on [a, b], which ends the proof.…”

Section: Proof By Definition Of Incommensurate Caputo Derivative We Havesupporting

confidence: 63%

Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems

Ndaïrou,

Torres

2023

Mathematics

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We introduce a new optimal control problem where the controlled dynamical system depends on multi-order (incommensurate) fractional differential equations. The cost functional to be maximized is of Bolza type and depends on incommensurate Caputo fractional-orders derivatives. We establish continuity and differentiability of the state solutions with respect to perturbed trajectories. Then, we state and prove a Pontryagin maximum principle for incommensurate Caputo fractional optimal control problems. Finally, we give an example, illustrating the applicability of our Pontryagin maximum principle.

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“…where E α,1 is the Mittag-Leffler function of parameter ᾱ [29]. Hence, we obtain that x θ − x * converges to zero on [a, b], which ends the proof.…”

Section: Proof By Definition Of Incommensurate Caputo Derivative We Havesupporting

confidence: 63%

Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems

Ndaïrou,

Torres

2023

Mathematics

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“…where a is the particle's radius, ν is the fluid's viscosity, 1 σ is the friction coefficient for unit mass, F(t, x(t)) = − 1 σ x ′ (t) + 1 m R(t) and R(t) is a random force. FLEs have attracted many scholars to study properties of solutions for FLEs-for instance, the existence and uniqueness of solutions for FLEs with Caputo or Riemann-Liouville fractional derivatives [7,8], boundary value problems for FLEs [9][10][11][12], etc. Baghani and Nieto [13] studied the following Langevin differential equation with two different fractional orders: c D ξ ( c D ν + λ)x(t) = h(t, x(t)).…”

Section: Introductionmentioning

confidence: 99%

Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space

Li,

Yang,

Wang

2024

Axioms

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In this work, analytic solutions of initial value problems for fractional Langevin equations involving Hilfer fractional derivatives and variable coefficients are studied. Firstly, the equivalence of an initial value problem to an integral equation is proved. Secondly, the existence and uniqueness of solutions for the above initial value problem are obtained without a contractive assumption. Finally, a formula of explicit solutions for the proposed problem is derived. By using similar arguments, corresponding conclusions for the case involving Riemann–Liouville fractional derivatives and variable coefficients are obtained. Moreover, the nonlinear case is discussed. Two examples are provided to illustrate theoretical results.

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“…In [7], the authors established Lyapunov-type inequalities for fractional boundary value problems with the Hilfer fractional derivative under multi-point boundary conditions. In [8,9], utilizing the fixed-point theorems and properties of the Mittag-Leffler function, the authors established the existence/uniqueness and stability results for fractional Langevin equations and nonlinear fractional hybrid differential equations, respectively. In [10], the authors presented analytical solutions for some fractional diffusion boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

Fixed Point Method for Nonlinear Fractional Differential Equations with Integral Boundary Conditions on Tetramethyl-Butane Graph

Nieto,

Yadav,

Mathur

et al. 2024

Symmetry

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Until now, little investigation has been done to examine the existence and uniqueness of solutions for fractional differential equations on star graphs. In the published articles on the subject, the authors used a star graph with one junction node that has edges with the other nodes, although there are no edges between them. These graph structures do not cover more generic non-star graph structures; they are specific examples. The purpose of this study is to prove the existence and uniqueness of solutions to a new family of fractional boundary value problems on the tetramethylbutane graph that have more than one junction node after presenting a labeling mechanism for graph vertices. The chemical compound tetramethylbutane has a highly symmetrical structure, due to which it has a very high melting point and a short liquid range; in fact, it is the smallest saturated acyclic hydrocarbon that appears as a solid at a room temperature of 25 °C. With vertices designated by 0 or 1, we propose a fractional-order differential equation on each edge of tetramethylbutane graph. Employing the fixed-point theorems of Schaefer and Banach, we demonstrate the existence and uniqueness of solutions for the suggested fractional differential equation satisfying the integral boundary conditions. In addition, we examine the stability of the system. Lastly, we present examples that illustrate our findings.

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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

Cited by 5 publications

References 26 publications

Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems

Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems

Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space

Fixed Point Method for Nonlinear Fractional Differential Equations with Integral Boundary Conditions on Tetramethyl-Butane Graph

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