Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense

Luchko, Yuri

doi:10.3390/math10060849

Cited by 43 publications

(25 citation statements)

References 36 publications

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“…where T αβ (x) is the energy-momentum tensor defined by (135), and S µ αβ (x) is the spin angular-momentum tensor…”

Section: General Orbital and Spin Angular-momentum Tensorsmentioning

confidence: 99%

“…where B(ϕ, ∂ϕ, ∂ 2 ϕ) cannot be represented in the form ( 153) and q = p, in general. Using (135) and field Equation ( 158), general energy-momentum tensor of the scalar field is…”

Section: Example Of Field Equations For Real Scalar Fieldmentioning

confidence: 99%

“…To describe a wider type of non-localities, we can use the general fractional calculus (GFC), which is based on Sonin's ideas [128,129]. The most convenient form of GFC is the Luchko's form of GFC [130][131][132][133][134][135][136][137][138]. The Luchko approach is also developed and appied to various sciences in [139][140][141][142][143][144][145][146][147][148][149].…”

Section: Introductionmentioning

confidence: 99%

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General Fractional Noether Theorem and Non-Holonomic Action Principle

Tarasov

2023

Mathematics

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Using general fractional calculus (GFC) of the Luchko form and non-holonomic variational equations of Sedov type, generalizations of the standard action principle and first Noether theorem are proposed and proved for non-local (general fractional) non-Lagrangian field theory. The use of the GFC allows us to take into account a wide class of nonlocalities in space and time compared to the usual fractional calculus. The use of non-holonomic variation equations allows us to consider field equations and equations of motion for a wide class of irreversible processes, dissipative and open systems, non-Lagrangian and non-Hamiltonian field theories and systems. In addition, the proposed GF action principle and the GF Noether theorem are generalized to equations containing general fractional integrals (GFI) in addition to general fractional derivatives (GFD). Examples of field equations with GFDs and GFIs are suggested. The energy–momentum tensor, orbital angular-momentum tensor and spin angular-momentum tensor are given for general fractional non-Lagrangian field theories. Examples of application of generalized first Noether’s theorem are suggested for scalar end vector fields of non-Lagrangian field theory.

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“…where T αβ (x) is the energy-momentum tensor defined by (135), and S µ αβ (x) is the spin angular-momentum tensor…”

Section: General Orbital and Spin Angular-momentum Tensorsmentioning

confidence: 99%

“…where B(ϕ, ∂ϕ, ∂ 2 ϕ) cannot be represented in the form ( 153) and q = p, in general. Using (135) and field Equation ( 158), general energy-momentum tensor of the scalar field is…”

Section: Example Of Field Equations For Real Scalar Fieldmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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General Fractional Noether Theorem and Non-Holonomic Action Principle

Tarasov

2023

Mathematics

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“…In fractional calculus, the derivative and integral operators are generalized to fractional orders, such as 1/2 or 3/4. These operators can be defined using fractional calculus techniques, such as the Caputo operators [5][6][7][8] and Riemann-Liouville [9,10]. One of the key features of fractional calculus is that it allows for the study of phenomena that exhibit long-term memory, such as anomalous diffusion, fractional Brownian motion, and viscoelasticity.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of fractional epidemic model for two infected classes incorporating hospitalization impact

Santra,

Mahapatra,

Basu

2024

Phys. Scr.

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This article presents an epidemic disease propagation mathematical model in fractional order. The epidemiological characteristics are presented based on the susceptible, exposed, unknown infected, known infected, hospitalized population and the population in the secure zone. Both the disease endemic equilibrium and the disease-free equilibrium's stability characteristics have been examined using the basic reproduction number. Variation of basic reproduction number based on the different sensitive parameters has been discussed. It has been disputed whether the fractional model provides a uniform, reliable solution. An analysis of the time history of unknown and known infected populations, hospitalized populations and recovered populations at different values of various sensitive parameters has been carried out. To support the key theoretical conclusions, some numerical simulations are completed using MATLAB. The impact of various populations on the propagation of the illness has also been investigated, as well as how specific state variables change over time for various fractional order values.

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“…The study of linear systems of fractional differential equations with Riemann-Liouville-type fractional derivatives was considered in [1], nonlinear systems in [2], existence and Ulam stability was studied in [3], and for basic concepts on stability for Riemann-Liouville fractional differential equations, we refer the reader to [4]. A general fractional derivative of arbitrary order in the Riemann-Liouville sense was defined and applied to Cauchy problems for single-and multi-term linear fractional differential equations by Luchko in [5]. In addition, generalized proportional fractional integrals and derivatives were defined, studied, and applied in [6,7].…”

Section: Introductionmentioning

confidence: 99%

Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory

Agarwal,

Hristova,

O’Regan

2023

Mathematics

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In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained.

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Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense

Cited by 43 publications

References 36 publications

General Fractional Noether Theorem and Non-Holonomic Action Principle

General Fractional Noether Theorem and Non-Holonomic Action Principle

Stability analysis of fractional epidemic model for two infected classes incorporating hospitalization impact

Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory

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