A survey on fractional variational calculus

Almeida, Ricardo; Torres, Delfim F. M.

doi:10.1515/9783110571622-014

Cited by 6 publications

(5 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The standard Schwinger action principle (see in [2] (Section 2.1, pp. [60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78], and [174]) can be generalized for spatial nonlocality and memory. It should be emphasized that to take into account dissipative and irreversible processes, quantum mechanics should be described by Lindblad equations and its generalizations instead of the Schrodinger equations for the wave function.…”

Section: Discussionmentioning

confidence: 99%

“…(A1) The holonomic functionals can be considered as definite integer-order integrals involving functions and their fractional derivatives of non-integer orders. This type of generalization is called "Fractional calculus of variations" (FCofV) or "Fractional variational calculus" [59][60][61][62][63][64][65][66][67][68]. Note that in paper [61], the fractional integrals are considered in addition to fractional derivatives in FCofV.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

General Fractional Noether Theorem and Non-Holonomic Action Principle

Tarasov

2023

Mathematics

View full text Add to dashboard Cite

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.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

General Fractional Noether Theorem and Non-Holonomic Action Principle

Tarasov

2023

Mathematics

View full text Add to dashboard Cite

show abstract

“…In recent years, considerable attention has been directed towards the burgeoning field of "Fractional Calculus" by numerous researchers, driven by its diverse applications across various scientific and engineering domains (1)(2)(3)(4)(5) . Introducing fractional operators into classical differential equations results in complex challenges when solving resulting fractional-order partial differential equations (6) . As a consequence, researchers tend to favor numerical methods over analytical ones.…”

Section: Introductionmentioning

confidence: 99%

Solving Time-Fractional Fitzhugh–Nagumo Equation using Homotopy Perturbation Method

Dhumal,

Sontakke

2024

IJST

View full text Add to dashboard Cite

Objectives: This study aims to explore solutions to the time-fractional Fitzhugh-Nagumo equation, a nonlinear reaction-diffusion equation. Method: We utilize the Homotopy Perturbation Method (HPM) as a proficient analytical approach for addressing the time-fractional Fitzhugh-Nagumo equation. The HPM offers a structured method for deriving approximate solutions in the shape of convergent series, enabling accurate solutions even for intricate nonlinear fractional equations. Finding: The series solution obtained is validated by comparing it with numerical methods, showcasing its precision and effectiveness. Additionally, we assessed the error across various time and space values. Our analysis and computations reveal that the Homotopy Perturbation Method (HPM) stands out for providing precise approximations while maintaining computational efficiency. It's clear that this method presents a robust alternative to conventional numerical techniques, particularly in situations where analytical solutions are difficult to obtain. Novelty: The application of the Homotopy Perturbation Method to the Time-fractional Fitzhugh-Nagumo Equation has been effectively explored, with specific examples showing a strong agreement between the exact solution and the obtained solution. Keywords: Time-Fractional Fitzhugh–Nagumo Equation, Homotopy Perturbation Method, Riemann-Liouville fractional integral, Caputo fractional derivative, Fractional Homotopy Perturbation Method

show abstract

“…The calculus of variations is a field of mathematical analysis that uses variations, which are small perturbations in functions to find maxima and minima of functionals. The Euler-Lagrange equation is the main tool for solving such optimization problems, and they have been developed in the context of fractional calculus to better describe non-conservative systems in mechanics [10]. Necessary optimality condition of Euler-Lagrange type for distributed-order arXiv:2108.03600v1 [math.OC] 8 Aug 2021 problems of the calculus of variations were first introduced and developed in Reference [11].…”

Section: Introductionmentioning

confidence: 99%

Pontryagin Maximum Principle for Distributed-Order Fractional Systems

Ndaïrou

Torres

2021

Mathematics

Self Cite

View full text Add to dashboard Cite

We consider distributed-order non-local fractional optimal control problems with controls taking values on a closed set and prove a strong necessary optimality condition of Pontryagin type. The possibility that admissible controls are subject to pointwise constraints is new and requires more sophisticated techniques to include a maximality condition. We start by proving results on continuity of solutions due to needle-like control perturbations. Then, we derive a differentiability result on the state solutions with respect to the perturbed trajectories. We end by stating and proving the Pontryagin maximum principle for distributed-order fractional optimal control problems, illustrating its applicability with an example.

show abstract

A survey on fractional variational calculus

Cited by 6 publications

References 37 publications

General Fractional Noether Theorem and Non-Holonomic Action Principle

General Fractional Noether Theorem and Non-Holonomic Action Principle

Solving Time-Fractional Fitzhugh–Nagumo Equation using Homotopy Perturbation Method

Pontryagin Maximum Principle for Distributed-Order Fractional Systems

Contact Info

Product

Resources

About