scholarly journals Reflection symmetry properties of generalized fractional derivatives

2012 ◽  
Vol 11 (3) ◽  
pp. 71-80 ◽  
Author(s):  
Malgorzata Klimek ◽  
◽  
Maria Lupa ◽  
Keyword(s):  
Fractional Derivatives ◽  
Reflection Symmetry ◽  
Symmetry Properties ◽  
Generalized Fractional Derivatives

Reflection symmetric formulation of generalized fractional variational calculus

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Małgorzata Klimek ◽  
Maria Lupa
Keyword(s):  
Fractional Derivatives ◽  
Finite Interval ◽  
Equations Of Motion ◽  
Variational Calculus ◽  
Reflection Symmetry ◽  
Lagrange Equations ◽  
Representation Formulas ◽  
Symmetry Properties ◽  
Symmetric Formulation ◽  
Generalized Fractional Derivatives

AbstractWe define generalized fractional derivatives (GFDs) symmetric and anti-symmetric w.r.t. the reflection symmetry in a finite interval. Arbitrary functions are split into parts with well defined reflection symmetry properties in a hierarchy of intervals [0, b/2m], m ∈ ℕ0. For these parts — [J]-projections of function, we derive the representation formulas for generalized fractional operators (GFOs) and examine integration properties. It appears that GFOs can be reduced to operators determined in subintervals [0, b/2m]. The results are applied in the derivation of Euler-Lagrange equations for action dependent on Riemann-Liouville type GFDs. We show that for Lagrangian being a sum (finite or not) of monomials, the obtained equations of motion can be localized in arbitrary short subinterval [0, b/2m].

scholarly journals Variational Problems with Time Delay and Higher-Order Distributed-Order Fractional Derivatives with Arbitrary Kernels

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1665
Author(s):  
Fátima Cruz ◽  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Time Delay ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Fractional Operator ◽  
Different Types ◽  
Generalized Fractional Derivatives ◽  
Necessary And Sufficient ◽  
Distributed Order

In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.

scholarly journals Zur Lösung des quantenmechanischen Drei-Körper-Problems

1968 ◽  
Vol 23 (4) ◽  
pp. 579-596
Author(s):  
H. Ruder
Keyword(s):  
Ground State ◽  
Physical Meaning ◽  
State Energy ◽  
Differential Operators ◽  
Perturbation Expansion ◽  
Reflection Symmetry ◽  
Body System ◽  
Rapid Convergence ◽  
Symmetry Properties ◽  
Three Body

For the description of a three-body-system 6 coordinates are introduced which take full advantage of the symmetry properties of the system. The Schrödinger equation in these coordinates is derived. Using rotational and reflection symmetry one obtains a set of l respectively l+1 coupled differential equations containing only the 3 coordinates of the triangle formed by the 3 masses. Solutions are given for special potentials and arbitrary l. The physical meaning of the differential operators appearing in the equations becomes evident from their application to the solution functions. This leads to a rearrangement of the Hamiltonian in a very transparent form and gives a hint how to get the most effective perturbation expansion. A simple example is worked out. For systems with Coulomb interaction a modification of the method is suggested by physical considerations. The calculation of the ground state energy of the Helium atom shows the rapid convergence of the procedure.

scholarly journals Opial typeLp-inequalities for fractional derivatives

2002 ◽  
Vol 31 (2) ◽  
pp. 85-95 ◽  
Author(s):  
G. A. Anastassiou ◽  
J. J. Koliha ◽  
J. Pecaric
Keyword(s):  
Fractional Derivatives ◽  
Continuous Functions ◽  
Taylor’S Formula ◽  
Integrable Functions ◽  
Generalized Fractional Derivatives ◽  
Taylor's Formula

This paper presents a class ofLp-type Opial inequalities for generalized fractional derivatives for integrable functions based on the results obtained earlier by the first author for continuous functions (1998). The novelty of our approach is the use of the index law for fractional derivatives in lieu of Taylor's formula, which enables us to relax restrictions on the orders of fractional derivatives.

Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives

2014 ◽  
Vol 17 (1) ◽  
Author(s):  
Myong-Ha Kim ◽  
Guk-Chol Ri ◽  
Hyong-Chol O
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Fractional Derivatives ◽  
Operational Calculus ◽  
Operational Method ◽  
Initial Value ◽  
Constant Coefficients ◽  
Generalized Fractional Derivatives ◽  
Fractional Differential

AbstractThis paper provides results on the existence and representation of solution to an initial value problem for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients by using operational calculus of Mikusinski’s type. We prove that the initial value problem has the solution if and only if some initial values are zero.

scholarly journals Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives

2019 ◽  
Vol 3 (3) ◽  
pp. 39 ◽  
Author(s):  
Ndolane Sene ◽  
José Francisco Gómez Aguilar
Keyword(s):  
Asymptotic Stability ◽  
Fractional Derivative ◽  
Analytical Solutions ◽  
Global Asymptotic Stability ◽  
Fractional Derivatives ◽  
Mass Damper ◽  
Generalized Fractional Derivatives ◽  
Fractional Mass ◽  
Mass Spring ◽  
Generalized Fractional Derivative

This paper proposes novel analytical solutions of the mass-spring-damper systems described by certain generalized fractional derivatives. The Liouville–Caputo left generalized fractional derivative and the left generalized fractional derivative were used. The behaviors of the analytical solutions of the mass-spring-damper systems described by the left generalized fractional derivative and the Liouville–Caputo left generalized fractional derivative were represented graphically and the effect of the orders of the fractional derivatives analyzed. We finish by analyzing the global asymptotic stability and the converging-input-converging-state of the unforced mass-damper system, the unforced spring-damper, the spring-damper system, and the mass-damper system.

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