scholarly journals Hamilton’s principle with variable order fractional derivatives

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Teodor Atanackovic ◽  
Stevan Pilipovic
Keyword(s):  
Fractional Derivatives ◽  
Necessary Conditions ◽  
Lagrange Equation ◽  
Variable Order ◽  
Hamilton’S Principle ◽  
Lagrange Equations ◽  
Hamilton's Principle ◽  
Special Cases ◽  
Admissible Functions ◽  
Variable Order Fractional Derivatives

AbstractWe propose a generalization of Hamilton’s principle in which the minimization is performed with respect to the admissible functions and the order of the derivation. The Euler-Lagrange equations for such minimization are derived. They generalize the classical Euler-Lagrange equation. Also, a new variational problem is formulated in the case when the order of the derivative is defined through a constitutive equation. Necessary conditions for the existence of the minimizer are obtained. They imply various known results in a special cases.

scholarly journals Noether’s theorems for the relative motion systems on time scales

2018 ◽  
Vol 3 (2) ◽  
pp. 513-526
Author(s):  
Sheng-nan Gong ◽  
Jing-li Fu
Keyword(s):  
Time Scales ◽  
Relative Motion ◽  
Conserved Quantities ◽  
Noether Symmetries ◽  
Hamilton’S Principle ◽  
Lagrange Equations ◽  
Hamilton's Principle ◽  
Generalized Coordinates ◽  
Delta Derivatives ◽  
Infinitesimal Transformations

AbstractThis paper propose Noether symmetries and the conserved quantities of the relative motion systems on time scales. The Lagrange equations with delta derivatives on time scales are presented for the system. Based upon the invariance of Hamilton action on time scales, under the infinitesimal transformations with respect to the time and generalized coordinates, the Hamilton’s principle, the Noether theorems and conservation quantities are given for the systems on time scales. Lastly, an example is given to show the application the conclusion.

scholarly journals ‘Uncertainty’ principle in two fluid–mechanics

2020 ◽  
Vol 69 ◽  
pp. 47-55
Author(s):  
Sergey Gavrilyuk
Keyword(s):  
Potential Energy ◽  
Second Law Of Thermodynamics ◽  
Coupled System ◽  
Degree Of Freedom ◽  
Hamilton’S Principle ◽  
Governing Equations ◽  
Lagrange Equations ◽  
Hamilton's Principle ◽  
Stationary Action ◽  
Energy Equations

Hamilton’s principle (or principle of stationary action) is one of the basic modelling tools in finite-degree-of-freedom mechanics. It states that the reversible motion of mechanical systems is completely determined by the corresponding Lagrangian which is the difference between kinetic and potential energy of our system. The governing equations are the Euler-Lagrange equations for Hamil- ton’s action. Hamilton’s principle can be naturally extended to both one-velocity and multi-velocity continuum mechanics (infinite-degree-of-freedom systems). In particular, the motion of multi–velocity continuum is described by a coupled system of ‘Newton’s laws’ (Euler-Lagrange equations) for each component. The introduction of dissipative terms compatible with the second law of thermodynamics and a natural restriction on the behaviour of potential energy (convexity) allows us to derive physically reasonable and mathematically well posed governing equations. I will consider a simplest example of two-velocity fluids where one of the phases is incompressible (for example, flow of dusty air, or flow of compressible bubbles in an incompressible fluid). A very surprising fact is that one can obtain different governing equations from the same Lagrangian. Different types of the governing equations are due to the choice of independent variables and the corresponding virtual motions. Even if the total momentum and total energy equations are the same, the equations for individual components differ from each other by the presence or absence of gyroscopic forces (also called ‘lift’ forces). These forces have no influence on the hyperbolicity of the governing equations, but can drastically change the distribution of density and velocity of components. To the best of my knowledge, such an uncertainty in obtaining the governing equations of multi- phase flows has never been the subject of discussion in a ‘multi-fluid’ community.

scholarly journals Explicit Finite-Difference Scheme for the Numerical Solution of the Model Equation of Nonlinear Hereditary Oscillator with Variable-Order Fractional Derivatives

2016 ◽  
Vol 26 (3) ◽  
pp. 429-435 ◽  
Author(s):  
Roman I. Parovik
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Fractional Derivatives ◽  
Stability And Convergence ◽  
Variable Order ◽  
The Difference ◽  
Explicit Finite Difference ◽  
The Stability ◽  
Variable Order Fractional Derivatives

Abstract The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is stable and converges to the exact solution.

A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Behrouz Parsa Moghaddam ◽  
José António Tenreiro Machado
Keyword(s):  
Differential Equation ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Spline Interpolation ◽  
Computational Approach ◽  
Variable Order ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Variable Order Fractional Derivatives

AbstractA new computational approach for approximating of variable-order fractional derivatives is proposed. The technique is based on piecewise cubic spline interpolation. The method is extended to a class of nonlinear variable-order fractional integro-differential equation with weakly singular kernels. Illustrative examples are discussed, demonstrating the performance of the numerical scheme.

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