scholarly journals On linear viscoelasticity within general fractional derivatives without singular kernel

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 335-342 ◽  
Author(s):  
Feng Gao ◽  
Xiao-Jun Yang ◽  
Syed Mohyud-Din
Keyword(s):  
Laplace Transform ◽  
Mathematical Models ◽  
Present Article ◽  
Linear Viscoelasticity ◽  
Fractional Derivatives ◽  
Singular Kernel ◽  
The Real ◽  
The Laplace Transform ◽  
Mechanical Models

The Riemann-Liouville and Caputo-Liouville fractional derivatives without singular kernel are proposed as mathematical tools to describe the mathematical models in line viscoelasticity in the present article. The fractional mechanical models containing the Maxwell and Kelvin-Voigt elements are graphically discussed with the Laplace transform. The results are accurate and efficient to reveal the complex behaviors of the real materials.

scholarly journals Solution of mathematical models by localization

2002 ◽  
Vol 123 (27) ◽  
pp. 1-17 ◽  
Author(s):  
Bogoljub Stankovic
Keyword(s):  
Boundary Conditions ◽  
Laplace Transform ◽  
Mathematical Models ◽  
Fractional Derivatives ◽  
Linear Equations ◽  
Generalized Solutions ◽  
The Laplace Transform ◽  
Definition Of

A definition of the Laplace transform of elements of D'?(?) of a subspace of distributions is given which can successfully be ap?plied to solve in a prescribed domain linear equations with derivatives, par?tial derivatives fractional derivatives and convolutions, all with initial or boundary conditions, regardless of the existence of classical or generalized solutions.

scholarly journals Inversion of the Laplace Transform from the Real Axis Using an Adaptive Iterative Method

2009 ◽  
Vol 2009 ◽  
pp. 1-38 ◽  
Author(s):  
Sapto W. Indratno ◽  
Alexander G. Ramm
Keyword(s):  
Approximate Solution ◽  
Iterative Method ◽  
Laplace Transform ◽  
Real Axis ◽  
Quadrature Formula ◽  
Compact Support ◽  
Unknown Function ◽  
Stopping Rule ◽  
The Real ◽  
The Laplace Transform

A new method for inverting the Laplace transform from the real axis is formulated. This method is based on a quadrature formula. We assume that the unknown function is continuous with (known) compact support. An adaptive iterative method and an adaptive stopping rule, which yield the convergence of the approximate solution to , are proposed in this paper.

POTENTIALS OF ARBITRARY FORCES WITH FRACTIONAL DERIVATIVES

2004 ◽  
Vol 19 (17n18) ◽  
pp. 3083-3092 ◽  
Author(s):  
EQAB M. RABEI ◽  
TAREQ S. ALHALHOLY ◽  
AKRAM ROUSAN
Keyword(s):  
Laplace Transform ◽  
General Formula ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Hamiltonian Mechanics ◽  
Dissipative Effects ◽  
Special Cases ◽  
Exact Agreement ◽  
The Laplace Transform ◽  
Lagrangian And Hamiltonian Mechanics

The Laplace transform of fractional integrals and fractional derivatives is used to develop a general formula for determining the potentials of arbitrary forces: conservative and nonconservative in order to introduce dissipative effects (such as friction) into Lagrangian and Hamiltonian mechanics. The results are found to be in exact agreement with Riewe's results of special cases. Illustrative examples are given.

scholarly journals The Mathematical Models of Lattice Functions in Modelling of Control System

2021 ◽  
Vol 2096 (1) ◽  
pp. 012149
Author(s):  
V Kramar
Keyword(s):  
Mathematical Model ◽  
Laplace Transform ◽  
Mathematical Models ◽  
Control Systems ◽  
Time Domain ◽  
Discrete Control ◽  
Automatic Control Systems ◽  
The Laplace Transform ◽  
Lattice Function ◽  
The Time Domain

Abstract The paper proposes an approach to constructing a mathematical model of lattice functions, which are mainly used in the study of discrete control systems in the time and domain of the Laplace transform. The proposed approach is based on the assumption of the physical absence of an impulse element. An alternative to the classical approach to the description of discrete data acquisition - the process of quantization in time, is considered. As a result, models of the lattice function in the time domain and the domain of the discrete Laplace transform are obtained. Based on the obtained mathematical models of lattice functions, a mathematical model of the time quantization element of the system is obtained. This will allow in the future to proceed to the construction of mathematical models of various discrete control systems, incl. expanding the proposed approaches to the construction of mathematical models of multi-cycle continuous-discrete automatic control systems

scholarly journals On applications of Caputo k-fractional derivatives

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ghulam Farid ◽  
Naveed Latif ◽  
Matloob Anwar ◽  
Ali Imran ◽  
Muhammad Ozair ◽  
...  
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Chebyshev Inequality ◽  
Absolutely Continuous ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Type Variable

Abstract This research explores Caputo k-fractional integral inequalities for functions whose nth order derivatives are absolutely continuous and possess Grüss type variable bounds. Using Chebyshev inequality (Waheed et al. in IEEE Access 7:32137–32145, 2019) for Caputo k-fractional derivatives, several integral inequalities are derived. Further, Laplace transform of Caputo k-fractional derivative is presented and Caputo k-fractional derivative and Riemann–Liouville k-fractional integral of an extended generalized Mittag-Leffler function are calculated. Moreover, using the extended generalized Mittag-Leffler function, Caputo k-fractional differential equations are presented and their solutions are proposed by applying the Laplace transform technique.

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