scholarly journals Mathematical Modeling of Linear Fractional Oscillators

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1879 ◽  
Author(s):  
Roman Parovik
Keyword(s):  
Cauchy Problem ◽  
Fractional Derivatives ◽  
Model Equation ◽  
Initial Conditions ◽  
Memory Function ◽  
Analytical Formula ◽  
Forced Oscillations ◽  
Caputo Fractional Derivatives ◽  
Specific Test ◽  
The Cauchy Problem

In this work, based on Newton’s second law, taking into account heredity, an equation is derived for a linear hereditary oscillator (LHO). Then, by choosing a power-law memory function, the transition to a model equation with Gerasimov–Caputo fractional derivatives is carried out. For the resulting model equation, local initial conditions are set (the Cauchy problem). Numerical methods for solving the Cauchy problem using an explicit non-local finite-difference scheme (ENFDS) and the Adams–Bashforth–Moulton (ABM) method are considered. An analysis of the errors of the methods is carried out on specific test examples. It is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method. Forced oscillations of linear fractional oscillators (LFO) are investigated. Using the ABM method, the amplitude–frequency characteristics (AFC) were constructed, which were compared with the AFC obtained by the analytical formula. The Q-factor of the LFO is investigated. It is shown that the orders of fractional derivatives are responsible for the intensity of energy dissipation in fractional vibrational systems. Specific mathematical models of LFOs are considered: a fractional analogue of the harmonic oscillator, fractional oscillators of Mathieu and Airy. Oscillograms and phase trajectories were constructed using the ABM method for various values of the parameters included in the model equation. The interpretation of the simulation results is carried out.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

scholarly journals Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation

2021 ◽  
Vol 5 (3) ◽  
pp. 66
Author(s):  
Azmat Ullah Khan Niazi ◽  
Jiawei He ◽  
Ramsha Shafqat ◽  
Bilal Ahmed
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
The Cauchy Problem ◽  
Fractional Differential ◽  
Ulam Type Stability ◽  
Fuzzy Fractional Differential Equations ◽  
Fuzzy Fractional Differential Equation

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.

scholarly journals The nonlocal solvability conditions for a system of two quasilinear equations of the first order with absolute terms

2019 ◽  
Vol 21 (3) ◽  
pp. 317-328
Author(s):  
Marina V. Dontsova
Keyword(s):  
Cauchy Problem ◽  
Finite Number ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Local Solution ◽  
Quasilinear Equations ◽  
Solvability Conditions ◽  
First Order ◽  
The Cauchy Problem ◽  
Global Estimates

The Cauchy problem for a system of two first-order quasilinear equations with absolute terms is considered. The study of this problem’s solvability in original coordinates is based on the method of an additional argument. The existence of the local solution of the problem with smoothness which is not lower than the smoothness of the initial conditions, is proved. Sufficient conditions of existence are determined for the nonlocal solution that is continued by a finite number of steps from the local solution. The proof of the nonlocal resolvability of the Cauchy problem relies on original global estimates.

scholarly journals Numerical investigation of the Carleman system

Vestnik MGSU ◽  
2015 ◽  
pp. 7-15
Author(s):  
Ol’ga Aleksandrovna Vasil’eva
Keyword(s):  
Cauchy Problem ◽  
Equilibrium State ◽  
Numerical Investigation ◽  
Initial Conditions ◽  
Practical Interest ◽  
Statistical Characteristics ◽  
Stabilization Time ◽  
System Solution ◽  
Stochastic Solution ◽  
The Cauchy Problem

In the article the Cauchy problem of the Carleman equation is considered. The Carleman system of equations is a model problem of the kinetic theory of gases. It is a discrete kinetic model of one-dimensional gas consisting of identical monatomic molecules. The molecules can have one of two speeds, which have equal values and opposite directions. This system of the equations is quasi-linear hyperbolic system of partial differential equations. There is no analytic solution for this problem in general case. So, the numerical investigation of the Cauchy problem of the Carleman system solution is very important.The paper presents and discusses the results of the numerical investigation of the Cauchy problem for the studied system solution with periodic initial conditions. The dependence of the stabilization time of the solution and the time dependence of energy exchange from small parameter are obtained.The second point of the paper is numerical investigation of the solution of the Cauchy problem with non-periodic initial conditions. The solution stabilization to the equilibrium state is obtained. The solution stabilization time is compared with stabilization time in periodic case.The final point of the paper is numerical investigation of the Cauchy problem with stationary normal processes as initial conditions. The solution to this problem is two stationary stochastic processes for any fixed value of time variable. As a rule, the practical interest is not a stochastic solution but its statistical characteristics. The stochastic solution realization is presented and discussed. The dependence of the mathematical expectation of the solution deviation modulus from equilibrium state is obtained. It demonstrates the process of the solution stabilization.

scholarly journals SOLVABILITY OF THE CAUCHY PROBLEM FOR EQUATIONS WITH RIEMANN–LIOUVILLE’S FRACTIONAL DERIVATIVES

2018 ◽  
Vol 62 (4) ◽  
pp. 391-397 ◽  
Author(s):  
Petr P. Zabreiko ◽  
Svetlana V. Ponomareva
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Unique Solution ◽  
Metric Space ◽  
Fractional Derivatives ◽  
Volterra Equation ◽  
Complete Metric Space ◽  
The Cauchy Problem ◽  
The Right

In this article we study the solvability of the analogue of the Cauchy problem for ordinary differential equations with Riemann–Liouville’s fractional derivatives with a nonlinear restriction on the right-hand side of functions in certain spaces. The conditions for solvability of the problem under consideration in given function spaces, as well as the conditions for existence of a unique solution are given. The study uses the method of reducing the problem to the second-kind Volterra equation, the Schauder principle of a fixed point in a Banach space, and the Banach-Cachoppoli principle of a fixed point in a complete metric space.

scholarly journals Solution of the singular Cauchy problem for a general inhomogeneous Euler–Poisson–Darboux equation

2018 ◽  
Vol 34 (2) ◽  
pp. 255-267
Author(s):  
ELINA SHISHKINA ◽  
Keyword(s):  
Cauchy Problem ◽  
Initial Conditions ◽  
Bessel Operator ◽  
Second Derivative ◽  
Singular Cauchy Problem ◽  
Classical Formulation ◽  
The Cauchy Problem ◽  
Darboux Equation

In this paper, we solve Cauchy problem for a general form of an inhomogeneous Euler–Poisson–Darboux equation, where Bessel operator acts instead of the each second derivative. In the classical formulation, the Cauchy problem for this equation is not correct. However, for a specially selected form of the initial conditions, the equation has a solution. The general form of the Euler–Poisson–Darboux equation with such conditions we will call the singular Cauchy problem.

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