scholarly journals Mathematical models of the pneumatic cascade and a mem-brane pneumatic actuator described by the fractional calculus

2018 ◽  
Vol 19 (12) ◽  
pp. 526-531
Author(s):  
Mirosław Luft ◽  
Artur Nowocień ◽  
Daniel Pietruszczak
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Dynamic Systems ◽  
Dynamic Properties ◽  
Pneumatic Actuator ◽  
Programming Tool ◽  
Analysis And Synthesis ◽  
Continuous Dynamic ◽  
Derivatives Of

The paper presents the analysis of dynamic properties of pneumatic systems such aa pneumatic cascade and a membrane pneumatic actuator using differential equations of integer orders and differential equations with derivatives of non-integer orders. The analyzed systems were described in the domain of time by means of step characteristics and in terms of frequency with the help of Bode characteristics, i.e. logarithmic amplitude and phase characteristics. Each characteristic was determined on the basis of a differential equation with derivatives of non-integer order. To determine the characteristics, an irreplaceable programming tool was the interactive Simulink package built on the basis of the MATLAB programme, which allows the analysis and synthesis of continuous dynamic systems.

scholarly journals Fuzzy Conformable Fractional Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Atimad Harir ◽  
Said Melliani ◽  
Lalla Saadia Chadli
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Uniqueness Theorem ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Existence And Uniqueness Theorem ◽  
Fractional Differential ◽  
Fractional Differentiability ◽  
Derivatives Of ◽  
Fuzzy Fractional Differential Equation

In this study, fuzzy conformable fractional differential equations are investigated. We study conformable fractional differentiability, and we define fractional integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy fractional differential equation by using the concept of conformable differentiability. This concept is based on the enlargement of the class of differentiable fuzzy mappings; for this, we consider the lateral Hukuhara derivatives of order q ∈ 0,1 .

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

XIV.—On General Dynamics:—I. Hamilton's Partial Differential Equations and the Determination of their Complete Integrals

1913 ◽  
Vol 32 ◽  
pp. 164-174
Author(s):  
A. Gray
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Equations Of Motion ◽  
Partial Differential ◽  
General Dynamics ◽  
Elementary Manner ◽  
Derivatives Of

The present paper contains the first part of a series of notes on general dynamics which, if it is found worth while, may be continued. In § 1 I have shown how the first Hamiltonian differential equation is led up to in a natural and elementary manner from the canonical equations of motion for the most general case, that in which the time t appears explicitly in the function usually denoted by H. The condition of constancy of energy is therefore not assumed. In § 2 it is proved that the partial derivatives of the complete integral of Hamilton's equation with respect to the constants which enter into the specification of that integral do not vary with the time, so that these derivatives equated to constants are the integral equations of motion of the system.*

Determining the Ordinary Differential Equation From Noisy Data

2011 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Simulated Annealing Algorithm ◽  
Uncertain Data ◽  
Global Search ◽  
Third Order ◽  
Data Points ◽  
Linear Differential ◽  
Derivatives Of ◽  
Smooth Data

A challenging inverse problem is to identify the smooth function and the differential equation it represents from uncertain data. This paper extends the procedure previously developed for smooth data. The approach involves two steps. In the first step the data is smoothed using a recursive Bezier filter. For smooth data a single application of the filter is sufficient. The final set of data points provides a smooth estimate of the solution. More importantly, it will also identify smooth derivatives of the function away from the edges of the domain. In the second step the values of the function and its derivatives are used to establish a specific form of the differential equation from a particular class of the same. Since the function and its derivatives are known, the only unknowns are parameters describing the structure of the differential equations. These parameters are of two kinds: the exponents of the derivatives and the coefficients of the terms in the differential equations. These parameters can be determined by defining an optimization problem based on the residuals in a reduced domain. To avoid the trivial solution a discrete global search is used to identify these parameters. An example involving a third order constant coefficient linear differential equation is presented. A basic simulated annealing algorithm is used for the global search. Once the differential form is established, the unknown initial and boundary conditions can be obtained by backward and forward numerical integration from the reduced region.

Stability of Continuous Dynamic Systems With Parametric Excitation

1969 ◽  
Vol 36 (2) ◽  
pp. 212-216 ◽  
Author(s):  
J. R. Dickerson ◽  
T. K. Caughey
Keyword(s):  
Partial Differential Equations ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Dynamic Systems ◽  
Parametric Excitation ◽  
Sufficient Conditions ◽  
Partial Differential ◽  
Continuous Dynamic

A Lyapunov-type approach is used to establish sufficient conditions guaranteeing the asymptotic stability of a class of partial differential equations with parametric excitation.

scholarly journals Explicit Solutions of Singular Differential Equation by Means of Fractional Calculus Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Resat Yilmazer ◽  
Okkes Ozturk
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Ordinary Differential Equations ◽  
Explicit Solutions ◽  
Singular Differential Equation ◽  
Linear Ordinary Differential Equations ◽  
Particular Solutions ◽  
Fractional Calculus Operators

Recently, several authors demonstrated the usefulness of fractional calculus operators in the derivation of particular solutions of a considerably large number of linear ordinary and partial differential equations of the second and higher orders. By means of fractional calculus techniques, we find explicit solutions of second-order linear ordinary differential equations.

scholarly journals Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2132
Author(s):  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Analytic Solutions ◽  
Fractional Integrals ◽  
Function Type ◽  
Fractional Differential ◽  
Derivatives Of

In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.

scholarly journals On the exact and approximate solutions of several differential equations with variational derivatives of the first and second orders

2020 ◽  
Vol 56 (1) ◽  
pp. 51-71
Author(s):  
M. V. Ignatenko ◽  
L. A. Yanovich
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Approximate Solution ◽  
Differential Equations ◽  
Interpolation Problem ◽  
Approximate Solutions ◽  
Hyperbolic Type ◽  
The Cauchy Problem ◽  
Variational Derivatives ◽  
Derivatives Of

In this paper, we consider the problem of the exact and approximate solutions of certain differential equations with variational derivatives of the first and second orders. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations with the first variational derivatives are given. An interpolation method for solving ordinary differential equations with variational derivatives is demonstrated. The general scheme of an approximate solution of the Cauchy problem for nonlinear differential equations with variational derivatives of the first order, based on the use of the operator interpolation apparatus, is presented. The exact solution of the differential equation of the hyperbolic type with variational derivatives, similar to the classical Dalamber solution, is obtained. The Hermite interpolation problem with the conditions of coincidence in the nodes of the interpolated and interpolation functionals, as well as their variational derivatives of the first and second orders, is considered for functionals defined on the sets of differentiable functions. The found explicit representation of the solution of the given interpolation problem is based on an arbitrary Chebyshev system of functions. This solution is generalized for the case of interpolation of functionals on one out of two variables and applied to construct an approximate solution of the Cauchy problem for the differential equation of the hyperbolic type with variational derivatives. The description of the material is illustrated by numerous examples.

A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives

1998 ◽  
Author(s):  
Lixia Yuan ◽  
Om P. Agrawal
Keyword(s):  
Differential Equations ◽  
Dynamic Systems ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Integral Formula ◽  
Analytical Technique ◽  
First Order ◽  
Numerical Studies ◽  
Fractional Differential ◽  
Derivatives Of

Abstract This paper presents a numerical scheme for dynamic analysis of mechanical systems subjected to damping forces which are proportional to fractional derivatives of displacements. In this scheme, a fractional differential equation governing the dynamic of a system is transformed into a set of differential equations with no fractional derivative terms. Using Laguerre integral formula, this set is converted to a set of first order ordinary differential equations, which are integrated using a numerical scheme to obtain the response of the system. Numerical studies show that the solution converges as the number of Laguerre node points are increased. Further, results obtained using this scheme agree well with those obtained using an analytical technique.

scholarly journals A Hybrid Function Approach to Solving a Class of Fredholm and Volterra Integro-Differential Equations

2020 ◽  
Vol 25 (2) ◽  
pp. 30
Author(s):  
Aline Hosry ◽  
Roger Nakad ◽  
Sachin Bhalekar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Numerical Method ◽  
Approximate Solutions ◽  
Function Approach ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Block Pulse Functions ◽  
Hybrid Function ◽  
Derivatives Of

In this paper, we use a numerical method that involves hybrid and block-pulse functions to approximate solutions of systems of a class of Fredholm and Volterra integro-differential equations. The key point is to derive a new approximation for the derivatives of the solutions and then reduce the integro-differential equation to a system of algebraic equations that can be solved using classical methods. Some numerical examples are dedicated for showing the efficiency and validity of the method that we introduce.

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