Application of Dynamic Fractional Differentiation to the Study of Oscillating Viscoelastic Medium With Cylindrical Cavity

2002 ◽  
Vol 124 (4) ◽  
pp. 642-645 ◽  
Author(s):  
D. Ingman ◽  
J. Suzdalnitsky
Keyword(s):  
Constitutive Equation ◽  
Fractional Derivative ◽  
Cylindrical Cavity ◽  
Viscoelastic Medium ◽  
Viscoelastic Behavior ◽  
Additional Term ◽  
Variable Order ◽  
Fractional Differentiation ◽  
Order Function ◽  
Damping Effect

Oscillations of a viscoelastic medium with a cylindrical cavity are studied. The viscosity is taken into account in the form of an additional term in the constitutive equation, proportional to a fractional derivative of variable order. In the considered examples the order function corresponds to dependences obtained for real materials. A damping effect is observed in the amplitude behavior. The field which determines the order function demonstrates the viscoelastic behavior of the material under load.

MIXED FRACTIONAL DIFFERENTIATION OPERATORS IN HÖLDER SPACES DEFINED BY USUAL HÖLDER CONDITION

2019 ◽  
Author(s):  
T. Mamatov ◽  
R. Sabirova ◽  
D. Barakaev
Keyword(s):  
Fractional Derivative ◽  
Fractional Differentiation ◽  
Hölder Condition ◽  
Hölder Spaces ◽  
Main Interest ◽  
Hölder Class ◽  
Holder Class ◽  
Function Of Two Variables ◽  
Holder Spaces

We study mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The main interest being in the evaluation of the latter for the mixed fractional derivative in the cases Hölder class defined by usual Hölder condition

scholarly journals Variable-Order Fractional Models for Wall-Bounded Turbulent Flows

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 782
Author(s):  
Fangying Song ◽  
George Em Karniadakis
Keyword(s):  
Fractional Derivative ◽  
Turbulent Flows ◽  
Fractional Order ◽  
Variable Order ◽  
Universal Curve ◽  
Reynolds Stresses ◽  
Continuous Change ◽  
Mean Velocity ◽  
Symmetric Variable ◽  
Fractional Models

Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30–50 y+ to 0.1–0.2 R+ in a pipe of radius R. Here, we propose a fundamentally new approach based on fractional calculus to model the entire mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of variable-order that decays with the distance from the wall. Surprisingly, we find that this variable fractional order has a universal form for all Reynolds numbers and for three different flow types, i.e., channel flow, Couette flow, and pipe flow. We first use existing databases from direct numerical simulations (DNSs) to lean the variable-order function and subsequently we test it against other DNS data and experimental measurements, including the Princeton superpipe experiments. Taken together, our findings reveal the continuous change in rate of turbulent diffusion from the wall as well as the strong nonlocality of turbulent interactions that intensify away from the wall. Moreover, we propose alternative formulations, including a divergence variable fractional (two-sided) model for turbulent flows. The total shear stress is represented by a two-sided symmetric variable fractional derivative. The numerical results show that this formulation can lead to smooth fractional-order profiles in the whole domain. This new model improves the one-sided model, which is considered in the half domain (wall to centerline) only. We use a finite difference method for solving the inverse problem, but we also introduce the fractional physics-informed neural network (fPINN) for solving the inverse and forward problems much more efficiently. In addition to the aforementioned fully-developed flows, we model turbulent boundary layers and discuss how the streamwise variation affects the universal curve.

scholarly journals On the sub–diffusion fractional initial value problem with time variable order

2021 ◽  
Vol 10 (1) ◽  
pp. 1301-1315
Author(s):  
Eduardo Cuesta ◽  
Mokhtar Kirane ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Fractional Derivative ◽  
Initial Value Problem ◽  
Regularity Result ◽  
Time Variable ◽  
Variable Order ◽  
Initial Value ◽  
Integration By Parts ◽  
Integration By Parts Formula ◽  
Variable Order Fractional Derivative

Abstract We consider a fractional derivative with order varying in time. Then, we derive for it a Leibniz' inequality and an integration by parts formula. We also study an initial value problem with our time variable order fractional derivative and present a regularity result for it, and a study on the asymptotic behavior.

Using the generalized Adams-Bashforth-Moulton method for obtaining the numerical solution of some variable-order fractional dynamical models

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohamed M. Khader
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Numerical Solution ◽  
Numerical Results ◽  
Convergence Order ◽  
Variable Order ◽  
Numerical Treatment ◽  
Dynamical Models ◽  
Fractional Modeling ◽  
Dynamics Problems

AbstractThis paper is devoted to introduce a numerical treatment using the generalized Adams-Bashforth-Moulton method for some of the variable-order fractional modeling dynamics problems, such as Riccati and Logistic differential equations. The fractional derivative is described in Caputo variable-order fractional sense. The obtained numerical results of the proposed models show the simplicity and efficiency of the proposed method. Moreover, the convergence order of the method is also estimated numerically.

scholarly journals Approximation and application of the Riesz-Caputo fractional derivative of variable order with fixed memory

Meccanica ◽  
2021 ◽  
Author(s):  
Tomasz Blaszczyk ◽  
Krzysztof Bekus ◽  
Krzysztof Szajek ◽  
Wojciech Sumelka
Keyword(s):  
Differential Operator ◽  
Fractional Derivative ◽  
Rate Of Convergence ◽  
Continuum Mechanics ◽  
Caputo Fractional Derivative ◽  
Polynomial Interpolation ◽  
Piecewise Linear ◽  
Variable Order ◽  
Piecewise Constant ◽  
Quadratic Interpolation

AbstractIn this paper, the Riesz-Caputo fractional derivative of variable order with fixed memory is considered. The studied non-integer differential operator is approximated by means of modified basic rules of numerical integration. The three proposed methods are based on polynomial interpolation: piecewise constant, piecewise linear, and piecewise quadratic interpolation. The errors generated by the described methods and the experimental rate of convergence are reported. Finally, an application of the Riesz-Caputo fractional derivative of space-dependent order in continuum mechanics is depicted.

Two-Dimensional Legendre Wavelets for Solving Variable-Order Fractional Nonlinear Advection-Diffusion Equation with Variable Coefficients

2018 ◽  
Vol 19 (7-8) ◽  
pp. 793-802 ◽  
Author(s):  
M. Hosseininia ◽  
M. H. Heydari ◽  
Z. Avazzadeh ◽  
F. M. Maalek Ghaini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Main Idea ◽  
Variable Coefficients ◽  
Variable Order ◽  
Two Dimensional ◽  
Legendre Wavelets ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Nonlinear Advection

AbstractThis article studies a numerical scheme for solving two-dimensional variable-order time fractional nonlinear advection-diffusion equation with variable coefficients, where the variable-order fractional derivative is in the Caputo type. The main idea is expanding the solution in terms of the 2D Legendre wavelets (2D LWs) where the variable-order time fractional derivative is discretized. We describe the method using the matrix operators and then implement it for solving various types of fractional advection-diffusion equations. The experimental results show the computational efficiency of the new approach.

A Molecular Structure-Informed Viscoelastic Constitutive Model for Natural Rubber Materials

2021 ◽  
Author(s):  
Jiwon Jung ◽  
Chanwook Park ◽  
myungshin RYU ◽  
Gunjin Yun
Keyword(s):  
Molecular Structure ◽  
Constitutive Equation ◽  
Molecular System ◽  
Loss Modulus ◽  
Low Frequency ◽  
Viscoelastic Behavior ◽  
Material Model ◽  
Coarse Grained ◽  
Diffusivity Coefficient ◽  
Element Analysis

Abstract This paper presents a molecular structure-informed viscoelastic constitutive equation that adopts the Doi-Edward’s tube model with coarse-grained molecular dynamics (MD) simulation and primitive path analysis. Since this model contains polymer physics-related parameters directly obtained from molecular simulations, it can reflect molecular information in predictions of the viscoelastic behavior of elastomers, unlike other empirical models. The proposed incremental formulations and constitutive stiffness matrix were implemented into implicit finite element analysis (FEA) codes as a user-supplied material model and viscoelastic properties (storage, loss modulus, and tan⁡δ) were calculated from the constitutive equation. While obtaining polymer dynamics parameter of the molecular system, a relationship between self-diffusivity coefficient (D_c) and the polymerization degree of the polymer was confirmed. Furthermore, a series of parametric studies showed that increase of the primitive path length (L) and decrease of D_c have led to the strengthening of moduli and decrease of tan⁡δ peak. Moreover, under the same condition, the shift of tan⁡δ peak to low-frequency domain was observed, which implies a decline in free volume in the molecular system and an increase in the glass transition temperature.

scholarly journals Consolidation Analysis of Ideal Sand-Drained Ground with Fractional-Derivative Merchant Model and Non-Darcian Flow Described by Non-Newtonian Index

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Zhongyu Liu ◽  
Penglu Cui ◽  
Jiachao Zhang ◽  
Yangyang Xia
Keyword(s):  
Fractional Derivative ◽  
Flow Model ◽  
Soft Soil ◽  
Viscoelastic Behavior ◽  
Vertical Flow ◽  
Vertical Drains ◽  
Saturated Clay ◽  
Implicit Finite Difference ◽  
Consolidation Analysis ◽  
The Ideal

To further investigate the rheological consolidation mechanism of soft soil ground with vertical drains, the fractional-derivative Merchant model (FDMM) is introduced to describe the viscoelastic behavior of saturated clay around the vertical drains, and the flow model with the non-Newtonian index is employed to describe the non-Darcian flow in the process of rheological consolidation. Accordingly, the governing partial differential equation of the ideal sand-drained ground with coupled radial-vertical flow is obtained under the assumption that the vertical strains develop freely. Then, the numerical solution to the consolidation system is conducted using the implicit finite difference method. The validity of this method is verified by comparing the results of Barron’s consolidation theory. Furthermore, the effects of the parameters of non-Darcian flow and FDMM on the rheological consolidation of ground with vertical drains are illustrated and discussed.

An efficient and novel technique for solving continuously variable fractional order mass-spring-damping system

2017 ◽  
Vol 34 (8) ◽  
pp. 2815-2835 ◽  
Author(s):  
S. Sahoo ◽  
S. Saha Ray ◽  
S. Das
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Dynamic Models ◽  
Variable Order ◽  
Recursive Method ◽  
Recursion Method ◽  
Content Type ◽  
Viscoelastic Dampers ◽  
Oscillating Systems ◽  
Mass Spring

Purpose In this paper, the formulation and analytic solutions for fractional continuously variable order dynamic models, namely, fractional continuously mass-spring damper (continuously variable fractional order) systems, have been presented. The authors will demonstrate via two cases where the frictional damping given by fractional derivative, the order of which varies continuously – while the mass moves in a guide. Here, the continuously changing nature of the fractional-order derivative for dynamic systems has been studied for the first time. The solutions of the fractional continuously variable order mass-spring damper systems have been presented here by using a successive recursive method, and the closed form of the solutions has been obtained. By using graphical plots, the nature of the solutions has been discussed for the different cases of continuously variable fractional order of damping force for oscillator. The purpose of the paper is to formulate the continuously variable order mass-spring damper systems and find their analytical solutions by successive recursion method. Design/methodology/approach The authors have used the viscoelastic and viscous – viscoelastic dampers for describing the damping nature of the oscillating systems, where the order of the fractional derivative varies continuously. Findings By using the successive recursive method, here, the authors find the solution of the fractional continuously variable order mass-spring damper systems, and then obtain close-form solutions. The authors then present and discuss the solutions obtained in the cases with the continuously variable order of damping for an oscillator through graphical plots. Originality/value Formulation of fractional continuously variable order dynamic models has been described. Fractional continuous variable order mass-spring damper systems have been analysed. A new approach to find solutions of the aforementioned dynamic models has been established. Viscoelastic and viscous – viscoelastic dampers are described. The discussed damping nature of the oscillating systems has not been studied yet.

Sign in / Sign up

Export Citation Format

Share Document