scholarly journals Recovering multiple fractional orders in time-fractional diffusion in an unknown medium

2021 ◽  
Vol 477 (2253) ◽  
pp. 20210468
Author(s):  
Bangti Jin ◽  
Yavar Kian
Keyword(s):  
Single Point ◽  
Numerical Procedure ◽  
Nonlinear Least Squares ◽  
Fractional Diffusion ◽  
Small Time ◽  
Initial Condition ◽  
Model Function ◽  
Multiple Orders ◽  
Full Knowledge ◽  
Fractional Orders

In this work, we investigate an inverse problem of recovering multiple orders in a time-fractional diffusion model from the data observed at one single point on the boundary. We prove the unique recovery of the orders together with their weights, which does not require a full knowledge of the domain or medium properties, e.g. diffusion and potential coefficients, initial condition and source in the model. The proof is based on Laplace transform and asymptotic expansion. Furthermore, inspired by the analysis, we propose a numerical procedure for recovering these parameters based on a nonlinear least-squares fitting with either fractional polynomials or rational approximations as the model function, and provide numerical experiments to illustrate the approach for small time t .

On Kelvin–Stuart vortices in a viscous fluid

2008 ◽  
Vol 366 (1876) ◽  
pp. 2717-2728 ◽  
Author(s):  
L.E Fraenkel
Keyword(s):  
Reynolds Number ◽  
Viscous Fluid ◽  
Small Time ◽  
Navier Stokes ◽  
Initial Condition ◽  
Small Reynolds Number ◽  
The One ◽  
Stuart Vortices ◽  
Reversed Time

When one contemplates the one-parameter family of steady inviscid shear flows discovered by J. T. Stuart in 1967, an obvious thought is that these flows resemble a row of vortices diffusing in a viscous fluid, with the parameter playing the role of a reversed time. In this paper, we ask how close this resemblance is. Accordingly, the paper begins to explore Navier–Stokes solutions having as initial condition the classical, irrotational flow due to a row of point vortices. However, since we seek explicit answers, such exploration seems possible only in two relatively easy cases: that of small time and arbitrary Reynolds number and that of small Reynolds number and arbitrary time.

Estimation of the Time-Dependent Body Force Needed to Exert on a Membrane to Reach a Desired State at the Final Time

2019 ◽  
Vol 19 (2) ◽  
pp. 323-339
Author(s):  
Lyubomir Boyadjiev ◽  
Kamal Rashedi ◽  
Mourad Sini
Keyword(s):  
Body Force ◽  
Single Point ◽  
Finite Size ◽  
Numerical Procedure ◽  
Time Dependent ◽  
Final Time ◽  
Initial Shape ◽  
Additional Information ◽  
Boundary Force ◽  
Hölder Stability

AbstractWe are concerned with the wave propagation in a homogeneous 2D or 3D membrane Ω of finite size. We assume that either the membrane is initially at rest or we know its initial shape (but not necessarily both) and its boundary is subject to a known boundary force. We address the question of estimating the needed time-dependent body force to exert on the membrane to reach a desired state at a given final time T. As an additional information, we ask for the displacement on the boundary. We consider the displacement either at a single point of the boundary or on the whole boundary. First, we show the uniqueness of solution of these inverse problems under natural conditions on the final time T. If, in addition, the displacement on the whole boundary is only time dependent (which means that the boundary moves with a constant speed), this condition on T is removed if Ω satisfies Schiffer’s property. Second, we derive a conditional Hölder stability inequality for estimating such a time-dependent force. Third, we propose a numerical procedure based on the application of the satisfier function along with the standard Fourier expansion of the solution to the problems. Numerical tests are given to illustrate the applicability of the proposed procedure.

Numerical Procedure for Identification of Constitutive Equations Based on Experimental Data

2013 ◽  
Vol 43 (3) ◽  
pp. 43-50 ◽  
Author(s):  
Anguel Baltov ◽  
Ana Yanakieva
Keyword(s):  
Experimental Data ◽  
Experimental Evidence ◽  
Constitutive Equations ◽  
Constitutive Relations ◽  
Numerical Procedure ◽  
Small Time ◽  
Metal Sheet ◽  
Thin Metal ◽  
Stress Strain ◽  
Thin Metal Sheet

Abstract In this research experimental data obtained by well-known methods of displacement measurement are considered, whereas a thin cell meshwork has been fixed to the surface of the observed loaded object (thin metal sheet, for instance) and a set of point displacements has been ob- tained. Based on such experimental evidence, an eight point numerical procedure is proposed in the paper to identify the constitutive relations for small time increments. The presented approach is also applicable in case of micro deformations and facilitates the derivation of nonlinear constitutive equations in stress/strain increments. Approximate error es- timation of the procedure is additionally performed, and a test example is given.

scholarly journals Numerical Inversion for the Multiple Fractional Orders in the Multiterm TFDE

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Chunlong Sun ◽  
Gongsheng Li ◽  
Xianzheng Jia
Keyword(s):  
Numerical Stability ◽  
Noisy Data ◽  
Fractional Diffusion ◽  
Exact Order ◽  
Inversion Problem ◽  
Inversion Algorithm ◽  
Regularization Algorithm ◽  
Fractional Orders ◽  
Diffusion Behaviors ◽  
Key Parameter

The fractional order in a fractional diffusion model is a key parameter which characterizes the anomalous diffusion behaviors. This paper deals with an inverse problem of determining the multiple fractional orders in the multiterm time-fractional diffusion equation (TFDE for short) from numerics. The homotopy regularization algorithm is applied to solve the inversion problem using the finite data at one interior point in the space domain. The inversion fractional orders with random noisy data give good approximations to the exact order demonstrating the efficiency of the inversion algorithm and numerical stability of the inversion problem.

scholarly journals A Gallery of Root Locus of Fractional Systems

2013 ◽  
Author(s):  
J. A. Tenreiro Machado
Keyword(s):  
Stability Analysis ◽  
Transfer Functions ◽  
Numerical Procedure ◽  
Root Locus ◽  
Closed Loop System ◽  
Fractional Systems ◽  
Finite Precision ◽  
Know How ◽  
The Stability ◽  
Fractional Orders

The root locus (RL) is a classical tool for the stability analysis of integer order linear systems, but its application in the fractional counterpart poses some difficulties. Therefore, researchers have mainly preferred to adopt frequency based methods. Nevertheless, recently the RL was considered for the stability analysis of fractional systems. One first method is by tacking advantage of commensurable expressions that occur when truncating fractional orders up to a finite precision. The second method consists of searching the complex plane for solutions of the characteristic equation using a numerical procedure. The resulting charts are insightful about the characteristics of the closed-loop system that outperform the frequency response methods. Given the limited know how in this particular topic and the shortage of literature, this study explores several types of fractional-order transfer functions and presents the corresponding RL.

Wall imperfections as a triggering mechanism for Stokes-layer transition

1994 ◽  
Vol 264 ◽  
pp. 107-135 ◽  
Author(s):  
P. Blondeaux ◽  
G. Vittori
Keyword(s):  
Reynolds Number ◽  
Numerical Procedure ◽  
Threshold Value ◽  
Numerical Approach ◽  
Small Time ◽  
Transition To Turbulence ◽  
Time Range ◽  
Layer Transition ◽  
Stokes Layer ◽  
Damping Effect

The boundary layer generated by the harmonic oscillations of a wavy wall in a fluid otherwise at rest is studied. First the wall waviness is assumed to be of small amplitude and large values of the Reynolds number are considered. The results obtained by means of a linear analysis, where the time variable appears only as a parameter, show that resonance may occur. Indeed it is found that when the Reynolds number is larger than a critical value, an instant within the decelerating part of the cycle exists such that a waviness of infinitesimal amplitude induces unbounded perturbations of the flow in the Stokes layer. The passage through resonance is then studied by means of a multiple-timescale approach, taking into account the damping effect of local acceleration within a small time range around resonance. The asymptotic approach fails beyond a threshold value of the Reynolds number, because the damping effect of the local acceleration terms spreads over the whole cycle. The problem is then tackled by means of an approach that takes into account the above damping effect throughout the whole cycle. Finally, a numerical procedure is used that also allows the inclusion of nonlinear terms and the study of the interactions among forced and free modes. The numerical approach reveals that, even for relatively large values of the amplitude of the wall waviness, nonlinear effects are negligible and the damping of resonance is mainly due to local acceleration effects. The relevance of the results to the understanding of transition to turbulence in Stokes layers is discussed.

Generalized fraction evolution equations with fractional Gross Laplacian

2016 ◽  
Vol 19 (2) ◽  
Author(s):  
Samah Horrigue ◽  
Habib Ouerdiane ◽  
Imen Salhi
Keyword(s):  
Diffusion Equation ◽  
Laplace Transform ◽  
Explicit Solution ◽  
Evolution Equations ◽  
Fractional Diffusion ◽  
Generalized Function ◽  
Initial Condition ◽  
Infinite Dimensions ◽  
The Laplace Transform ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, we define and consider the fractional Gross Laplacian which is characterized by the Laplace transform. As application, we study the generalized Riemann-Liouville time fractional diffusion equation in infinite dimensions. We show that the explicit solution is given as the convolution between the initial condition and a generalized function related to the Mittag-Leffler function.

Application of the Direct Stiffness Method to Plane Problems Involving Large, Time-Dependent Deformations

1966 ◽  
Vol 88 (4) ◽  
pp. 771-776 ◽  
Author(s):  
G. F. Gerstenkorn ◽  
A. S. Kobayashi
Keyword(s):  
Large Time ◽  
Numerical Procedure ◽  
Structural Response ◽  
Small Time ◽  
Time Dependent ◽  
Polar Coordinates ◽  
Structural Problems ◽  
Stiffness Method ◽  
Direct Stiffness ◽  
Creep Deformations

The direct stiffness method is used to formulate a numerical procedure for solving plane structural problems involving large, time-dependent deformations and nonhomogeneous, time-dependent material properties. The stiffness matrix in polar coordinates is derived for the state of plane strain. The nonlinear structural response is incrementally linearized by considering the deformation process to be linear within small time increments. The developed procedure is compared numerically with a known solution of creep deformations in a thick-walled cylinder subjected to internal pressure loading and elevated temperature.

scholarly journals Approximate solution of the non-linear diffusion equation of multiple orders

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 683-687 ◽  
Author(s):  
Fei Wu ◽  
Xiao-Jun Yang
Keyword(s):  
Approximate Solution ◽  
Diffusion Equation ◽  
Analytical Solutions ◽  
Integral Form ◽  
Fractional Diffusion ◽  
Experiment Data ◽  
Linear Diffusion ◽  
Multiple Orders ◽  
Two Parameters ◽  
Adomian Polynomial

In this paper, fractional diffusion equation of multiple orders is approximately solved. The equation is given in the equivalent integral form. The Adomian polynomial is adopted and analytical solutions are obtained. The result contains two parameters that can have more space for fitting the experiment data.

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