Research On The Applications Of Fractional Calculus In Anomalous Transport

In recent decades,fractional calculus has been successfully applied in various field-s of engineering and science,such as electromagnetics,fluid mechanics,viscoelasticity,anomalous diffusion and signal processing,showing a prosperous trend.This indicates that the theory of fractional calculus is playing unique and irreplaceable roles,therefore its applications have become an international hot issue.Compared with the classical integer differential operator,the fractional differential operator has global or non-local characteristics,which means it is more suitable to describe the long memory and hered-itary characteristics of various material,systems and process.This thesis mainly introduces the applications of fractional calculus in viscoelas-tic materials,fluid mechanics,bio-heat transfer and laser heating.In order to better analyze these abnormal phenomena,the optimal estimation of unknown parameters is obtained by suitable parameter estimation methods.Firstly,fractional constitutive re-lation models,represented by the fractional element networks,are used to describe the time-dependent creep behavior of viscoelastic materials.The creep compliances of the fractional element model,fractional Maxwell model,fractional Kelvin-Voigt model and fractional Poynting-Thomson model are obtained and presented in terms of Mittag-Leffler functions.Three sets of creep experimental data for polymers and rock are em-ployed to demonstrate the effectiveness of these fractional derivative models.To better analyze the viscoelastic behavior of materials,the interior-point algorithm is used to obtain the optimum parameters of these fractional derivative models.The comparison results demonstrate that the fractional Poynting-Thomson model is optimal in simulat-ing the creep behavior of viscoelastic materials.Secondly,the space fractional differential operator is considered to accurately describe the path dependence and long-range cor-relation of anomalous mechanical behaviors,while the space fractional Navier-Stokes equation is obtained by replacing the Laplacian operator in the Navier-Stokes equation with the Riesz fractional derivatives.Meanwhile,the pressure-driven flow between t-wo parallel plates is solved with the finite difference method of fractional differential equations.Based on the above research,the influences of model parameters on the characteristics of fluid flow are analyzed in detail with graphics.Further,the Levenberg-Marquardt algorithm is proposed to estimate model parameters,the results of which show that model parameters have strong influences on the velocity fields as well as that the Levenberg-Marquardt method is more effective to solve the inverse problem of the space fractional differential equation.Thirdly,the fractional dual-phase-lag model and the corresponding bio-heat transfer equation are established to explain the heat transfer phenomenon in processed meat,through which the analytical solution of the fractional dual-phase-lag equation is obtained by means of Laplace and Fourier transforms method and presented under the series form of H functions.On the other hand,the inverse fractional dual-phase-lag heat conduction problems for simultaneous estimation of t-wo relaxation times and orders of fractionality are solved by applying the nonlinear least square method.What's more,the measured and calculated temperatures versus time are compared and discussed.Some numerical examples are also given and dis-cussed.Finally,the short-pulse laser heating of a semi-infinite medium is considered.The time fractional dual-phase-lag model is used as the heat conduction model and the corresponding fractional heat equation with a volumetric heat source is established.The semi-analytical solution for the temperature distribution is obtained by using the Laplace transform method.At the end,the influence of pertinent parameters on the temperature distribution is studied graphically.The thesis is organized,as follows.In Chapter 1,the history of fractional calculus,the main problems to be studied in this thesis and some preliminaries are introduced.In Chapter 2,time-dependent creep behaviors of viscoelastic material based on frac-tional constitutive relation models are studied.The fractional constitutive relationship model is more flexible than the conventional one in describing the properties of viscoelas-tic materials.In this chapter,the fractional derivative models,i.e.fractional element model,fractional Maxwell model,fractional Kelvin-Voigt model and fractional Poynting-Thomson model,are used to study the creep behavior of viscoelastic materials(polymer and rock).The corresponding creep compliances of these models are listed as follows where Ep,q(z)is Mittag-Leffler function.Three sets of creep experimental data,corre-sponding to HDPE,PEEK and rock,are taken into account to illustrate the effectiveness of fractional viscoelastic models.The interior-point algorithm is used for the identifi-cation of model parameters.The results show that the fractional Poynting-Thomson model is optimal in all these fractional viscoelastic models.This model can well cap-ture the short-term and long-term creep behaviors of viscoelastic solid.Meanwhile,the interior-point algorithm is also proved to be effective in the inverse problem to estimate parameters of the fractional viscoelastic models.In Chapter 3,the space fractional Navier-Stokes equation is studied and the pressure-driven flow between two parallel plates is solved with the finite difference method of fractional differential equations.The following space fractional Navier-Stokes equation is obtained by replacing the Laplacian operator in the Navier-Stokes equation with the Riesz fractional derivativeThe unsteady pressure-driven flow between two parallel stationary plates is considered.The fluid,bounded by the plates at y ? and y = L,is initially at rest,but the motion starts suddenly due to a constant pressure gradient.The initial and boundary conditions areThe finite difference method is present for this fractional initial-boundary value problem.The influences of the fractional derivative and generalized Reynolds number on velocity profiles of fluid flow are discussed It is found that the above parameters have great influence on the velocity profiles.Finally,the Levenberg-Marquardt algorithm is used to estimate two model parameters in the space fractional N-S equation.The results also show that the Levenberg-Maxquardt method is effective for the inverse problem in the fractional differential equations.In Chapter 4,the time fractional dual-phase-lag heat conduction model and its application in bio-heat transfer are studied.First of all,based on the classical Fourier law,we propose the following time fractional dual-phase-lag model(?)where 0<?,?<1 and the fractional derivatives are defined in Caputo sense.The heat equation in biological systems is usually expressed by the following Pennes' equation(?)(?)where Q(r,t)= ?bCb(T?-T(r,t))+ qm +qr.Here,p,c and T are the density,specific heat and temperature of the skin tissue,respectively;cb is the specific heat of the blood,Wb is the blood perfusion rate;T? is the temperatures of the arterial blood;qm is the metabolic heat generation in the skin tissue and qr is the heat source due to spatial heating.The combination of Eqs.(24)and(25)provides a bio-heat transfer equation with two fractional parameters ? and ?,which includes the relaxation time ?q and the time delayed rT.Based on the experiments of Mitra et al.,the following initial and boundary conditions are proposed:T(x,0)= T0f(x),q(x,0)= 0(26)(?)(?)The exact solution of(24)-(27)is obtained with the help of integral transforms and H-function.This model is further used to predict the temperatures of processed meat with the nonlinear least square method and a consistent result is obtained.At the end of this chapter,the fractional dual-phase-lag model is proved to be capable of helping to promote understanding the heat conduction in biological tissue.In Chapter 5,the short-pulse laser heating of a semi-infinite medium is considered.The following time fractional dual-phase-lag model(?)is used as the heat conduction model and the corresponding fractional heat equation with a volumetric heat source(?)is established.Here g(x,t)=(1-rf)I0?f(t)e-?x.The corresponding initial and boundary conditions for the laser heating situation are given by(?)(?)(?)T(x,t)= T0,x ??,t>0.(32)The semi-analytical solution for the temperature distribution is obtained by using the Laplace transform method.Finally,the influence of pertinent parameters on the temper-ature distribution is studied graphically by the numerical inversion of Laplace transforms.In Chapter 6,the conclusion of this thesis and future work are discussed.