Research On Fractional Derivative Modeling And Their Applications In Complex Mechanical Problems

In recent years,it has been found that the integer order calculus theory always has limitations when describing anomalous phenomena in complex mechanical problems such as diffusion,fluid flow,and heat transfer,while fractional calculus in describing these abnormal physical phenomena shows superiority due to its memory and heredity,which makes the fractional calculus theory become an important tool to describe various complex mechanical and physical behaviors,and has been successfully applied in clinical medicine,biomedicine,environmental protection,mechanical processing,petroleum transportation,sewage treatment,electric power industry,blood circulation and other fields.At present,the research of fractionalorder calculus theory in various complex mechanical problems is still an international hot issue,and constructing a suitable fractional-order model to characterize its transport mechanism is an important topic of current research.However,the existence of fractional operators makes it difficult to solve fractional models.Therefore,it is a very meaningful work to study complex mechanical problems and develop efficient numerical solutions using fractional derivatives.This paper mainly studies the application of fractional calculus theory in pharmacokinetics,laser heating,flow and heat transfer of viscoelastic fluids,develops efficient and feasible numerical algorithms for various fractional models,and carries out parameter inversion with some experimental data to estimate the optimal values of model parameters,and analyze the anomalous transport mechanism in the process of diffusion and heat transfer.First,we studied the application of time fractional derivatives in pharmacokinetics and used the two compartment fractional model to explore the dynamic changes of drugs in body under different administration methods.We gave and verified the numerical method for solving the fractional compartment model,fitted to experimental data for amiodarone and validated the accuracy and applicability of fractional order model in describing drug diffusion in vivo.Afterwards,we used the time fractional derivative to study the heat conduction problem on the thin films heated by femtosecond laser,established a fractional dual-phase-lag heat conduction model to describe the abnormal heat transfer phenomenon on the film,and derived its semi-analytical and numerical solutions.The parameter estimation is carried out,and the experimental data of laser heating metal thin films are fitted.It is proved that the fractional dual-phase-lag heat conduction model can accurately describe the heat conduction problem on the laser heated metal film.Then,we consider the time distributed-order model and the variable fractional order model to describe the flow behavior of viscoelastic fluid in a circular tube driven by periodic pressure,and construct effective numerical methods to compare the differences in the physical behavior described by the two models.The transport process of viscoelastic fluid with different characteristics is further analyzed.In addition,we also studied the rotational flow and heat transfer of magnetic fluid on an infinite plate by using the time distributed-order derivative,proposed the Crank-Nicolson finite difference method to solve the numerical solution of the distributed-order model,and discussed the influence of model parameters on fluid flow and heat transfer behavior.Finally,we study the two-dimensional magnetohydrodynamic flow and heat transfer of generalized Maxwell in rectangular pipes under the action of pressure gradient and magnetic field by using time and space fractional derivatives,deduce the alternating direction implicit algorithm,and verify the validity of the numerical method.The specific content of this article is arranged as follows:In chapter 1:we introduce the origin and development history of fractional calculus,list several common definitions of fractional derivatives and their numerical discrete schemes,and mention the definitions and properties of Fourier transform and Laplace transform,which provide theoretical support for later research.In chapter 2:we introduce the fractional calculus theory into the two compartment model to describe the dynamic process of drug diffusion in the human body.Considering different fractional order transmission processes,we propose a two compartment fractional order model,And based on shifted Griinwald-Letnikov approximation formula and the L1 formula,we develope two numerical methods for solving the compartment model.The comparison between the numerical solution and the semi-analytical solution proves the effectiveness of the numerical algorithm,and then the influence of fractional order parameters on the dosage in human body is discussed.Finally,hybrid simplex search method and particle swarm optimization algorithm is used to estimate the parameters of the fractional order model.It indicates that the two compartment fractional order model can be used to fit the pharmacokinetic data of amiodarone.In chapter 3:we propose a time fractional order dual-phase-lag(DPL)heat conduction model for thin gold films heating by femtosecond laser pulses using the fractional Taylor series expansion.The fractional order DPL model is solved by analytical and numerical methods,respectively.First,in order to describe the effect of femtosecond laser pulses on the temperature distribution of the gold film,we derived a semi-analytical solution using the Laplace transform method and Fourier cosine transform method,and then we use the L1 approximation formula of Caputo fractional derivative to derive the implicit finite difference scheme.The effectiveness of the numerical algorithm is verified by comparing the numerical value with the semi-analytical solution.Then,based on the numerical algorithm and the experimental data set of thin films with different thicknesses heated by femtosecond laser pulses,we estimated the model parameters using the modified hybrid simplex search and particle swarm optimization algorithm,and obtained the optimal values,which again verified the consistency between the semi-analytical solution,numerical solution and data of the model,and proved the accuracy of the time fractional order DPL heat conduction model to characterize the temperature distribution on laser heated thin films.Finally,the varieties of temperature distribution with time and position are considered and then the effects of the model parameters on the temperature distribution are also discussed.In chapter 4:we use fractional calculus operators modeling to study the flow process of viscoelastic fluids.First,considering the complexity of material structure and multiscale effects in viscoelastic fluid flow,we systematically study the viscoelastic fluid flow of generalized Maxwell fluid in infinite straight pipe driven by periodic pressure gradient.The temporal inconsistency of the viscoelastic fluid is described by using the time distributed-order derivative and the variable fractional order derivative respectively,and the time distributedorder and time variable fractional order Maxwell governing equations are proposed.Based on the L1 approximation formula of Caputo’s fractional derivative,the implicit finite difference schemes of the distributed-order and variable fractional order governing equations are given,respectively,and the numerical solutions are derived.Numerical examples are used to compare the numerical and exact solutions of distributed-order and variable fractional differential equations,and the high agreement between them demonstrates the effectiveness of our numerical method.Then,the velocity distribution of the distributed-order and variable fractional order Maxwell governing equations under specific conditions is analyzed,and the effects of the weight coefficient in the distributed order time fractional derivative,the variable order function in the variable fractional order derivative,the relaxation time and the frequency of the periodic pressure gradient on the fluid flow velocity are discussed.Finally,the flow rate distributions of the fluid is studied.In chapter 5:we study the unsteady rotating magnetohydrodynamic flow and heat transfer phenomena of the generalized Maxwell fluid with distribution-order characteristics under the Hall effect on an infinite plate.Aiming at the multi-scale characteristics and non-uniformity of generalized Maxwell fluid flow and heat transfer,the time distributed-order derivative is introduced to accurately describe its flow and heat transfer mechanism.The time distributedorder momentum equation and the time fractional energy equation are derived to form a new set of coupled distributed-order governing equations.In order to calculate the numerical solution of the established coupling model,we proposed the Crank-Nicolson finite difference scheme based on the LI approximation formula,verified the validity and feasibility of the numerical method,and discussed the influence of the relevant parameters on the fluid velocity and temperature distributions.In chapter 6:we study the numerical methods of two-dimensional magnetohydrodynamics flow and heat transfer of generalized Maxwell fluid through rectangular pipes under the joint action of pressure gradient and magnetic field.Using time and space fractional derivatives to describe the complex flow and heat transfer phenomena of generalized Maxwell fluids,a set of coupled space-time fractional governing equations are proposed.Based on the finite difference method,numerical methods for fractional equations are developed,a coupled alternating direction implicit algorithm is established,and numerical solutions are derived.By comparing the exact solution of the numerical example with the numerical solution,it is found that the numerical solution fits well with the analytical solution,which verifies the validity and accuracy of the numerical method.The numerical solution is also compared with the results of previous work,and the agreement between them also indicates the feasibility of the numerical method.Finally,the effects of pressure gradients,fractional order parameters,and other relevant parameters on fluid velocity and temperature are discussed.In chapter 7:we briefly summarize the content of this paper and look forward to future research work.