Numerical analysis for viscoelastic fluid flow with distributed/variable order time fractional Maxwell constitutive models

Fractional calculus has been widely used to study the flow of viscoelastic fluids recently,and fractional differential equations have attracted a lot of attention.However,the research has shown that the fractional equation with constant order operators has certain limitations in characterizing some physical phenomena.In this paper,the viscoelastic fluid flow of generalized Maxwell fluids in an infinite straight pipe driven by a periodic pressure gradient is investigated systematically.Consider the complexity of the material structure and multi-scale effects in the viscoelastic fluid flow.The modified time fractional Maxwell models and the corresponding governing equations with distributed/variable order time fractional derivatives are proposed.Based on the L1-approximation formula of Caputo fractional derivatives,the implicit finite difference schemes for the distributed/variable order time fractional governing equations are presented,and the numerical solutions are derived.In order to test the correctness and availability of numerical schemes,two numerical examples are established to give the exact solutions.The comparisons between the numerical solutions and the exact solutions have been made,and their high consistency indicates that the present numerical methods are effective.Then,this paper analyzes the velocity distributions of the distributed/variable order fractional Maxwell governing equations under specific conditions,and discusses the effects of the weight coefficient(α)in distributed order time fractional derivatives,the orderα(r,t)in variable fractional order derivatives,the relaxation timeλ,and the frequencyωof the periodic pressure gradient on the fluid flow velocity.Finally,the flow rates of the distributed/variable order fractional Maxwell governing equations are also studied.

Controllability of Fractional Order Stochastic Differential Inclusions with Fractional Brownian Motion in Finite Dimensional Space

In this paper,sufficient conditions are formulated for controllability of fractional order stochastic differential inclusions with fractional Brownian motion(f Bm) via fixed point theorems,namely the Bohnenblust-Karlin fixed point theorem for the convex case and the Covitz-Nadler fixed point theorem for the nonconvex case.The controllability Grammian matrix is defined by using Mittag-Leffler matrix function.Finally,a numerical example is presented to illustrate the efficiency of the obtained theoretical results.

Fractional Order Proportional Integral Derivative Controller Design and Simulation for Bioengineering Systems

This paper deals with the study of fractional order system tuning method based on Factional Order Proportional Integral Derivative( FOPID) controller in allusion to the nonlinear characteristics and fractional order mathematical model of bioengineering systems. The main contents include the design of FOPID controller and the simulation for bioengineering systems. The simulation results show that the tuning method of fractional order system based on the FOPID controller outperforms the fractional order system based on Fractional Order Proportional Integral( FOPI) controller. As it can enhance control character and improve the robustness of the system.

Pfaff-Birkhoff Variational Problem and Noether Symmetry in Terms of RiemannLiouville Fractional Derivatives of Variable Order

The Pfaff-Birkhoff variational problem and its Noether symmetry are studied in terms of Riemann-Liouville fractional derivatives of variable order. Based on the combination of variational principle and fractional calculus of variable order,the Pfaff-Birkhoff variational principle with Riemann-Liouville fractional derivatives of variable order is proposed, and the fractional Birkhoff's equations of variable order are derived. Then,the Noether 's theorem for the fractional Birkhoffian system of variable order is given. At last,an example is expressed to illustrate the application of the results.

Existence of positive solutions for integral boundary value problem of fractional differential equations

In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator. By using the fixed point theorem in cone,the existence of positive solutions for the boundary value problem is obtained. Some examples are also presented to illustrate the application of our main results.

Research on Stick-Slip Vibration Suppression Method of Drill String Based on Machine Learning Optimization

During the drilling process,stick-slip vibration of the drill string is mainly caused by the nonlinear friction gen-erated by the contact between the drill bit and the rock.To eliminate the fatigue wear of downhole drilling tools caused by stick-slip vibrations,the Fractional-Order Proportional-Integral-Derivative(FOPID)controller is used to suppress stick-slip vibrations in the drill string.Although the FOPID controller can effectively suppress the drill string stick-slip vibration,its structure isflexible and parameter setting is complicated,so it needs to use the cor-responding machine learning algorithm for parameter optimization.Based on the principle of torsional vibration,a simplified model of multi-degree-of-freedom drill string is established and its block diagram is designed.The continuous nonlinear friction generated by cutting rock is described by the LuGre friction model.The adaptive learning strategy of genetic algorithm(GA),particle swarm optimization(PSO)and particle swarm optimization improved(IPSO)by arithmetic optimization(AOA)is used to optimize and adjust the controller parameters,and the drill string stick-slip vibration is suppressed to the greatest extent.The results show that:When slight drill string stick-slip vibration occurs,the FOPID controller optimized by machine learning algorithm has a good effect on suppressing drill string stick-slip vibration.However,the FOPID controller cannot get the drill string system which has fallen into serious stick-slip vibration(stuck pipe)out of trouble,and the machine learning algorithm is required to mark a large amount of data on adjacent Wells to train the model.Set a reasonable range of drilling parameters(weight on bit/drive torque)in advance to avoid severe stick-slip vibration(stuck pipe)in the drill string system.

The Self-similar Solution to Some Nonlinear Integro-differential Equations Corresponding to Fractional Order Time Derivative

In this paper we study the self-similar solution to a class of nonlinear integro-differential equations which correspond to fractional order time derivative and interpolate nonlinear heat and wave equation. Using the space-time estimates which were established by Hirata and Miao in [1] we prove the global existence of self-similar solution of Cauchy problem for the nonlinear integro-differential equation in C＊（[0,∞];B^8pp,∞（R^n）.

Control of a 3-RRR Planar Parallel Robot Using Fractional Order PID Controller

3-RRR planar parallel robots are utilized for solving precise material-handling problems in industrial automation applications.Thus,robust and stable control is required to deliver high accuracy in comparison to the state of the art.The operation of the mechanism is achieved based on three revolute(3-RRR)joints which are geometrically designed using an open-loop spatial robotic platform.The inverse kinematic model of the system is derived and analyzed by using the geometric structure with three revolute joints.The main variables in our design are the platform base positions,the geometry of the joint angles,and links of the 3-RRR planar parallel robot.These variables are calcula ted based on Cayley-Menger determinants and bilateration to det ermine the final position of the platform when moving and placing objects.Additionally,a proposed fractional order proportional integral derivative(FOPID)is optimized using the bat optimization algorithm to control the path tracking of the center of the 3-RRR planar parallel robot.The design is compared with the state of the art and simulated using the Matlab environment to validate the effectiveness of the proposed controller.Furthermore,real-time implementation has been tested to prove that the design performance is practical.

An Adaptive Strategy for the Restoration of Textured Images using Fractional Order Regularization

Total variation regularization has good performance in noise removal and edge preservation but lacks in texture restoration.Here we present a texture-preserving strategy to restore images contaminated by blur and noise.According to a texture detection strategy,we apply spatially adaptive fractional order diffusion.A fast algorithm based on the half-quadratic technique is used to minimize the resulting objective function.Numerical results show the effectiveness of our strategy.

Application of Fractional Order PID Controller for AGC Under Deregulated Environment

In this paper, a fractional order proportional integral derivative （FOPID） controller for multiarea automatic generation control （AGC） scheme has been designed. FOPID controller has five parameters and provides two additional degrees of flexibility in comparison to a proportional integral derivative （PID） controller. The optimal values of parameters of FOPID controller have been determined using Big Bang Big Crunch （BBBC） search algorithm. The designed controller regulates real power output of generators to achieve the best dynamic response of frequency and tie-line power on a load perturbation. The complete scheme for designing of the controllers has been developed and demonstrated on multiarea deregulated power system. The performance of the designed FOPID controllers has been compared with the optimally tuned PID controllers. It is observed from the results that the FOPID controller shows a considerable improvement in the performance as compared to the conventional PID controller.

Memory dependent anomalous diffusion in comb structure under distributed order time fractional dual-phase-lag model

This paper considers a novel distributed order time fractional dual-phase-lag model to analyze the anomalous diffusion in a comb structure,which has a widespread application in medicine and biology.The newly proposed constitution model is a generalization of the dual-phase-lag model,in which a spectrum of the time fractional derivatives with the memory characteristic governed by the weight coefficient is considered and the formulated governing equation contains both the diffusion and wave characteristics.With the L1-formula to discrete the time Caputo fractional derivatives,the finite difference method is used to discretize the model and the related numerical results are plotted graphically.By adding a source term,an exact solution is defined to verify the correctness of the numerical scheme and the convergence order of the error in spatial direction is presented.Finally,the dynamic characteristics of the particle distributions and the effects of involved parameters on the total number of particles in the x-direction are analyzed in detail.

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO FRACTIONAL BOUNDARY VALUE PROBLEMS

In this paper, using fixed point theorems of general β-concave operators in ordered Banach space, we obtain the existence and uniqueness of positive solutions to a class of fractional boundary problem. In the end, an example is given to illustrate our main result.

Soil texture prediction through stratification of a regional soil spectral library

Knowing the spatial distribution of soil texture,which is a physical property,is essential to support agricultural and environmental decision making.Soil texture can be estimated using visible,near infrared,and shortwave infrared(Vis-NIR-SWIR)spectroscopy.However,the performance of spectroscopic models is variable because of soil heterogeneity.Currently,few studies address the effects of soil sample variability on the performance of the models,especially for larger spectral libraries that include soils that are more heterogeneous.Therefore,the objectives of this study were to:i)apply Vis-based color parameters on the stratification of a regional soil spectral library;ii)evaluate the performance of the predictive models generated from the spectral library stratification;iii)compare the performance of stratified models(SMs)and the model without stratification(WSM),and iv)explain possible changes in prediction accuracy based on the SMs.Thus,a regional soil spectral library with 1535 samples from the State of Santa Catarina,Brazil was used.Soil reflectance data were obtained by Vis-NIR-SWIR spectroscopy in the laboratory using a spectroradiometer covering the 350–2500 nm spectral range.Sand,silt,and clay fractions were determined using the pipette method.Twenty-two components of color parameters were derived from the Vis spectrum using the colorimetric models.A cubist regression algorithm was used to assess the accuracy of the applicability of the initial models(SMs and WSM)and of the validation between the clusters.Fractional order derivatives(FODs)at 0.5,1.5,and 2 intervals were used to explain possible changes in the performance of the SMs.The SMs with higher contents of clay and iron oxides obtained the highest accuracy,and the most important spectral bands were identified,mainly in the 480–550 and 850–900 nm ranges and the 1400,1900,and 2200 nm bands.Therefore,stratification of soil spectral libraries is a good strategy to improve regional assessments of soil resources,reducing prediction errors in the qualitative determination of soil properties.