A Novel Method for Linear Systems of Fractional Ordinary Differential Equations with Applications to Time-Fractional PDEs

This paper presents an efficient numerical technique for solving multi-term linear systems of fractional ordinary differential equations(FODEs)which have been widely used in modeling various phenomena in engineering and science.An approximate solution of the system is sought in the formof the finite series over the Müntz polynomials.By using the collocation procedure in the time interval,one gets the linear algebraic system for the coefficient of the expansion which can be easily solved numerically by a standard procedure.This technique also serves as the basis for solving the time-fractional partial differential equations(PDEs).The modified radial basis functions are used for spatial approximation of the solution.The collocation in the solution domain transforms the equation into a system of fractional ordinary differential equations similar to the one mentioned above.Several examples have verified the performance of the proposed novel technique with high accuracy and efficiency.

Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

Discussion on the application prospect of the transient electromagnetic method based on the dual launcher and mine advanced detection

The dual transmitter implements the equivalent anti-magnetic flux transient electromagnetic method, which can effectively reduce the scope of the transient electromagnetic detection blind area. However, this method is rarely reported in the detection of pipelines in urban geophysical exploration and the application of coal mines. Based on this, this paper realizes the equivalent anti-magnetic flux transient electromagnetic method based on the dual launcher. The suppression effect of this method on the blind area is analyzed by physical simulation. And the detection experiment of underground pipelines is carried out outdoors. The results show that the dual launcher can significantly reduce the turn-off time, thereby effectively reducing the impact of the blind area on the detection results, and the pipeline detection results verify the device’s effectiveness. Finally, based on the ground experimental results, the application prospect of mine advanced detection is discussed. Compared with other detection fields, the formation of blind areas is mainly caused by the equipment. If the dual launcher can be used to reduce the blind area, the accuracy of advanced detection can be improved more effectively. The above research results are of great significance for improving the detection accuracy of the underground transient electromagnetic method.

Effects of Standing Time during Pretreatment on the Nitrite Concentration Detected by Spectrophotometric Method

Food safety problems caused by excessive nitrite addition have been frequently reported and the detection of nitrite in food is particularly important. The standing time during the pretreatment of primary sample has a great influence on the concentration of nitrite tested by spectrophotometric method. In this context, three kinds of food samples are prepared, including canned mustard, canned fish and home-made pickled water. A series of standing times are placed during the sample pretreatments and the corresponding nitrite contents in these samples are detected by spectrophotometric method based on N-ethylenediamine dihydrochloride. This study aims to find out a reasonable standing time during the pretreatment of food sample, providing influence factor for precise detection of nitrite.

Variational Iteration Method for Solving Time Fractional Burgers Equation Using Maple

The Time Fractional Burger equation was solved in this study using the Mabel software and the Variational Iteration approach. where a number of instances of the Time Fractional Burger Equation were handled using this technique. Tables and images were used to present the collected numerical results. The difference between the exact and numerical solutions demonstrates the effectiveness of the Mabel program’s solution, as well as the accuracy and closeness of the results this method produced. It also demonstrates the Mabel program’s ability to quickly and effectively produce the numerical solution.

Infinitesimal dividing modeling method for dual suppliers inventory model with random lead times

As one of the basic inventory cost models, the （Q, τ）inventory cost model of dual suppliers with random procurement lead time is mostly formulated by using the concepts of ＂effective lead time＂ and ＂lead time demand＂, which may lead to an imprecise inventory cost. Through the real-time statistic of the inventory quantities, this paper considers the precise （Q, τ） inventory cost model of dual supplier procurement by using an infinitesimal dividing method. The traditional modeling method of the inventory cost for dual supplier procurement includes complex procedures. To reduce the complexity effectively, the presented method investigates the statistics properties in real-time of the inventory quantities with the application of the infinitesimal dividing method. It is proved that the optimal holding and shortage costs of dual supplier procurement are less than those of single supplier procurement respectively. With the assumption that both suppliers have the same distribution of lead times, the convexity of the cost function per unit time is proved. So the optimal solution can be easily obtained by applying the classical convex optimization methods. The numerical examples are given to verify the main conclusions.

Real-time Synchronization Method Based on Dual Ring Fieldbus in CNC System

As the popularization of dual ring fieldbus, the optimized dual ring synchronization methods are still in short. The current synchronization methods are generally established in traditional industrial fieldbus, in which transmission is commonly considered in single track, the two-way transmitting cannot take full effect, and would result in unwanted idle load on equipment lines. In stamp-transferring part, the synchronizing algorithm is not properly processed to diminish the latency, so the real-time performance of entire system cannot be ensured. To support the synchronization control of stations in the CNC system, a real-time time synchronization method for dual ring fieldbus in the CNC system is designed in this paper. In this method a synchronizing message transmission scheme based on dual ring architecture and the synchronization algorithm between master and secondary stations are integrated. In the scheme, the clock models of both master and secondary stations are optimized with corresponding modules and the stamp data transmission based on the dual ring fieldbus is devised exclusively, so the transmitting efficiency improves with less idle work. In the algorithm, all the secondary stations can accomplish the consistent state with master station by updating clock discrepancy information in one communication cycle, and it takes the advantage of two-way transmitting and makes the best use of dual ring structure, so the real-time performance of the system can be promoted while retaining the precision of synchronization. To evaluate the performance, the costs of the method and errors during synchronizing are noted and analyzed based on the actual running environment in the industrial fieldbus. The results show that it reduces communication cost and ensures the smoothness of the system with low lag effects under heavy load. The proposed time synchronization method optimizes the architecture of sync message transmission in dual ring fieldbus, and improves the efficiency of time synchronization in the stations of CNC system.

A time domain induced polarization relaxation time spectrum inversion method based on a damping factor and residual correction

Relaxation time spectra （RTS） derived from time domain induced polarization data （TDIP） are helpful to assess oil reservoir pore structures. However, due to the sensitivity to the signal-to-noise ratio （SNR）, the inversion accuracy of the traditional singular value decomposition （SVD） inversion method reduces with a decrease of SNR. In order to enhance the inversion accuracy and improve robustness of the inversion method to the SNR, an improved inversion method, based on damping factor and spectrum component residual correction, is proposed in this study. The numerical inversion results show that the oscillation of the RTS derived from the SVD method increased with a decrease of SNR, which makes it impossible to get accurate inversion components. However, the SNR has little influence on inversion components of the improved method, and the RTS has high inversion accuracy and robustness. Moreover, RTS derived from core sample data is basically in accord with the pore-size distribution curve, and the RTS derived from the actual induced polarization logging data is smooth and continuous, which indicates that the improved method is practicable.

Equilibrium dividend strategies in the dual model with a random time horizon

This paper investigates the dividend problem with non-exponential discounting in a dual model.We assume that the dividends can only be paid at a bounded rate and that the surplus process is killed by an exponential random variable.Since the non-exponential discount function leads to a time inconsistent control problem,we study the equilibrium HJB-equation and give the associated verification theorem.For the case of a mixture of exponential discount functions and exponential gains,we obtain the explicit equilibrium dividend strategy and the corresponding equilibrium value function.Besides,numerical examples are shown to illustrate our results.

A Time Neighborhood Method for the Verification of Landfalling Typhoon Track Forecast

Landfalling typhoons can cause disasters over large regions.The government and emergency responders need to take measures to mitigate disasters according to the forecast of landfall position,while slight timing error can be ignored.The reliability of operational model forecasts of typhoon landfall position needs to be evaluated beforehand,according to the forecasts and observation of historical cases.In the evaluation of landfalling typhoon track,the traditional method based on point-to-point matching methods could be influenced by the predicted typhoon translation speed.Consequently,the traditional track evaluation method may result in a large track error even if the predicted landfall position is close to observation.The purpose of this paper is to address the above issue using a simple evaluation method of landfalling typhoon track forecast based on the time neighborhood approach.In this new method,the timing error was lessened to highlight the importance of the position error during the landfall of typhoon.The properties of the time neighborhood method are compared with the traditional method based on numerical forecast results of 12 landfalling typhoon cases.Results demonstrated that the new method is not sensitive to the sampling frequency,and that the difference between the time neighborhood and traditional method will be more obvious when the moving speed of typhoon is moderate(between 15−30 km h^(−1)).The time neighborhood concept can be easily extended to a broader context when one attempts to examine the position error more than the timing error.

Efficient Finite Difference/Spectral Method for the Time Fractional Ito Equation Using Fast Fourier Transform Technic

A finite difference/spectral scheme is proposed for the time fractional Ito equation.The mass conservation and stability of the numerical solution are deduced by the energy method in the L^(2)norm form.To reduce the computation costs,the fast Fourier transform technic is applied to a pair of equivalent coupled differential equations.The effectiveness of the proposed algorithm is verified by the first numerical example.The mass conservation property and stability statement are confirmed by two other numerical examples.

A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS

In this paper,a local discontinuous Galerkin(LDG)scheme for the time-fractional diffusion equation is proposed and analyzed.The Caputo time-fractional derivative(of orderα,with 0<α<1)is approximated by a finite difference method with an accuracy of order3-α,and the space discretization is based on the LDG method.For the finite difference method,we summarize and supplement some previous work by others,and apply it to the analysis of the convergence and stability of the proposed scheme.The optimal error estimate is obtained in the L2norm,indicating that the scheme has temporal(3-α)th-order accuracy and spatial(k+1)th-order accuracy,where k denotes the highest degree of a piecewise polynomial in discontinuous finite element space.The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.

Generation of endurance time excitation functions using spectral representation method

In this study,application of the spectral representation method for generation of endurance time excitation functions is introduced.Using this method,the intensifying acceleration time series is generated so that its acceleration response spectrum in any desired time duration is compatible with a time-scaled predefined acceleration response spectrum.For this purpose,simulated stationary acceleration time series is multiplied by the time dependent linear modulation function,then using a simple iterative scheme,it is forced to match a target acceleration response spectrum.It is shown that the generated samples have excellent conformity in low frequency,which is useful for nonlinear endurance time analysis.In the second part of this study,it is shown that this procedure can be extended to generate a set of spatially correlated endurance time excitation functions.This makes it possible to assess the performance of long structures under multi-support seismic excitation using endurance time analysis.

Towards a Unified Single Analysis Framework Embedded with Multiple Spatial and Time Discretized Methods for Linear Structural Dynamics

We propose a novel computational framework that is capable of employing different time integration algorithms and different space discretized methods such as the Finite Element Method,particle methods,and other spatial methods on a single body sub-dividedintomultiple subdomains.This is in conjunctionwithimplementing thewell known Generalized Single Step Single Solve(GS4)family of algorithms which encompass the entire scope of Linear Multistep algorithms that have been developed over the past 50 years or so and are second order accurate into the Differential Algebraic Equation framework.In the current state of technology,the coupling of altogether different time integration algorithms has been limited to the same family of algorithms such as theNewmarkmethods and the coupling of different algorithms usually has resulted in reduced accuracy in one or more variables including the Lagrange multiplier.However,the robustness and versatility of the GS4 with its ability to accurately account for the numerical shifts in various time schemes it encompasses,overcomes such barriers and allows a wide variety of arbitrary implicit-implicit,implicit-explicit,and explicit-explicit pairing of the various time schemes while maintaining the second order accuracy in time for not only all primary variables such as displacement,velocity and acceleration but also the Lagrange multipliers used for coupling the subdomains.By selecting an appropriate spatialmethod and time scheme on the area with localized phenomena contrary to utilizing a single process on the entire body,the proposed work has the potential to better capture the physics of a given simulation.The method is validated by solving 2D problems for the linear second order systems with various combination of spatial methods and time schemes with great flexibility.The accuracy and efficacy of the present work have not yet been seen in the current field,and it has shown significant promise in its capabilities and effectiveness for general linear dynamics through numerical examples.

Adomian Decomposition Method for Solving Time Fractional Burgers Equation Using Maple

In this paper, the Adomian decomposition method was used to solve the Time Fractional Burger equation using Mabel program. This method was applied to a number of examples of the Time Fractional Burger Equation. The obtained numerical results were presented in the form of tables and graphics. The difference between the exact solutions and the numerical solutions shows us the effectiveness of the solution using the Mabel program and that this method gave accurate results and was close to the exact solution, in addition to its ability to obtain the numerical solution quickly and efficiently using the Mabel program.

A strategy for strengthening chaotic mixing of dual shaft eccentric mixers by changing non-Newtonian fluids kinetic energy distribution

Efficiently modulating the velocity distribution and flow pattern of non-Newtonian fluids is a critical challenge in the context of dual shaft eccentric mixers for process intensification,posing a significant barrier for the existing technologies.Accordingly,this work reports a convenient strategy that changes the kinetic energy to controllably regulate the flow patterns from radial flow to axial flow.Results showed that the desired velocity distribution and flow patterns could be effectively obtained by varying the number and structure of baffles to change kinetic energy,and a more uniform velocity distribution,which could not be reached normally in standard baffle dual shaft mixers,was easily obtained.Furthermore,a comparative analysis of velocity and shear rate distributions is employed to elucidate the mechanism behind the generation of flow patterns in various dual-shaft eccentric mixers.Importantly,there is little difference in the power number of the laminar flow at the same Reynolds number,meaning that the baffle type has no effect on the power consumption,while the power number of both unbaffle and U-shaped baffle mixing systems decreases compared with the standard baffle mixing system in the transition flow.Finally,at the same rotational condition,the dimensionless mixing time of the U-shaped baffle mixing system is 15.3%and 7.9%shorter than that of the standard baffle and the unbaffle mixing system,respectively,which shows the advantage of the U-shaped baffle in stirring rate.

Scattering of Water Waves by Dual Symmetric Inclined Floating Porous Barriers Using the DBEM

The scattering of normally incident water waves by two surface-piercing inclined perforated barriers in water with a uniform finite depth is investigated within the framework of linear water wave theory.Considering that thin barriers are zero-thickness,a novel numerical method involving the the coupling of the dual boundary element method(DBEM)with damping layers is applied.In order to effectively damp out the reflected waves,two damping layers,instead of pseudoboundaries are implemented near the two side boundaries of the computational domain.Thus,the modified linearized free surface boundary conditions are formulated and used for solving both the ordinary boundary integral equation as well as the hypersingular boundary integral equation for degenerate boundaries.The newly developed numerical method is validated against analytical methods using the matched eigenfunction expansion method for the special case of two vertical barriers or the inclined angle to the vertical being zero.The influence of the length of the two damping layers has been discussed.Moreover,these findings are also validated against previous results for several cases.After validation,the numerical results for the reflection coefficient,transmission coefficient and dissipation coefficient are obtained by varying the inclination angle and porosity-effect parameter.The effects of both the inclination angle and the porosity on the amplitudes of wave forces acting on both the front and rear barriers are also investigated.It is found that the effect of the inclination angle mainly shifts the location of the extremal values of the reflection and the transmission coefficients.Additionally,a moderate value of the porosity-parameter is quite effective at dissipating wave energy and mitigating the wave loads on dual barriers.

A spatiotemporal evolution model of a short-circuit arc to a secondary arc based on the improved charge simulation method

The initial shape of the secondary arc considerably influences its subsequent shape.To establish the model for the arcing time of the secondary arc and modify the single-phase reclosing sequence,theoretical and experimental analysis of the evolution process of the short-circuit arc to the secondary arc is critical.In this study,an improved charge simulation method was used to develop the internal-space electric-field model of the short-circuit arc.The intensity of the electric field was used as an independent variable to describe the initial shape of the secondary arc.A secondary arc evolution model was developed based on this model.Moreover,the accuracy of the model was evaluated by comparison with physical experimental results.When the secondary arc current increased,the arcing time and dispersion increased.There is an overall trend of increasing arc length with increasing arcing time.Nevertheless,there is a reduction in arc length during arc ignition due to short circuits between the arc columns.Furthermore,the arcing time decreased in the range of 0°-90°as the angle between the wind direction and the x-axis increased.This work investigated the method by which short-circuit arcs evolve into secondary arcs.The results can be used to develop the secondary arc evolution model and to provide both a technical and theoretical basis for secondary arc suppression.

Particle residence time distribution and axial dispersion coefficient in a pressurized circulating fluidized bed by using multiphase particle-in-cell simulation

The particle residence time distribution(RTD)and axial dispersion coefficient are key parameters for the design and operation of a pressurized circulating fluidized bed(PCFB).In this study,the effects of pressure(0.1-0.6 MPa),fluidizing gas velocity(2-7 m·s^(-1)),and solid circulation rate(10-90 kg·m^(-2)·s^(-1))on particle RTD and axial dispersion coefficient in a PCFB are numerically investigated based on the multiphase particle-in-cell(MP-PIC)method.The details of the gas-solid flow behaviors of PCFB are revealed.Based on the gas-solid flow pattern,the particles tend to move more orderly under elevated pressures.With an increase in either fluidizing gas velocity or solid circulation rate,the mean residence time of particles decreases while the axial dispersion coefficient increases.With an increase in pressure,the core-annulus flow is strengthened,which leads to a wider shape of the particle RTD curve and a larger mean particle residence time.The back-mixing of particles increases with increasing pressure,resulting in an increase in the axial dispersion coefficient.

TIME ACCURATE 2D NAVIER-STOKES CALCULATIONS WITH A DUAL-TIME STEPPING METHOD

With the cell vertex finite volume discretization in space and second order backward implicit discretization in time, 2D unsteady Navier Stokes equations are solved by a dual time stepping method to simulate compressible viscous flow around rigid airfoils in arbitrary unsteady motion. The selection of physical time step is not restricted by stability condition any more, and most of the successful acceleration techniques used in steady calculations can be implemented to increase the computation efficiency.