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.

A Numerical Algorithm Based on Quadratic Finite Element for Two-Dimensional Nonlinear Time Fractional Thermal Diffusion Model

In this article,a high-order scheme,which is formulated by combining the quadratic finite element method in space with a second-order time discrete scheme,is developed for looking for the numerical solution of a two-dimensional nonlinear time fractional thermal diffusion model.The time Caputo fractional derivative is approximated by using the L2-1formula,the first-order derivative and nonlinear term are discretized by some second-order approximation formulas,and the quadratic finite element is used to approximate the spatial direction.The error accuracy O(h3+
t2)is obtained,which is verified by the numerical results.

Dynamics of the HBV model with diffusion and time delay

Chronic hepatitis B infection is a major health problem,with approximately 350 million virus carriers worldwide.In Africa,about 30%-60% of children and 60%-100% of adults have

Analysis of an Implicit Finite Difference Scheme for Time Fractional Diffusion Equation

Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α∈(0, 1). In this paper, an implicit finite difference scheme for solving the time fractional diffusion equation with source term is presented and analyzed, where the fractional derivative is described in the Caputo sense. Stability and convergence of this scheme are rigorously established by a Fourier analysis. And using numerical experiments illustrates the accuracy and effectiveness of the scheme mentioned in this paper.

Influence of diffusion time on property of steel-aluminum solid to liquid bonding

The bonding of solid steel to liquid aluminum was conducted using rapidsolidification. The influence of diffusion time on interfacial shear strength was studied. Theresults show that when the temperature of aluminum liquid is 700℃ and the preheat temperature ofsteel plate is 250℃, the relationship between diffusion time (t) and interfacial shear strength(σ) is σ =15.1+8.14t-037t^2 +0.005t^3, and the maximum interfacial shear strength is 71.1 MPa.

Low-field, high-gradient NMR shows diffusion contrast consistent with localization or motional averaging of water near surfaces

Nuclear magnetic resonance(NMR)measurements of water diffusion have been extensively used to probe microstructure in porous materials,such as biological tissue,however primarily using pulsed gradient spin echo(PGSE)methods.Low-field single-sided NMR systems have built-in static gradients(SG)much stronger than typical PGSE maximum gradient strengths,which allows for the signal attenuation at extremely high b-values to be explored.Here,we perform SG spin echo(SGSE)and SG stimulated echo(SGSTE)diffusion measurements on biological cells,tissues,and gels.Measurements on fixed and live neonatal mouse spinal cord,lobster ventral nerve cord,and starved yeast cells all show multiexponential signal attenuation on a scale of b with significant signal fractions observed at b×Do>1 with b as high as 400 ms/um2.These persistent signal fractions trend with surface-to-volume ratios for these systems,as expected from porous media theory.An exception found for the case of fixed vs.live spinal cords was attributed to faster exchange or permeability in live spinal cords than in fixed spinal cords on the millisecond timescale.Data suggests the existence of multiple exchange processes in neural tissue,which may be relevant to the modeling of time-dependent diffusion in gray matter.The observed multi-exponential attenuation is from protons on water and not macromolecules because it remains proportional to the normalized signal when a specimen is washed with D20.The signal that persists to b×Do>1 is also drastically reduced after delipidation,indicating that it originates from lipid membranes that restrict water diffusion.The multiexponential or stretched exponential character of the signal attenuation at b×Do>1 appears mono-exponential when viewed on a scale of(b×Do)/3,suggesting it may originate from localization or motional averaging of water near membranes on sub-micron length scales.To try to disambiguate these two contributions,signal attenuation curves were compared at varying temperatures.While the curves align when normalizing them using the localization length scale,they separate on a motional averaging length scale.This supports localization as the source of non-Gaussian displacements,but this interpretation is still provisional due to the possible confounds of heterogeneity,exchange,and relaxation.Measurements on two types of gel phantoms designed to mimic extracellular matrix.one with charged functional groups synthesized from polyacrylic acid(PAC)and another with uncharged functional groups synthesized from polyacrylamide(PAM),both exhibit signal at b×Do>1,potentially due to water interacting with macromolecules.These preliminary finding motivate future research into contrast and attenuation mechanisms in tissue with low-field,high-gradient NMR。

Green Function of Generalized Time Fractional Diffusion Equation Using Addition Formula of Mittag-Leffler Function

In this paper, we use the Mittag-Leffler addition formula to solve the Green function of generalized time fractional diffusion equation in the whole plane and prove the convergence of the Green function.

Temporal Second-order Scheme for a Hidden-memory Variable Order Time Fractional Diffusion Equation with an Initial Singularity

In this work,a novel time-stepping L1 formula is developed for a hidden-memory variable-order Caputo’s fractional derivative with an initial singularity.This formula can obtain second-order accuracy and an error estimate is analyzed strictly.As an application,a fully discrete difference scheme is established for the initial-boundary value problem of a hidden-memory variable-order time fractional diffusion model.Numerical experiments are provided to support our theoretical results.

A DIRECT DISCONTINUOUS GALERKIN METHOD FOR TIME FRACTIONAL DIFFUSION EQUATIONS WITH FRACTIONAL DYNAMIC BOUNDARY CONDITIONS

This paper deals with the numerical approximation for the time fractional diffusion problem with fractional dynamic boundary conditions.The well-posedness for the weak solutions is studied.A direct discontinuous Galerkin approach is used in spatial direction under the uniform meshes,together with a second-order Alikhanov scheme is utilized in temporal direction on the graded mesh,and then the fully discrete scheme is constructed.Furthermore,the stability and the error estimate for the full scheme are analyzed in detail.Numerical experiments are also given to illustrate the effectiveness of the proposed method.

An occupation time related potential measure for diffusion processes

In this paper, for homogeneous diffusion processes, the approach of Y. Li and X. Zhou [Statist. Probab. Lett., 2014, 94： 48-55] is adopted to find expressions of potential measures that are discounted by their joint occupation times over semi-infinite intervals （-∞, a） and （a, ∞）. The results are expressed in terms of solutions to the differential equations associated with the diffusions generator. Applying these results, we obtain more explicit expressions for Brownian motion with drift, skew Brownian motion, and Brownian motion with two-valued drift, respectively.

Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem

An inverse problem of reconstructing the initial condition for a time fractional diffusion equation is investigated.On the basis of the optimal control framework,the uniqueness and first order necessary optimality condition of the minimizer for the objective functional are established,and a time-space spectral method is proposed to numerically solve the resulting minimization problem.The contribution of the paper is threefold:1)a priori error estimate for the spectral approximation is derived;2)a conjugate gradient optimization algorithm is designed to efficiently solve the inverse problem;3)some numerical experiments are carried out to show that the proposed method is capable to find out the optimal initial condition,and that the convergence rate of the method is exponential if the optimal initial condition is smooth.

Large Deviations for Parameter Estimators of Some Time Inhomogeneous Diffusion Process

The goal of this paper is to study large deviations for estimator and score function of some time inhomogeneous diffusion process. Large deviation in the non-steepness case with explicit rate functions is obtained by using parameter-dependent change of measure.

Finite element multigrid method for multi-term time fractional advection diffusion equations

In this paper,a class of multi-term time fractional advection diffusion equations(MTFADEs)is considered.By finite difference method in temporal direction and finite element method in spatial direction,two fully discrete schemes of MTFADEs with different definitions on multi-term time fractional derivative are obtained.The stability and convergence of these numerical schemes are discussed.Next,a V-cycle multigrid method is proposed to solve the resulting linear systems.The convergence of the multigrid method is investigated.Finally,some numerical examples are given for verification of our theoretical analysis.

Therapeutic imaging window of cerebral infarction revealed by multisequence magnetic resonance imaging An animal and clinical study

In this study, we established a Wistar rat model of right middle cerebral artery occlusion and observed pathological imaging changes （T2-weighted imaging [T2WI], T2FLAIR, and diffusion-weighted imaging [DWI]） following cerebral infarction. The pathological changes were divided into three phases： early cerebral infarction, middle cerebral infarction, and late cerebral infarction. In the early cerebral infarction phase （less than 2 hours post-infarction）, there was evidence of intracellular edema, which improved after reperfusion. This improvement was defined as the ischemic penumbra. In this phase, a high DWI signal and a low apparent diffusion coefficient were observed in the right basal ganglia region. By contrast, there were no abnormal T2WI and T2FLAIR signals. For the middle cerebral infarction phase （2-4 hours post-infarction）, a mixed edema was observed. After reperfusion, there was a mild improvement in cell edema, while the angioedema became more serious. A high DWI signal and a low apparent diffusion coefficient signal were observed, and some rats showed high T2WI and T2FLAIR signals. For the late cerebral infarction phase （4-6 hours post-infarction）, significant angioedema was visible in the infarction site. After reperfusion, there was a significant increase in angioedema, while there was evidence of hemorrhage and necrosis. A mixed signal was observed on DWI, while a high apparent diffusion coefficient signal, a high T2WI signal, and a high T2FLAIR signal were also observed. All 86 cerebral infarction patients were subjected to T2WI, T2FLAIR, and DWI. MRI results of clinic data similar to the early infarction phase of animal experiments were found in 51 patients, for which 10 patients （10/51） had an onset time greater than 6 hours. A total of 35 patients had MRI results similar to the middle and late infarction phase of animal experiments, of which eight patients （8/35） had an onset time less than 6 hours. These data suggest that defining the ＂therapeutic time window＂ as the time 6 hours after infarction may not be suitable for all patients. Integrated application of MRI sequences including T2WI, T2FLAIR, DW-MRI, and apparent diffusion coefficient mapping should be used to examine the ischemic penumbra, which may provide valuable information for identifying the ＂therapeutic time window＂.

Numerical Methods for Semilinear Fractional Diffusion Equations with Time Delay

In this paper,we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations(RSFDEs)with time delay,which constitute an important class of differential equations of practical significance.We develop a novel implicit alternating direction method that can effectively and efficiently tackle the RSFDEs in both two and three dimensions.The numerical method is proved to be uniquely solvable,stable and convergent with second order accuracy in both space and time.Numerical results are presented to verify the accuracy and efficiency of the proposed numerical scheme.

Interfacial mechanical property of steel-mushy Al-20Sn bonding

The bonding of a steel plate to Al-20Sn slurry was conducted using thecasting rolling technique. The surface of the steel plate was defatted, descaled, immersed (inK_2ZrF_6 flux aqueous solution) and stoved. Al-20Sn slurry was prepared using the electromagneticmechanical starring method. The interfacial mechanical property of the bonding plate was researchedto determine the relationship between the diffusion time and the interfacial shear strength. Inorder to identify the mechanism of bonding, the interfacial structure of the bonding plate wasstudied. The results show that at a prebeat temperature of the steel plate of 505 deg C and a solidfraction of Al-20Sn slurry of 35 percent, the relationship between the interfacial shear strength Sand the diffusion time t is S=28.8+4.3t-0.134t^2 +0.0011t^3. When the diffusion time is 22 s, thelargest interfacial shear strength is 70.3 MPa, and the corresponding interface is a new one whichis made up of Fe-Al compound and Fe-Al solid solution alternatively and in a right proportion. Inthis interfacial structure, the interfacial embrittlement does not happen and Fe-Al compound canplay its role in strong combination adequately.

Scaling Properties of Runaway Electrons

Runaway electrons in tokamaks have been widely studied theoretically and experimentally. The runaway confinement time τ1 in ohmic and additionally heated tokamak plasmas presents an anomalous behavior when compared with theoretical predictions based on neoclassical models. Runaway electrons have received lately a great attention due to several reasons： （a） the possibility to study electromagnetic turbulence by measuring the runaway flux fluctuations and its energy spectra, and （ b ） the runaway electrons are powerful diagnostics capable of yielding valuable information on the actual distribution function of fusion experiments.

Application of low-dimensional finite element method to fractional diffusion equation

Classical finite element method(FEM)has been applied to solve some fractional differential equations,but its scheme has too many degrees of freedom.In this paper,a low-dimensional FEM,whose number of basis functions is reduced by the theory of proper orthogonal decomposition(POD)technique,is proposed for the time fractional diffusion equation in two-dimensional space.The presented method has the properties of low dimensions and high accuracy so that the amount of computation is decreased and the calculation time is saved.Moreover,error estimation of the method is obtained.Numerical example is given to illustrate the feasibility and validity of the low-dimensional FEM in comparison with traditional FEM for the time fractional differential equations.

An effective description of a periodic one-dimensional hopping model

The periodic one-dimensional hopping model is useful for studying the motion of microscopic particles in the thermal noise environment. Based on the explicit formulations of mean velocity, mean first passage time and effective diffusion constant, a general N internal states or even infinite internal states model can be approximated by a one state model that retains the basic properties of the original process. This effective description aids the analysis of biochemical and biophysical problems. This effective description also implies that, to some extent, many processes can be well described by simple two-state models, or even one-state models.