The field of diffusion micro structural magnetic resonance(MR)aims to probe timedependent diffusion,i.e.,an ensemble-averaged mean-squared displacement that is not linear in time.This time-dependence contains rich inf...The field of diffusion micro structural magnetic resonance(MR)aims to probe timedependent diffusion,i.e.,an ensemble-averaged mean-squared displacement that is not linear in time.This time-dependence contains rich information about the surrounding microenvironment.MR methods to measure time-dependent diffusion quantitatively,however,require either non-standard pulse sequences,such as oscillating gradients,or make non-physical assumptions,such as infinitely narrow gradient pulses.Here,we argue that standard spin echo and stimulated echo MR sequences can be used to probe directly.In particular,we propose a framework in which the log-signal ratio obtained from a pair of measurements with different inter-pulse spacingΔis proportional to the MSD between these twoΔvalues along the gradient direction x:-.The framework is quantitative for short,finite-duration gradient pulses and under the Gaussian phase approximation(GPA).To validate the framework,we consider onedimensional diffusion between impermeable,parallel planes,as well as periodicallyspaced,permeable planes.Excellent agreement is obtained between the estimation and the ground truth in the regime where the GPA is expected to hold.Importantly,the GPA can be made to hold for any underlying microstructure,making the proposed framework widely applicable.展开更多
The main result of this paper is that when the coefficients of the time-dependent divergence form operators are Hlder continuous in time with order not too much smaller than （1/2）,the distance of the semigroups of t...The main result of this paper is that when the coefficients of the time-dependent divergence form operators are Hlder continuous in time with order not too much smaller than （1/2）,the distance of the semigroups of two operators is bounded by the L<sub>2</sub> distance of the coefficients of their corresponding operators.展开更多
Increasingly,attention is being directed towards time-dependent diffusion magnetic resonance imaging(TDDMRI),a method that reveals time-related changes in the diffusional behavior of water molecules in biological tiss...Increasingly,attention is being directed towards time-dependent diffusion magnetic resonance imaging(TDDMRI),a method that reveals time-related changes in the diffusional behavior of water molecules in biological tissues,thereby enabling us to probe related microstructure events.With ongoing improvements in hardware and advanced pulse sequences,significant progress has been made in applying TDDMRI to clinical research.The development of accurate mathematical models and computational methods has bolstered theoretical support for TDDMRI and elevated our understanding of molecular diffusion.In this review,we introduce the concept and basic physics of TDDMRI,and then focus on the measurement strategies and modeling approaches in short-and long-diffusion-time domains.Finally,we discuss the challenges in this field,including the requirement for efficient scanning and data processing technologies,the development of more precise models depicting time-dependent molecular diffusion,and critical clinical applications.展开更多
In this paper, two finite difference streamline diffusion (FDSD) schemes for solving two-dimensional time-dependent convection-diffusion equations are constructed. Stability and optimal order error estimati-ions for c...In this paper, two finite difference streamline diffusion (FDSD) schemes for solving two-dimensional time-dependent convection-diffusion equations are constructed. Stability and optimal order error estimati-ions for considered schemes are derived in the norm stronger than L^2-norm.展开更多
Time-dependent diffusion coefficient and conventional diffusion constant are calculated and analyzed to study diffusion of nanoparticles in polymer melts. A generalized Langevin equa- tion is adopted to describe the d...Time-dependent diffusion coefficient and conventional diffusion constant are calculated and analyzed to study diffusion of nanoparticles in polymer melts. A generalized Langevin equa- tion is adopted to describe the diffusion dynamics. Mode-coupling theory is employed to calculate the memory kernel of friction. For simplicity, only microscopic terms arising from binary collision and coupling to the solvent density fluctuation are included in the formalism. The equilibrium structural information functions of the polymer nanocomposites required by mode-coupling theory are calculated on the basis of polymer reference interaction site model with Percus-Yevick closure. The effect of nanoparticle size and that of the polymer size are clarified explicitly. The structural functions, the friction kernel, as well as the diffusion coefficient show a rich variety with varying nanoparticle radius and polymer chain length. We find that for small nanoparticles or short chain polymers, the characteristic short time non-Markov diffusion dynamics becomes more prominent, and the diffusion coefficient takes longer time to approach asymptotically the conventional diffusion constant. This constant due to the microscopic contributions will decrease with the increase of nanoparticle size, while increase with polymer size. Furthermore, our result of diffusion constant from mode- coupling theory is compared with the value predicted from the Stokes-Einstein relation. It shows that the microscopic contributions to the diffusion constant are dominant for small nanoparticles or long chain polymers. Inversely, when nanonparticle is big, or polymer chain is short, the hydrodynamic contribution might play a significant role.展开更多
A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretizatio...A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.展开更多
In this paper,we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term f satisfies critical exponential growth and the external force g(x)∈L^(2)...In this paper,we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term f satisfies critical exponential growth and the external force g(x)∈L^(2)(Ω).In the framework of time-dependent spaces,we verify the existence of absorbing sets and the asymptotic compactness of the process,then we obtain the existence of the time-dependent global attractor A={A_t}t∈Rin Mt.Furthermore,we achieve the regularity of A,that is,A_(t) is bounded in M_(t)^(1) with a bound independent of t.展开更多
A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect ...A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.展开更多
The stability of coal pillar dams is crucial for the long-term service of underground reservoirs storing water or heat.Chemi-cal damage of coal dams induced by ions-atttacking in coal is one of the main reasons for th...The stability of coal pillar dams is crucial for the long-term service of underground reservoirs storing water or heat.Chemi-cal damage of coal dams induced by ions-atttacking in coal is one of the main reasons for the premature failure of coal dams.However,the diffusion process of harmful ions in coal is far from clear,limiting the reliability and durability of coal dam designs.This paper investigates sulfate diffusion in coal pillar through experimental and analytical methods.Coal specimens are prepared and exposed to sulfate solutions with different concentrations.The sulfate concentrations at different locations and time are measured.Based on experimental data and Fick's law,the time-dependent surface concentration of sulfate and diffusion coefficient are determined and formulated.Further,an analytical model for predicting sulfate diffusion in coal pillar is developed by considering dual time-dependent characteristics and Laplace transformations.Through comparisons with experimental data,the accuracy of the analytical model for predicting sulfate diffusion is verified.Further,sulfate diffusions in coal dams for different concentrations of sulfate in mine water are investigated.It has been found that the sulfate concen-tration of exposure surface and diffusion coefficient in coal are both time-dependent and increase with time.Conventional Fick's law is not able to predict the sulfate diffusion in coal pillar due to the dual time-dependent characteristics.The sulfate attacking makes the coal dam a typical heterogeneous gradient structure.For sulfate concentrations 0.01-0.20 mol/L in mine water,it takes almost 1.5 and 4 years for sulfate ions to diffuse 9.46 and 18.92 m,respectively.The experimental data and developed model provide a practical method for predicting sulfate diffusion in coal pillar,which helps the service life design of coal dams.展开更多
The element of pesedospectral-multiwavelet-Galerkin method, and how tocombine it with penalty method for treating boundary conditions are given. Multiwavelet bases don'toverlap on the given scale, and possess the ...The element of pesedospectral-multiwavelet-Galerkin method, and how tocombine it with penalty method for treating boundary conditions are given. Multiwavelet bases don'toverlap on the given scale, and possess the same compact set in a group of several functions, sothey can be directly used to the numerical discretion on the finite interval. Numerical tests showthat general boundary conditions can be enforced with the penalty method, and thatpesedospectral-multiwavelet-Galerkin method can well track the solutions' development. This alsoproves that pesedospectral-multiwavelet-Galerkin method is effective.展开更多
This paper presents a new numerical technique for solving initial and bound-ary value problems with unsteady strongly nonlinear advection diffusion reaction(ADR)equations.The method is based on the use of the radial b...This paper presents a new numerical technique for solving initial and bound-ary value problems with unsteady strongly nonlinear advection diffusion reaction(ADR)equations.The method is based on the use of the radial basis functions(RBF)for the approximation space of the solution.The Crank-Nicolson scheme is used for approximation in time.This results in a sequence of stationary nonlinear ADR equations.The equations are solved sequentially at each time step using the proposed semi-analytical technique based on the RBFs.The approximate solution is sought in the form of the analytical expansion over basis functions and contains free parameters.The basis functions are constructed in such a way that the expansion satisfies the boundary conditions of the problem for any choice of the free parameters.The free parameters are determined by substitution of the expansion in the equation and collocation in the solution domain.In the case of a nonlinear equation,we use the well-known procedure of quasilinearization.This transforms the original equation into a sequence of the linear ones on each time layer.The numerical examples confirm the high accuracy and robustness of the proposed numerical scheme.展开更多
This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant(x,t)∈R^+×R^+,with the null-Dirichlet boundary condition or the null-Neumann bou...This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant(x,t)∈R^+×R^+,with the null-Dirichlet boundary condition or the null-Neumann boundary condition on u. We show that the corresponding initial-boundary value problem admits a unique global smooth solution which tends timeasymptotically to the nonlinear diffusion wave. Compared with the previous work about Euler equations with constant coefficient damping, studied by Nishihara and Yang(1999), and Jiang and Zhu(2009, Discrete Contin Dyn Syst), we obtain a general result when the initial perturbation belongs to the same space. In addition,our main novelty lies in the fact that the cut-off points of the convergence rates are different from our previous result about the Cauchy problem. Our proof is based on the classical energy method and the analyses of the nonlinear diffusion wave.展开更多
This paper studies existence of mild solution to a sharp cut off model for contact driven tumor growth.Analysis is based on application of the Crandall-Liggett theorem for w-quas-contractive semigroups on the Banach s...This paper studies existence of mild solution to a sharp cut off model for contact driven tumor growth.Analysis is based on application of the Crandall-Liggett theorem for w-quas-contractive semigroups on the Banach space L^(1)(Ω).Furthermore,numerical computations are provided which compare the sharp cut off model with the tumor growth model of Perthame,Quiros,and Vazquez[13].展开更多
基金supported by the intramural research program(IRP)of the Eunice Kennedy Shriver National Institute of Child Health and Human Development。
文摘The field of diffusion micro structural magnetic resonance(MR)aims to probe timedependent diffusion,i.e.,an ensemble-averaged mean-squared displacement that is not linear in time.This time-dependence contains rich information about the surrounding microenvironment.MR methods to measure time-dependent diffusion quantitatively,however,require either non-standard pulse sequences,such as oscillating gradients,or make non-physical assumptions,such as infinitely narrow gradient pulses.Here,we argue that standard spin echo and stimulated echo MR sequences can be used to probe directly.In particular,we propose a framework in which the log-signal ratio obtained from a pair of measurements with different inter-pulse spacingΔis proportional to the MSD between these twoΔvalues along the gradient direction x:-.The framework is quantitative for short,finite-duration gradient pulses and under the Gaussian phase approximation(GPA).To validate the framework,we consider onedimensional diffusion between impermeable,parallel planes,as well as periodicallyspaced,permeable planes.Excellent agreement is obtained between the estimation and the ground truth in the regime where the GPA is expected to hold.Importantly,the GPA can be made to hold for any underlying microstructure,making the proposed framework widely applicable.
基金Research partially supported by N.S.F.Grants DMS-9625642
文摘The main result of this paper is that when the coefficients of the time-dependent divergence form operators are Hlder continuous in time with order not too much smaller than （1/2）,the distance of the semigroups of two operators is bounded by the L<sub>2</sub> distance of the coefficients of their corresponding operators.
基金supported by the Ministry of Science and Technology of the People’s Republic of China(No.2021ZD0200202)the National Natural Science Foundation of China(No.82122032)the Science and Technology Department of Zhejiang Province(Nos.202006140 and 2022C03057).
文摘Increasingly,attention is being directed towards time-dependent diffusion magnetic resonance imaging(TDDMRI),a method that reveals time-related changes in the diffusional behavior of water molecules in biological tissues,thereby enabling us to probe related microstructure events.With ongoing improvements in hardware and advanced pulse sequences,significant progress has been made in applying TDDMRI to clinical research.The development of accurate mathematical models and computational methods has bolstered theoretical support for TDDMRI and elevated our understanding of molecular diffusion.In this review,we introduce the concept and basic physics of TDDMRI,and then focus on the measurement strategies and modeling approaches in short-and long-diffusion-time domains.Finally,we discuss the challenges in this field,including the requirement for efficient scanning and data processing technologies,the development of more precise models depicting time-dependent molecular diffusion,and critical clinical applications.
基金Project supported by National Natural Science Foundation of China and China State Key project for Basic Researchcs.
文摘In this paper, two finite difference streamline diffusion (FDSD) schemes for solving two-dimensional time-dependent convection-diffusion equations are constructed. Stability and optimal order error estimati-ions for considered schemes are derived in the norm stronger than L^2-norm.
基金This work was supported by the National Natural Science Foundation of China （No.21173152）, the Ministry of Education of China （No.NCET-11-0359 and No.2011SCU04B31）, and the Science and Technology Department of Sichuan Province （No.2011HH0005）.
文摘Time-dependent diffusion coefficient and conventional diffusion constant are calculated and analyzed to study diffusion of nanoparticles in polymer melts. A generalized Langevin equa- tion is adopted to describe the diffusion dynamics. Mode-coupling theory is employed to calculate the memory kernel of friction. For simplicity, only microscopic terms arising from binary collision and coupling to the solvent density fluctuation are included in the formalism. The equilibrium structural information functions of the polymer nanocomposites required by mode-coupling theory are calculated on the basis of polymer reference interaction site model with Percus-Yevick closure. The effect of nanoparticle size and that of the polymer size are clarified explicitly. The structural functions, the friction kernel, as well as the diffusion coefficient show a rich variety with varying nanoparticle radius and polymer chain length. We find that for small nanoparticles or short chain polymers, the characteristic short time non-Markov diffusion dynamics becomes more prominent, and the diffusion coefficient takes longer time to approach asymptotically the conventional diffusion constant. This constant due to the microscopic contributions will decrease with the increase of nanoparticle size, while increase with polymer size. Furthermore, our result of diffusion constant from mode- coupling theory is compared with the value predicted from the Stokes-Einstein relation. It shows that the microscopic contributions to the diffusion constant are dominant for small nanoparticles or long chain polymers. Inversely, when nanonparticle is big, or polymer chain is short, the hydrodynamic contribution might play a significant role.
基金supported by the National Natural Science Foundation of China(No.10771150)the National Basic Research Program of China(No.2005CB321701)+1 种基金the Program for New Century Excellent Talents in University(No.NCET-07-0584)the Natural Science Foundation of Sichuan Province(No.07ZB087)
文摘A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.
基金supported by the National Natural Science Foundation of China(No.12171082)the Fundamental Research Funds for the Central Universities(No.2232023G-13).
文摘In this paper,we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term f satisfies critical exponential growth and the external force g(x)∈L^(2)(Ω).In the framework of time-dependent spaces,we verify the existence of absorbing sets and the asymptotic compactness of the process,then we obtain the existence of the time-dependent global attractor A={A_t}t∈Rin Mt.Furthermore,we achieve the regularity of A,that is,A_(t) is bounded in M_(t)^(1) with a bound independent of t.
基金Supported by NSFC Grant(11401100,10601021)the foundation of Fujian Education Department(JB14021)the innovation foundation of Fujian Normal University(IRTL1206)
文摘A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.
基金supported by Hunan Provincial Education Department Funded Research Projects (Grant No.22C0221)Open Fund of State Key Laboratory of Water Resource Protection and Utilization in Coal Mining (Grant No.GJNY-18-73.11).
文摘The stability of coal pillar dams is crucial for the long-term service of underground reservoirs storing water or heat.Chemi-cal damage of coal dams induced by ions-atttacking in coal is one of the main reasons for the premature failure of coal dams.However,the diffusion process of harmful ions in coal is far from clear,limiting the reliability and durability of coal dam designs.This paper investigates sulfate diffusion in coal pillar through experimental and analytical methods.Coal specimens are prepared and exposed to sulfate solutions with different concentrations.The sulfate concentrations at different locations and time are measured.Based on experimental data and Fick's law,the time-dependent surface concentration of sulfate and diffusion coefficient are determined and formulated.Further,an analytical model for predicting sulfate diffusion in coal pillar is developed by considering dual time-dependent characteristics and Laplace transformations.Through comparisons with experimental data,the accuracy of the analytical model for predicting sulfate diffusion is verified.Further,sulfate diffusions in coal dams for different concentrations of sulfate in mine water are investigated.It has been found that the sulfate concen-tration of exposure surface and diffusion coefficient in coal are both time-dependent and increase with time.Conventional Fick's law is not able to predict the sulfate diffusion in coal pillar due to the dual time-dependent characteristics.The sulfate attacking makes the coal dam a typical heterogeneous gradient structure.For sulfate concentrations 0.01-0.20 mol/L in mine water,it takes almost 1.5 and 4 years for sulfate ions to diffuse 9.46 and 18.92 m,respectively.The experimental data and developed model provide a practical method for predicting sulfate diffusion in coal pillar,which helps the service life design of coal dams.
基金This project is supported by National Natural Science Foundation of China(No. 19971020) Multidiseipline Scientific Research Foundation of Harbin Institute of Technology, China(No.HIT.MD2001.26).
文摘The element of pesedospectral-multiwavelet-Galerkin method, and how tocombine it with penalty method for treating boundary conditions are given. Multiwavelet bases don'toverlap on the given scale, and possess the same compact set in a group of several functions, sothey can be directly used to the numerical discretion on the finite interval. Numerical tests showthat general boundary conditions can be enforced with the penalty method, and thatpesedospectral-multiwavelet-Galerkin method can well track the solutions' development. This alsoproves that pesedospectral-multiwavelet-Galerkin method is effective.
基金supported by the Fundamental Research Funds for the Central Universities(No.2018B16714)the National Natural Science Foundation of China(Nos.11702083,11572111,51679150,51579153,51739008,51527811)+5 种基金the State Key Laboratory of Mechanics and Control of Mechanical Structures(Nanjing University of Aeronautics and Astronautics)(No.MCMS-0218G01)the China Postdoctoral Science Foundation(No.2017M611669)the China Postdoctoral Science Special Foundation(No.2018T110430)the Postdoctoral Foundation of Jiangsu Province(No.1701059C)the National Key R&D Program of China(No.2016YFC0401902)the Fund Project of NHRI(Nos.Y417002,Y417015).
文摘This paper presents a new numerical technique for solving initial and bound-ary value problems with unsteady strongly nonlinear advection diffusion reaction(ADR)equations.The method is based on the use of the radial basis functions(RBF)for the approximation space of the solution.The Crank-Nicolson scheme is used for approximation in time.This results in a sequence of stationary nonlinear ADR equations.The equations are solved sequentially at each time step using the proposed semi-analytical technique based on the RBFs.The approximate solution is sought in the form of the analytical expansion over basis functions and contains free parameters.The basis functions are constructed in such a way that the expansion satisfies the boundary conditions of the problem for any choice of the free parameters.The free parameters are determined by substitution of the expansion in the equation and collocation in the solution domain.In the case of a nonlinear equation,we use the well-known procedure of quasilinearization.This transforms the original equation into a sequence of the linear ones on each time layer.The numerical examples confirm the high accuracy and robustness of the proposed numerical scheme.
基金supported by National Natural Science Foundation of China (Grant Nos. 11331005,11771150,11601164 and 11601165)
文摘This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant(x,t)∈R^+×R^+,with the null-Dirichlet boundary condition or the null-Neumann boundary condition on u. We show that the corresponding initial-boundary value problem admits a unique global smooth solution which tends timeasymptotically to the nonlinear diffusion wave. Compared with the previous work about Euler equations with constant coefficient damping, studied by Nishihara and Yang(1999), and Jiang and Zhu(2009, Discrete Contin Dyn Syst), we obtain a general result when the initial perturbation belongs to the same space. In addition,our main novelty lies in the fact that the cut-off points of the convergence rates are different from our previous result about the Cauchy problem. Our proof is based on the classical energy method and the analyses of the nonlinear diffusion wave.
基金This work was supported in part by National Research Foundation of Korea(NRF-2017R1A2B2010398)The authors thank Profs.L.C.Evans and W.Strauss for their valuable suggestions.
文摘This paper studies existence of mild solution to a sharp cut off model for contact driven tumor growth.Analysis is based on application of the Crandall-Liggett theorem for w-quas-contractive semigroups on the Banach space L^(1)(Ω).Furthermore,numerical computations are provided which compare the sharp cut off model with the tumor growth model of Perthame,Quiros,and Vazquez[13].
基金Project supported by NSFC(No.19871051No.19601038)+1 种基金the Outstanding Young People SF of Henan Province(No.0016)the NSF of Henan Province(No.004052000).
