The method of matrix continued fraction is used to investigate stochastic resonance (SR) in the biasedsubdiffusive Smoluchowski system within linear response range.Numerical results of linear dynamic susceptibility an...The method of matrix continued fraction is used to investigate stochastic resonance (SR) in the biasedsubdiffusive Smoluchowski system within linear response range.Numerical results of linear dynamic susceptibility andspectral amplification factor are presented and discussed in two-well potential and mono-well potential with differentsubdiffusion exponents.Following our observation,the introduction of a bias in the potential weakens the SR effect inthe subdiffusive system just as in the normal diffusive case.Our observation also discloses that the subdiffusion inhibitsthe low-frequency SR,but it enhances the high-frequency SR in the biased Smoluchowski system,which should reflect a'flattening' influence of the subdiffusion on the linear susceptibility.展开更多
Thermally driven diffusive motion of a particle underlies many physical and biological processes. In the presence of traps and obstacles, the spread of the particle is substantially impeded, leading to subdiffusive sc...Thermally driven diffusive motion of a particle underlies many physical and biological processes. In the presence of traps and obstacles, the spread of the particle is substantially impeded, leading to subdiffusive scaling at long times. The statistical mechanical treatment of diffusion in a disordered environment is often quite involved. In this short review, we present a simple and unified view of the many quantitative results on anomalous diffusion in the literature, including the scaling of the diffusion front and the mean first-passage time. Varioust analytic calculations and physical arguments are examined to highlight the role of dimensionality, energy landscape, and rare events in affecting the particle trajectory statistics. The general understanding that emerges will aid the interpretation of relevant experimental and simulation results.展开更多
The dynamics of tracers in crowded matrix is of interest in various areas of physics, such as the diffusion of proteins in living cells. By using two-dimensional （2D） Langevin dynamics simulations, we investigate th...The dynamics of tracers in crowded matrix is of interest in various areas of physics, such as the diffusion of proteins in living cells. By using two-dimensional （2D） Langevin dynamics simulations, we investigate the diffusive properties of a tracer of a diameter in crowded environments caused by randomly distributed crowders of a diameter. Results show that the emergence of subdiffusion of a tracer at intermediate time scales depends on the size ratio of the tracer to crowders a. If a falls between a lower critical size ratio and a upper one, the anomalous diffusion occurs purely due to the molecular crowding, tqlrther analysis indicates that the physical origin of subdiffusion is the ＂cage effect＂. Moreover, the subdiffusion exponent α decreases with the increasing medium viscosity and the degree of crowding, and gets a minimum αmin=0.75 at δ=1. At long time scales, normal diffusion of a tracer is recovered. For δ≤1, the relative mobility of tracers is independent of the degree of crowding. Meanwhile, it is sensitive to the degree of crowding for δ〉l. Our results are helpful in deepening the understanding of the diffusive properties of biomacromolecules that lie within crowded intracellular environments, such as proteins, DNA and ribosomes.展开更多
Three-dimensional （3D） Fick＇s diffusion equation and fractional diffusion equation are solved for different reflecting boundaries. We use the continuous time random walk model （CTRW） to investigate the time-avera...Three-dimensional （3D） Fick＇s diffusion equation and fractional diffusion equation are solved for different reflecting boundaries. We use the continuous time random walk model （CTRW） to investigate the time-averaged mean square dis- placement （MSD） of a 3D single particle trajectory. Theoretical results show that the ensemble average of the time-averaged MSD can be expressed analytically by a Mittag-Leffler function. Our new expression is in agreement with previous formu- las in two limiting cases： （^-δ2） ~ △1 in short lag time and （^-δ2} ~ △1 -α in long lag time. We also simulate the experimental data of mRNA diffusion in living E. coli using a 3D CTRW model under confined and crowded conditions. The simulation results are well consistent with experimental results. The calculations of power spectral density （PSD） further indicate the subdiffsive behavior of an individual trajectory.展开更多
To describe the energy-dependent characteristics of the reaction-subdiffusion process, we analyze the simple reaction A--→B under subdiffsion with waiting time depending on the preceding jump length, and derive the c...To describe the energy-dependent characteristics of the reaction-subdiffusion process, we analyze the simple reaction A--→B under subdiffsion with waiting time depending on the preceding jump length, and derive the corresponding master equations in the Fourier Laplace space for the distribution of A and B particles in a continuous time random walk scheme. Moreover, the generalizations of the reaction-diffusion equation for the Gaussian jump length with the probability density function of waiting time being quadratically dependent on the preceding jump length are obtained by applying the derived master equations.展开更多
Considering the discrete nonlinear Schrodinger model with dipole-dipole interactions （DDIs）, we comparatively and numerically study the effects of contact interaction, DDI and disorder on the properties of diffusion...Considering the discrete nonlinear Schrodinger model with dipole-dipole interactions （DDIs）, we comparatively and numerically study the effects of contact interaction, DDI and disorder on the properties of diffusion of dipolar condensate in one-dimensional quasi-periodic potentials. Due to the coupled effects of the contact interaction and the DDI, some new and interesting mechanisms are found： both the DDI and the contact interaction can destroy localization and lead to a subdiffusive growth of the second moment of the wave packet. However, compared with the contact interaction, the effect of DDI on the subdiffusion is stronger. Furthermore and interestingly, we find that when the contact interaction （λ1） and DDI （A2） satisfy λ1 ≥ 2λ2, the property of the subdiffusion depends only on contact interaction; when λ1 ≤ 2λ2, the property of the subdiffusion is completely determined by DDI. Remarkably, we numerically give the critical value of disorder strength v＊ for different values of contact interaction and DDI. When the disorder strength v ≥ v＊, the wave packet is localized. On the contrary, when the disorder strength v ≤ v＊, the wave packet is subdiffusive.展开更多
Thanks to the singularity of the solution of linear subdiffusion problems,most time-stepping methods on uniform meshes can result in O(τ)accuracy whereτ denotes the time step.The present work aims to discover the re...Thanks to the singularity of the solution of linear subdiffusion problems,most time-stepping methods on uniform meshes can result in O(τ)accuracy whereτ denotes the time step.The present work aims to discover the reason why some type of Crank-Nicolson schemes(the averaging Crank-Nicolson(ACN)scheme)for the subdiffusion can only yield O(τ^(α))(α<1)accuracy,which is much lower than the desired.The existing well-developed error analysis for the subdiffusion,which has been successfully applied to many time-stepping methods such as the fractional BDF-p,(1≤p≤ 6),requires singular points to be out of the path of contour integrals involved.The ACN scheme in this work is quite natural but fails to meet this requirement.By resorting to the residue theorem,some novel sharp error analysis is developed in this study,upon which correction methods are further designed to obtain the optimal O(τ^(2))accuracy.All results are verified by numerical tests.展开更多
In this paper, we design and analyze a space-time spectral method for the subdiffusion equation.Here, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the init...In this paper, we design and analyze a space-time spectral method for the subdiffusion equation.Here, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the initial time. Consequently, traditional high-order numerical methods in time are inefficient. The second obstacle is that the resulting system of the space-time spectral approach is usually large and time-consuming to solve. We aim at overcoming the first difficulty by proposing a novel approach in time, which is based on variable transformation techniques. Suitable ψ-fractional Sobolev spaces and a new variational framework are introduced to establish the well-posedness of the associated variational problem. This allows us to construct our space-time spectral method using a combination of temporal generalized Jacobi polynomials(GJPs) and spatial Legendre polynomials. For the second difficulty, we propose a fast algorithm to effectively solve the resulting linear system. The fast algorithm makes use of a matrix diagonalization in space and QZ decomposition in time. Our analysis and numerical experiments show that the proposed method is exponentially convergent with respect to the polynomial degrees in both space and time directions, even though the exact solution has very limited regularity.展开更多
Fractional initial-value problems(IVPs) and time-fractional initial-boundary value problems(IBVPs), each with a Caputo temporal derivative of order α ∈(0, 1), are considered. An averaged variant of the well-known L1...Fractional initial-value problems(IVPs) and time-fractional initial-boundary value problems(IBVPs), each with a Caputo temporal derivative of order α ∈(0, 1), are considered. An averaged variant of the well-known L1 scheme is proved to be O(N^(-2)) convergent for IVPs on suitably graded meshes with N points, thereby improving the O(N^(-(2-α))) convergence rate of the standard L1 scheme. The analysis relies on a delicate decomposition of the temporal truncation error that yields a sharp dependence of the order of convergence on the degree of mesh grading used. This averaged L1 scheme can be combined with a finite difference or piecewise linear finite element discretization in space for IBVPs, and under a restriction on the temporal mesh width, one gets again O(N^(-2)) convergence in time, together with O(h^(2)) convergence in space,where h is the spatial mesh width. Numerical experiments support our results.展开更多
An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity p...An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity parameterσ∈(0,1)∪(1,2).Under this general regularity assumption,we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results,i.e.,a refined discrete fractional-type Grönwall inequality(DFGI).After that,we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations.The present results fill the gap on some interesting convergence results of L1 scheme onσ∈(0,α)∪(α,1)∪(1,2].Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.展开更多
In this paper, we design a semi-implicit scheme for the scalar time fractional reaction-diffusion equation. We theoretically prove that the numerical scheme is stable without the restriction on the ratio of the time a...In this paper, we design a semi-implicit scheme for the scalar time fractional reaction-diffusion equation. We theoretically prove that the numerical scheme is stable without the restriction on the ratio of the time and space stepsizes, and numerically show that the convergence orders are 1 in time and 2 in space. As a concrete model, the subdiffusive predator-prey system is discussed in detail. First, we prove that the analytical solution to the system is positive and bounded. Then, we use the provided numerical scheme to solve the subdiffusive predator-prey system, and theoretically prove and numerically verify that the numerical scheme preserves the positivity and boundedness.展开更多
In this work, we study a system of autonomous fractional differential equations. The differential operator is taken in the Caputo sense. Using the monotone iterative technique combined with the method of upper and low...In this work, we study a system of autonomous fractional differential equations. The differential operator is taken in the Caputo sense. Using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence and uniqueness of solutions for coupled system which are nonlinear fractional differential equations, moreover, we obtain the dependence of the solution on the initial values. In addition, we give an important example that is a two-patch subdiffusive predator-prey metapopulation model, investigate the solvability and give the numerical results with this model. The numerical simulation indicates that the results of the suhdiffusive model approximate to the two-patch predator-prey metapopulation model with the order a approach to 1.展开更多
In this paper,we study a high-order compact difference scheme for the fourth-order fractional subdiffusion system.We consider the situation in which the unknown function and its first-order derivative are given at the...In this paper,we study a high-order compact difference scheme for the fourth-order fractional subdiffusion system.We consider the situation in which the unknown function and its first-order derivative are given at the boundary.The scheme is shown to have high order convergence.Numerical examples are given to verify the theoretical results.展开更多
Advances in nanotechnology enable scientists for the first time to study biological pro-cesses on a nanoscale molecule-by-molecule basis.They also raise challenges and opportunities for statisticians and applied proba...Advances in nanotechnology enable scientists for the first time to study biological pro-cesses on a nanoscale molecule-by-molecule basis.They also raise challenges and opportunities for statisticians and applied probabilists.To exemplify the stochastic inference and modeling problems in the field,this paper discusses a few selected cases,ranging from likelihood inference,Bayesian data augmentation,and semi-and non-parametric inference of nanometric biochemical systems to the uti-lization of stochastic integro-differential equations and stochastic networks to model single-molecule biophysical processes.We discuss the statistical and probabilistic issues as well as the biophysical motivation and physical meaning behind the problems,emphasizing the analysis and modeling of real experimental data.展开更多
We present the finite difference/element method for a two-dimensional modified fractional diffusion equation.The analysis is carried out first for the time semi-discrete scheme,and then for the full discrete scheme.Th...We present the finite difference/element method for a two-dimensional modified fractional diffusion equation.The analysis is carried out first for the time semi-discrete scheme,and then for the full discrete scheme.The time discretization is based on the L1-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term.We use finite element method for the spatial approximation in full discrete scheme.We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent.Moreover,the optimal convergence rate is obtained.Finally,some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.展开更多
基金Supported by the National Natural Science Foundation of China under Grant Nos.10602041 and 10972170
文摘The method of matrix continued fraction is used to investigate stochastic resonance (SR) in the biasedsubdiffusive Smoluchowski system within linear response range.Numerical results of linear dynamic susceptibility andspectral amplification factor are presented and discussed in two-well potential and mono-well potential with differentsubdiffusion exponents.Following our observation,the introduction of a bias in the potential weakens the SR effect inthe subdiffusive system just as in the normal diffusive case.Our observation also discloses that the subdiffusion inhibitsthe low-frequency SR,but it enhances the high-frequency SR in the biased Smoluchowski system,which should reflect a'flattening' influence of the subdiffusion on the linear susceptibility.
基金supported by the National Natural Science Foundation of China (Grant No. 11175013)the Research Grants Council of the Hong Kong Special Administrative Region,China (Grant No. N HKBU 213/10)
文摘Thermally driven diffusive motion of a particle underlies many physical and biological processes. In the presence of traps and obstacles, the spread of the particle is substantially impeded, leading to subdiffusive scaling at long times. The statistical mechanical treatment of diffusion in a disordered environment is often quite involved. In this short review, we present a simple and unified view of the many quantitative results on anomalous diffusion in the literature, including the scaling of the diffusion front and the mean first-passage time. Varioust analytic calculations and physical arguments are examined to highlight the role of dimensionality, energy landscape, and rare events in affecting the particle trajectory statistics. The general understanding that emerges will aid the interpretation of relevant experimental and simulation results.
基金This work is supported by the National Natural Science Foundation of China （No.21225421 and No.21474099）, the National Basic Research Program of China （No.2014CB845605）.
文摘The dynamics of tracers in crowded matrix is of interest in various areas of physics, such as the diffusion of proteins in living cells. By using two-dimensional （2D） Langevin dynamics simulations, we investigate the diffusive properties of a tracer of a diameter in crowded environments caused by randomly distributed crowders of a diameter. Results show that the emergence of subdiffusion of a tracer at intermediate time scales depends on the size ratio of the tracer to crowders a. If a falls between a lower critical size ratio and a upper one, the anomalous diffusion occurs purely due to the molecular crowding, tqlrther analysis indicates that the physical origin of subdiffusion is the ＂cage effect＂. Moreover, the subdiffusion exponent α decreases with the increasing medium viscosity and the degree of crowding, and gets a minimum αmin=0.75 at δ=1. At long time scales, normal diffusion of a tracer is recovered. For δ≤1, the relative mobility of tracers is independent of the degree of crowding. Meanwhile, it is sensitive to the degree of crowding for δ〉l. Our results are helpful in deepening the understanding of the diffusive properties of biomacromolecules that lie within crowded intracellular environments, such as proteins, DNA and ribosomes.
基金supported by the National Natural Science Foundation of China(Grant No.21153002)the Fundamental Research Funds for the Central Universities of China(Grant No.2013zzts151)
文摘Three-dimensional （3D） Fick＇s diffusion equation and fractional diffusion equation are solved for different reflecting boundaries. We use the continuous time random walk model （CTRW） to investigate the time-averaged mean square dis- placement （MSD） of a 3D single particle trajectory. Theoretical results show that the ensemble average of the time-averaged MSD can be expressed analytically by a Mittag-Leffler function. Our new expression is in agreement with previous formu- las in two limiting cases： （^-δ2） ~ △1 in short lag time and （^-δ2} ~ △1 -α in long lag time. We also simulate the experimental data of mRNA diffusion in living E. coli using a 3D CTRW model under confined and crowded conditions. The simulation results are well consistent with experimental results. The calculations of power spectral density （PSD） further indicate the subdiffsive behavior of an individual trajectory.
基金Supported by the National Natural Science Foundation of China under Grant No 11626047the Foundation for Young Key Teachers of Chengdu University of Technology under Grant No KYGG201414
文摘To describe the energy-dependent characteristics of the reaction-subdiffusion process, we analyze the simple reaction A--→B under subdiffsion with waiting time depending on the preceding jump length, and derive the corresponding master equations in the Fourier Laplace space for the distribution of A and B particles in a continuous time random walk scheme. Moreover, the generalizations of the reaction-diffusion equation for the Gaussian jump length with the probability density function of waiting time being quadratically dependent on the preceding jump length are obtained by applying the derived master equations.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11274255 and 11305132the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grand No 20136203110001+1 种基金the Natural Science Foundation of Gansu Province under Grant No 2011GS04358the Creation of Science and Technology of Northwest Normal University under Grant Nos NWNU-KJCXGC-03-48 and NWNU-LKQN-12-12
文摘Considering the discrete nonlinear Schrodinger model with dipole-dipole interactions （DDIs）, we comparatively and numerically study the effects of contact interaction, DDI and disorder on the properties of diffusion of dipolar condensate in one-dimensional quasi-periodic potentials. Due to the coupled effects of the contact interaction and the DDI, some new and interesting mechanisms are found： both the DDI and the contact interaction can destroy localization and lead to a subdiffusive growth of the second moment of the wave packet. However, compared with the contact interaction, the effect of DDI on the subdiffusion is stronger. Furthermore and interestingly, we find that when the contact interaction （λ1） and DDI （A2） satisfy λ1 ≥ 2λ2, the property of the subdiffusion depends only on contact interaction; when λ1 ≤ 2λ2, the property of the subdiffusion is completely determined by DDI. Remarkably, we numerically give the critical value of disorder strength v＊ for different values of contact interaction and DDI. When the disorder strength v ≥ v＊, the wave packet is localized. On the contrary, when the disorder strength v ≤ v＊, the wave packet is subdiffusive.
基金supported by the Autonomous Region Level High-Level Talent Introduction Research Support Program in 2022(No.12000-15042224 to B.Y.)the National Natural Science Foundation of China(No.12201322 to B.Y.,12061053 to Y.L.,and 12161063 to H.L.)the Natural Science Foundation of Inner Mongolia,China(No.2020MS01003 to Y.L.and 2021MS01018 to H.L.).
文摘Thanks to the singularity of the solution of linear subdiffusion problems,most time-stepping methods on uniform meshes can result in O(τ)accuracy whereτ denotes the time step.The present work aims to discover the reason why some type of Crank-Nicolson schemes(the averaging Crank-Nicolson(ACN)scheme)for the subdiffusion can only yield O(τ^(α))(α<1)accuracy,which is much lower than the desired.The existing well-developed error analysis for the subdiffusion,which has been successfully applied to many time-stepping methods such as the fractional BDF-p,(1≤p≤ 6),requires singular points to be out of the path of contour integrals involved.The ACN scheme in this work is quite natural but fails to meet this requirement.By resorting to the residue theorem,some novel sharp error analysis is developed in this study,upon which correction methods are further designed to obtain the optimal O(τ^(2))accuracy.All results are verified by numerical tests.
基金supported by National Natural Science Foundation of China (Grant No. 11971408)。
文摘In this paper, we design and analyze a space-time spectral method for the subdiffusion equation.Here, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the initial time. Consequently, traditional high-order numerical methods in time are inefficient. The second obstacle is that the resulting system of the space-time spectral approach is usually large and time-consuming to solve. We aim at overcoming the first difficulty by proposing a novel approach in time, which is based on variable transformation techniques. Suitable ψ-fractional Sobolev spaces and a new variational framework are introduced to establish the well-posedness of the associated variational problem. This allows us to construct our space-time spectral method using a combination of temporal generalized Jacobi polynomials(GJPs) and spatial Legendre polynomials. For the second difficulty, we propose a fast algorithm to effectively solve the resulting linear system. The fast algorithm makes use of a matrix diagonalization in space and QZ decomposition in time. Our analysis and numerical experiments show that the proposed method is exponentially convergent with respect to the polynomial degrees in both space and time directions, even though the exact solution has very limited regularity.
基金supported by National Natural Science Foundation of China (Grant Nos. 12101509, 12171283, 12171025 and NSAF-U1930402)the Science Foundation Program for Distinguished Young Scholars of Shandong (Overseas) (Grant No. 2022HWYQ-045)。
文摘Fractional initial-value problems(IVPs) and time-fractional initial-boundary value problems(IBVPs), each with a Caputo temporal derivative of order α ∈(0, 1), are considered. An averaged variant of the well-known L1 scheme is proved to be O(N^(-2)) convergent for IVPs on suitably graded meshes with N points, thereby improving the O(N^(-(2-α))) convergence rate of the standard L1 scheme. The analysis relies on a delicate decomposition of the temporal truncation error that yields a sharp dependence of the order of convergence on the degree of mesh grading used. This averaged L1 scheme can be combined with a finite difference or piecewise linear finite element discretization in space for IBVPs, and under a restriction on the temporal mesh width, one gets again O(N^(-2)) convergence in time, together with O(h^(2)) convergence in space,where h is the spatial mesh width. Numerical experiments support our results.
基金supported by the National Natural Science Foundation of China under grants 11771162,11771035,12171376 and 2020-JCJQ-ZD-029.
文摘An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity parameterσ∈(0,1)∪(1,2).Under this general regularity assumption,we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results,i.e.,a refined discrete fractional-type Grönwall inequality(DFGI).After that,we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations.The present results fill the gap on some interesting convergence results of L1 scheme onσ∈(0,α)∪(α,1)∪(1,2].Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.
基金supported by New Century Excellent Talents in University(Grant No.NCET-09-0438)National Natural Science Foundation of China(Grant Nos.10801067 and 11271173)the Fundamental Research Funds for the Central Universities(Grant Nos.lzujbky-2010-63 and lzujbky-2012-k26)
文摘In this paper, we design a semi-implicit scheme for the scalar time fractional reaction-diffusion equation. We theoretically prove that the numerical scheme is stable without the restriction on the ratio of the time and space stepsizes, and numerically show that the convergence orders are 1 in time and 2 in space. As a concrete model, the subdiffusive predator-prey system is discussed in detail. First, we prove that the analytical solution to the system is positive and bounded. Then, we use the provided numerical scheme to solve the subdiffusive predator-prey system, and theoretically prove and numerically verify that the numerical scheme preserves the positivity and boundedness.
文摘In this work, we study a system of autonomous fractional differential equations. The differential operator is taken in the Caputo sense. Using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence and uniqueness of solutions for coupled system which are nonlinear fractional differential equations, moreover, we obtain the dependence of the solution on the initial values. In addition, we give an important example that is a two-patch subdiffusive predator-prey metapopulation model, investigate the solvability and give the numerical results with this model. The numerical simulation indicates that the results of the suhdiffusive model approximate to the two-patch predator-prey metapopulation model with the order a approach to 1.
基金the Macao Science and Technology Development Fund FDCT/001/2013/A and the grant MYRG086(Y2-L2)-FST12-VSW from the University of Macao.
文摘In this paper,we study a high-order compact difference scheme for the fourth-order fractional subdiffusion system.We consider the situation in which the unknown function and its first-order derivative are given at the boundary.The scheme is shown to have high order convergence.Numerical examples are given to verify the theoretical results.
基金supported by the United States National Science Fundation Career Award (Grant No. DMS-0449204)
文摘Advances in nanotechnology enable scientists for the first time to study biological pro-cesses on a nanoscale molecule-by-molecule basis.They also raise challenges and opportunities for statisticians and applied probabilists.To exemplify the stochastic inference and modeling problems in the field,this paper discusses a few selected cases,ranging from likelihood inference,Bayesian data augmentation,and semi-and non-parametric inference of nanometric biochemical systems to the uti-lization of stochastic integro-differential equations and stochastic networks to model single-molecule biophysical processes.We discuss the statistical and probabilistic issues as well as the biophysical motivation and physical meaning behind the problems,emphasizing the analysis and modeling of real experimental data.
基金This research was partly supported by the National Basic Research Program of China973 Program under Grant No.2011CB706903+3 种基金the Program for New Century Excellent Talents in University under Grant No.NCET-09-0438the National Natural Science Foundation of China under Grant No.10801067the Fundamental Research Funds for the Central Universities under Grant No.lzujbky-2010-63No.lzujbky-2012-k26。
文摘We present the finite difference/element method for a two-dimensional modified fractional diffusion equation.The analysis is carried out first for the time semi-discrete scheme,and then for the full discrete scheme.The time discretization is based on the L1-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term.We use finite element method for the spatial approximation in full discrete scheme.We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent.Moreover,the optimal convergence rate is obtained.Finally,some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.
