The motion of pore water directly influences mechanical properties of soils, which are variable during creep. Accurate description of the evolution of mechanical properties of soils can help to reveal the internal beh...The motion of pore water directly influences mechanical properties of soils, which are variable during creep. Accurate description of the evolution of mechanical properties of soils can help to reveal the internal behavior of pore water. Based on the idea of using the fractional order to reflect mechanical properties of soils, a fractional creep model is proposed by introducing a variable-order fractional operator, and realized on a series of creep responses in soft soils. A comparative analysis illustrates that the evolution of mechanical properties, shown through the simulated results, exactly corresponds to the motion of pore water and the solid skeleton. This demonstrates that the proposed variable-order fractional model can be employed to characterize the evolution of mechanical properties of and the pore water motion in soft soils during creep. It is observed that the fractional order from the proposed model is related to the dissipation rate of pore water pressure.展开更多
In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order a...In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order accuracy, while the exponential-sum-approximation (ESA) is used to approximate the variable-order Caputo fractional derivative in the temporal direction, and a novel spatial sixth-order hybrid ESA-CCD method is implemented successfully. Finally, the accuracy of the proposed method is verified by numerical experiments.展开更多
In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators...In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators.The existence,uniqueness,and stability of the proposed model are discussed.Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator is developed to study the proposed models.Comparative studies with generalized fifth-order Runge-Kutta method are given.Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations are presented.We have showcased the efficiency of the proposed method and garnered robust empirical support for our theoretical findings.展开更多
Understanding the variations in microscopic pore-fracture structures(MPFS) during coal creep under pore pressure and stress coupling is crucial for coal mining and effective gas treatment. In this manuscript, a triaxi...Understanding the variations in microscopic pore-fracture structures(MPFS) during coal creep under pore pressure and stress coupling is crucial for coal mining and effective gas treatment. In this manuscript, a triaxial creep test on deep coal at various pore pressures using a test system that combines in-situ mechanical loading with real-time nuclear magnetic resonance(NMR) detection was conducted.Full-scale quantitative characterization, online real-time detection, and visualization of MPFS during coal creep influenced by pore pressure and stress coupling were performed using NMR and NMR imaging(NMRI) techniques. The results revealed that seepage pores and microfractures(SPM) undergo the most significant changes during coal creep, with creep failure gradually expanding from dense primary pore fractures. Pore pressure presence promotes MPFS development primarily by inhibiting SPM compression and encouraging adsorption pores(AP) to evolve into SPM. Coal enters the accelerated creep stage earlier at lower stress levels, resulting in more pronounced creep deformation. The connection between the micro and macro values was established, demonstrating that increased porosity at different pore pressures leads to a negative exponential decay of the viscosity coefficient. The Newton dashpot in the ideal viscoplastic body and the Burgers model was improved using NMR experimental results, and a creep model that considers pore pressure and stress coupling using variable-order fractional operators was developed. The model’s reasonableness was confirmed using creep experimental data. The damagestate adjustment factors ω and β were identified through a parameter sensitivity analysis to characterize the effect of pore pressure and stress coupling on the creep damage characteristics(size and degree of difficulty) of coal.展开更多
In this paper,we first prove the existence and uniqueness theorem of the solution of nonlinear variable-order fractional stochastic differential equations(VFSDEs).We futher constructe the Euler-Maruyama method to solv...In this paper,we first prove the existence and uniqueness theorem of the solution of nonlinear variable-order fractional stochastic differential equations(VFSDEs).We futher constructe the Euler-Maruyama method to solve the equations and prove the convergence in mean and the strong convergence of the method.In particular,when the fractional order is no longer varying,the conclusions obtained are consistent with the relevant conclusions in the existing literature.Finally,the numerical experiments at the end of the article verify the correctness of the theoretical results obtained.展开更多
This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in R<sup>N</sup>, s(⋅)...This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in R<sup>N</sup>, s(⋅) ∈ C (R<sup>N</sup> × R<sup>N</sup>, (0,1)), (-Δ)<sup>s(⋅)</sup> is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.展开更多
In this paper,we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation,and then develop an optimal Galerkin finite element scheme without any regularity ass...In this paper,we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation,and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution.We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time t=0.More precisely,we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to C 2([0,T])in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness,otherwise the solution exhibits the same singular behavior like its constant-order counterpart.Based on these regularity results,we prove optimalorder convergence rate of the Galerkin finite element scheme.Furthermore,we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives.Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.展开更多
In this paper,we propose a fast second-order approximation to the variable-order(VO)Caputo fractional derivative,which is developed based on L2-1σformula and the exponential-sum-approximation technique.The fast evalu...In this paper,we propose a fast second-order approximation to the variable-order(VO)Caputo fractional derivative,which is developed based on L2-1σformula and the exponential-sum-approximation technique.The fast evaluation method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO Caputo fractional derivative.This fast algorithm is applied to construct a relevant fast temporal second-order and spatial fourth-order scheme(F L2-1σscheme)for the multi-dimensional VO time-fractional sub-diffusion equations.Theoretically,F L2-1σscheme is proved to fulfill the similar properties of the coefficients as those of the well-studied L2-1σscheme.Therefore,F L2-1σscheme is strictly proved to be unconditionally stable and convergent.A sharp decrease in the computational cost and the acting memory is shown in the numerical examples to demonstrate the efficiency of the proposed method.展开更多
We develop fractional buffer layers(FBLs)to absorb propagating waves without reflection in bounded domains.Our formulation is based on variable-order spatial fractional derivatives.We select a proper variable-order fu...We develop fractional buffer layers(FBLs)to absorb propagating waves without reflection in bounded domains.Our formulation is based on variable-order spatial fractional derivatives.We select a proper variable-order function so that dissipation is induced to absorb the coming waves in the buffer layers attached to the domain.In particular,we first design proper FBLs for the one-dimensional one-way and two-waywave propagation.Then,we extend our formulation to two-dimensional problems,wherewe introduce a consistent variable-order fractionalwave equation.In each case,we obtain the fully discretized equations by employing a spectral collocation method in space and Crank-Nicolson or Adams-Bashforth method in time.We compare our results with a finely tuned perfectly matched layer(PML)method and show that the proposed FBL is able to suppress reflected waves including corner reflections in a two-dimensional rectangular domain.We also demonstrate that our formulation is more robust and uses less number of equations.展开更多
The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems.The scheme is discretized by a weighted-shifted Grünwald f...The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems.The scheme is discretized by a weighted-shifted Grünwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction.The stability and the L^(2)-convergence of the scheme are proved for all variable-orderα(t)∈(0,1).The proposed method is of accuracy-order O(τ^(3)+h^(k+1)),whereτ,h,and k are the temporal step size,the spatial step size,and the degree of piecewise P^(k)polynomials,respectively.Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.展开更多
基金supported by the Natural Science Foundation of Jiangsu Province of China(Grant No.BK2012810)the Fundamental Research Funds for the Central Universities(Grant No.2009B15114)
文摘The motion of pore water directly influences mechanical properties of soils, which are variable during creep. Accurate description of the evolution of mechanical properties of soils can help to reveal the internal behavior of pore water. Based on the idea of using the fractional order to reflect mechanical properties of soils, a fractional creep model is proposed by introducing a variable-order fractional operator, and realized on a series of creep responses in soft soils. A comparative analysis illustrates that the evolution of mechanical properties, shown through the simulated results, exactly corresponds to the motion of pore water and the solid skeleton. This demonstrates that the proposed variable-order fractional model can be employed to characterize the evolution of mechanical properties of and the pore water motion in soft soils during creep. It is observed that the fractional order from the proposed model is related to the dissipation rate of pore water pressure.
文摘In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order accuracy, while the exponential-sum-approximation (ESA) is used to approximate the variable-order Caputo fractional derivative in the temporal direction, and a novel spatial sixth-order hybrid ESA-CCD method is implemented successfully. Finally, the accuracy of the proposed method is verified by numerical experiments.
文摘In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators.The existence,uniqueness,and stability of the proposed model are discussed.Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator is developed to study the proposed models.Comparative studies with generalized fifth-order Runge-Kutta method are given.Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations are presented.We have showcased the efficiency of the proposed method and garnered robust empirical support for our theoretical findings.
基金supported by the National Natural Science Foundation of China(Nos.52121003,51827901 and 52204110)China Postdoctoral Science Foundation(No.2022M722346)+1 种基金the 111 Project(No.B14006)the Yueqi Outstanding Scholar Program of CUMTB(No.2017A03).
文摘Understanding the variations in microscopic pore-fracture structures(MPFS) during coal creep under pore pressure and stress coupling is crucial for coal mining and effective gas treatment. In this manuscript, a triaxial creep test on deep coal at various pore pressures using a test system that combines in-situ mechanical loading with real-time nuclear magnetic resonance(NMR) detection was conducted.Full-scale quantitative characterization, online real-time detection, and visualization of MPFS during coal creep influenced by pore pressure and stress coupling were performed using NMR and NMR imaging(NMRI) techniques. The results revealed that seepage pores and microfractures(SPM) undergo the most significant changes during coal creep, with creep failure gradually expanding from dense primary pore fractures. Pore pressure presence promotes MPFS development primarily by inhibiting SPM compression and encouraging adsorption pores(AP) to evolve into SPM. Coal enters the accelerated creep stage earlier at lower stress levels, resulting in more pronounced creep deformation. The connection between the micro and macro values was established, demonstrating that increased porosity at different pore pressures leads to a negative exponential decay of the viscosity coefficient. The Newton dashpot in the ideal viscoplastic body and the Burgers model was improved using NMR experimental results, and a creep model that considers pore pressure and stress coupling using variable-order fractional operators was developed. The model’s reasonableness was confirmed using creep experimental data. The damagestate adjustment factors ω and β were identified through a parameter sensitivity analysis to characterize the effect of pore pressure and stress coupling on the creep damage characteristics(size and degree of difficulty) of coal.
基金supported by the National Natural Science Foundation of China(No.12071403)the Scientific Research Foundation of Hunan Provincial Education Department of China(No.18A049).
文摘In this paper,we first prove the existence and uniqueness theorem of the solution of nonlinear variable-order fractional stochastic differential equations(VFSDEs).We futher constructe the Euler-Maruyama method to solve the equations and prove the convergence in mean and the strong convergence of the method.In particular,when the fractional order is no longer varying,the conclusions obtained are consistent with the relevant conclusions in the existing literature.Finally,the numerical experiments at the end of the article verify the correctness of the theoretical results obtained.
文摘This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in R<sup>N</sup>, s(⋅) ∈ C (R<sup>N</sup> × R<sup>N</sup>, (0,1)), (-Δ)<sup>s(⋅)</sup> is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.
基金the National Natural Science Foundation of China(No.11971482)the Natural Science Foundation of Shandong Province(No.ZR2017MA006)+2 种基金the National Science Foundation(No.DMS-1620194)the China Postdoctoral Science Foundation(Nos.2020M681136,2021TQ0017,2021T140129)the International Postdoctoral Exchange Fellowship Program(Talent-Introduction Program)(No.YJ20210019).
文摘In this paper,we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation,and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution.We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time t=0.More precisely,we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to C 2([0,T])in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness,otherwise the solution exhibits the same singular behavior like its constant-order counterpart.Based on these regularity results,we prove optimalorder convergence rate of the Galerkin finite element scheme.Furthermore,we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives.Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.
基金supported in part by research grants of the Science and Technology De-velopment Fund,Macao SAR(0122/2020/A3)University of Macao(MYRG2020-00224-FST).
文摘In this paper,we propose a fast second-order approximation to the variable-order(VO)Caputo fractional derivative,which is developed based on L2-1σformula and the exponential-sum-approximation technique.The fast evaluation method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO Caputo fractional derivative.This fast algorithm is applied to construct a relevant fast temporal second-order and spatial fourth-order scheme(F L2-1σscheme)for the multi-dimensional VO time-fractional sub-diffusion equations.Theoretically,F L2-1σscheme is proved to fulfill the similar properties of the coefficients as those of the well-studied L2-1σscheme.Therefore,F L2-1σscheme is strictly proved to be unconditionally stable and convergent.A sharp decrease in the computational cost and the acting memory is shown in the numerical examples to demonstrate the efficiency of the proposed method.
基金the support by the China Scholarship Council(No.201906890040)partial support by the Department of Energy by the PhILMs project(DE-SC0019453)by the MURI/ARO,USA on Fractional PDEs for Conservation Laws and Beyond:Theory,Numerics and Applications(W911NF-15-1-0562).
文摘We develop fractional buffer layers(FBLs)to absorb propagating waves without reflection in bounded domains.Our formulation is based on variable-order spatial fractional derivatives.We select a proper variable-order function so that dissipation is induced to absorb the coming waves in the buffer layers attached to the domain.In particular,we first design proper FBLs for the one-dimensional one-way and two-waywave propagation.Then,we extend our formulation to two-dimensional problems,wherewe introduce a consistent variable-order fractionalwave equation.In each case,we obtain the fully discretized equations by employing a spectral collocation method in space and Crank-Nicolson or Adams-Bashforth method in time.We compare our results with a finely tuned perfectly matched layer(PML)method and show that the proposed FBL is able to suppress reflected waves including corner reflections in a two-dimensional rectangular domain.We also demonstrate that our formulation is more robust and uses less number of equations.
基金This work was supported by the Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology(2018RCJH10)the Training Plan of Young Backbone Teachers in Henan University of Technology(21420049)+2 种基金the Training Plan of Young Backbone Teachers in Colleges and Universities of Henan Province(2019GGJS094)the innovative Funds Plan of Henan University of Technology,Foundation of Henan Educational Committee(19A110005)the National Natural Science Foundation of China(11861068).
文摘The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems.The scheme is discretized by a weighted-shifted Grünwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction.The stability and the L^(2)-convergence of the scheme are proved for all variable-orderα(t)∈(0,1).The proposed method is of accuracy-order O(τ^(3)+h^(k+1)),whereτ,h,and k are the temporal step size,the spatial step size,and the degree of piecewise P^(k)polynomials,respectively.Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.
