This paper deals with a semi-linear parabolic system with nonlinear nonlocal sources and nonlocal boundaries. By using super-and sub-solution techniques, we first give the sufficient conditions that the classical solu...This paper deals with a semi-linear parabolic system with nonlinear nonlocal sources and nonlocal boundaries. By using super-and sub-solution techniques, we first give the sufficient conditions that the classical solution exists globally and blows up in a finite time respectively, and then give the necessary and sufficient conditions that two components u and ν blow up simultaneously. Finally, the uniform blow-up profiles in the interior are presented.展开更多
This paper deals with propagations of singularities in solutions to a parabolic system coupled with nonlocal nonlinear sources. The estimates for the four possible blow-up rates as well as the boundary layer profiles ...This paper deals with propagations of singularities in solutions to a parabolic system coupled with nonlocal nonlinear sources. The estimates for the four possible blow-up rates as well as the boundary layer profiles are established. The critical exponent of the system is determined also.展开更多
In this paper,we consider a nonlinear wave equation with nonlocal damping and nonlinear boundary damping.We prove a general energy decay property for solutions in terms of coefficient of the frictional boundary dampin...In this paper,we consider a nonlinear wave equation with nonlocal damping and nonlinear boundary damping.We prove a general energy decay property for solutions in terms of coefficient of the frictional boundary damping by using of the multiplier technique from the idea of Martinez[1],Our result extends and improves the result in the literature such as the work by Louredo,Ferreira de Araujo and Mi-randain[2]in which only exponential energy decay is considered.Furthermore,we get also the energy decay for the equation with nonlocal damping only but without nonlinear boundary damping.展开更多
This paper is concerned with numerical approximations of a nonlocal heat equation define on an infinite domain.Two classes of artificial boundary conditions(ABCs)are designed,namely,nonlocal analog Dirichlet-to-Neuman...This paper is concerned with numerical approximations of a nonlocal heat equation define on an infinite domain.Two classes of artificial boundary conditions(ABCs)are designed,namely,nonlocal analog Dirichlet-to-Neumann-type ABCs(global in time)and high-order Pad´e approximate ABCs(local in time).These ABCs reformulate the original problem into an initial-boundary-value(IBV)problem on a bounded domain.For the global ABCs,we adopt a fast evolution to enhance computational efficiency and reduce memory storage.High order fully discrete schemes,both second-order in time and space,are given to discretize two reduced problems.Extensive numerical experiments are carried out to show the accuracy and efficiency of the proposed methods.展开更多
In this work,the size-dependent buckling of functionally graded(FG)Bernoulli-Euler beams under non-uniform temperature is analyzed based on the stressdriven nonlocal elasticity and nonlocal heat conduction.By utilizin...In this work,the size-dependent buckling of functionally graded(FG)Bernoulli-Euler beams under non-uniform temperature is analyzed based on the stressdriven nonlocal elasticity and nonlocal heat conduction.By utilizing the variational principle of virtual work,the governing equations and the associated standard boundary conditions are systematically extracted,and the thermal effect,equivalent to the induced thermal load,is explicitly assessed by using the nonlocal heat conduction law.The stressdriven constitutive integral equation is equivalently transformed into a differential form with two non-standard constitutive boundary conditions.By employing the eigenvalue method,the critical buckling loads of the beams with different boundary conditions are obtained.The numerically predicted results reveal that the growth of the nonlocal parameter leads to a consistently strengthening effect on the dimensionless critical buckling loads for all boundary cases.Additionally,the effects of the influential factors pertinent to the nonlocal heat conduction on the buckling behavior are carefully examined.展开更多
Transonic shocks play a pivotal role in designation of supersonic inlets and ramjets.For the three-dimensional steady non-isentropic compressible Euler system with frictions,we constructe a family of transonic shock s...Transonic shocks play a pivotal role in designation of supersonic inlets and ramjets.For the three-dimensional steady non-isentropic compressible Euler system with frictions,we constructe a family of transonic shock solutions in rectilinear ducts with square cross-sections.In this article,we are devoted to proving rigorously that a large class of these transonic shock solutions are stable,under multidimensional small perturbations of the upcoming supersonic flows and back pressures at the exits of ducts in suitable function spaces.This manifests that frictions have a stabilization effect on transonic shocks in ducts,in consideration of previous works which shown that transonic shocks in purely steady Euler flows are not stable in such ducts.Except its implications to applications,because frictions lead to a stronger coupling between the elliptic and hyperbolic parts of the three-dimensional steady subsonic Euler system,we develop the framework established in previous works to study more complex and interesting Venttsel problems of nonlocal elliptic equations.展开更多
The nonlocal singularly perturbed nonlinear problem is considered. Underthe appropriate assumptions the author proved that there exists a solution andthe estimation of solution is obtained using the method of differe...The nonlocal singularly perturbed nonlinear problem is considered. Underthe appropriate assumptions the author proved that there exists a solution andthe estimation of solution is obtained using the method of differential inequalities and a class of boundary layer functions.展开更多
We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics,a nonlocal formulation of continuum mechanics.We prove that the methods are opt...We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics,a nonlocal formulation of continuum mechanics.We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation.A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero.We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible.We fnd that,for polynomial degrees greater than or equal to two,there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible.We verify these fndings through extensive numerical experiments.展开更多
This paper is concerned with the spatial propagation of an SIR epidemic model with nonlocal diffusion and free boundaries describing the evolution of a disease.This model can be viewed as a nonlocal version of the fre...This paper is concerned with the spatial propagation of an SIR epidemic model with nonlocal diffusion and free boundaries describing the evolution of a disease.This model can be viewed as a nonlocal version of the free boundary problem studied by Kim et al.(An SIR epidemic model with free boundary.Nonlinear Anal RWA,2013,14:1992-2001).We first prove that this problem has a unique solution defined for all time,and then we give sufficient conditions for the disease vanishing and spreading.Our result shows that the disease will not spread if the basic reproduction number R_(0)<1,or the initial infected area h_(0),expanding ability μ and the initial datum S_(0) are all small enough when 1<R_(0)<1+d/μ_(2)+α.Furthermore,we show that if 1<R_(0)<1+d/μ_(2)+α,the disease will spread when h_(0) is large enough or h_(0) is small but μ is large enough.It is expected that the disease will always spread when R_(0)≥1+d/μ_(2)+α which is different from the local model.展开更多
Free vibration of elastically constrained rectangular single-layered M0S_(2)is investigated by using a nonlocal Kirchhoff plate model with an initial stress.The variationally consistent elastically constrained boundar...Free vibration of elastically constrained rectangular single-layered M0S_(2)is investigated by using a nonlocal Kirchhoff plate model with an initial stress.The variationally consistent elastically constrained boundary conditions are obtained by using the weighted residual method,while the governing equations of the nonlocal Kirchhoff plate model are known.A modified Fourier series method is applied to study the vibrational behaviors of elastically constrained nonlocal Kirchhoff plate models.The convergence and reliability of the modified Fourier series method is verified via comparison with the finite element method.A comprehensive parametric study is performed to show the influences of the boundary elastic constant,nonlocal parameter and initial stress on the vibrational behaviors of single-layered M0S2.The results should be good for the design of nanoresonators.展开更多
In this paper,we are concerned with the numerical solutions for the parabolic and hyperbolic partial differential equations with nonlocal boundary conditions.Thus,we presented a new iterative algorithm based on the Re...In this paper,we are concerned with the numerical solutions for the parabolic and hyperbolic partial differential equations with nonlocal boundary conditions.Thus,we presented a new iterative algorithm based on the Restarted Adomian Decomposition Method(RADM)for solving the two equations of different types involving dissimilar boundary and nonlocal conditions.The algorithm presented transforms the given nonlocal initial boundary value problem to a local Dirichlet one and then employs the RADM for the numerical treatment.Numerical comparisons were made between our proposed method and the Adomian Decomposition Method(ADM)to demonstrate the efficiency and performance of the proposed method.展开更多
In this article, we will investigate the viscous Burgers equation with boundary feedback. The existence of the solution is proved by constructing a convergence sequence inductively. Moreover, the decay property of the...In this article, we will investigate the viscous Burgers equation with boundary feedback. The existence of the solution is proved by constructing a convergence sequence inductively. Moreover, the decay property of the solution is shown based on the maximum principle for nonlinear parabolic equations.展开更多
基金This work was supported by the National Natural Science Foundation of China (Grant Nos.10471013,10471022)the Ministry of Education of China Science and Technology Major Projects (Grant No.104090)
文摘This paper deals with a semi-linear parabolic system with nonlinear nonlocal sources and nonlocal boundaries. By using super-and sub-solution techniques, we first give the sufficient conditions that the classical solution exists globally and blows up in a finite time respectively, and then give the necessary and sufficient conditions that two components u and ν blow up simultaneously. Finally, the uniform blow-up profiles in the interior are presented.
基金supported by the National Natural Science Foundation of China (Grant No. 10771024)
文摘This paper deals with propagations of singularities in solutions to a parabolic system coupled with nonlocal nonlinear sources. The estimates for the four possible blow-up rates as well as the boundary layer profiles are established. The critical exponent of the system is determined also.
基金supported by National Natural Science Foundation of China(Nos.11601122,11801145)。
文摘In this paper,we consider a nonlinear wave equation with nonlocal damping and nonlinear boundary damping.We prove a general energy decay property for solutions in terms of coefficient of the frictional boundary damping by using of the multiplier technique from the idea of Martinez[1],Our result extends and improves the result in the literature such as the work by Louredo,Ferreira de Araujo and Mi-randain[2]in which only exponential energy decay is considered.Furthermore,we get also the energy decay for the equation with nonlocal damping only but without nonlinear boundary damping.
基金This research is supported in part by the U.S.NSF grants DMS-1318586 and DMS-1315259AFOSR MURI Center for Material Failure Prediction Through Peridynamics,OSD/ARO/MURI W911NF-15-1-0562 on Fractional PDEs for Conservation Laws and Beyond:Theory,Numerics and Applicationsthe NSFC under grants 91430216 and the NSFC program for Scientific Research Center under program No.:U1530401.
文摘This paper is concerned with numerical approximations of a nonlocal heat equation define on an infinite domain.Two classes of artificial boundary conditions(ABCs)are designed,namely,nonlocal analog Dirichlet-to-Neumann-type ABCs(global in time)and high-order Pad´e approximate ABCs(local in time).These ABCs reformulate the original problem into an initial-boundary-value(IBV)problem on a bounded domain.For the global ABCs,we adopt a fast evolution to enhance computational efficiency and reduce memory storage.High order fully discrete schemes,both second-order in time and space,are given to discretize two reduced problems.Extensive numerical experiments are carried out to show the accuracy and efficiency of the proposed methods.
基金Project supported by the National Natural Science Foundation of China(Nos.51435008 and 51705247)the China Postdoctoral Science Foundation(No.2020M671474)
文摘In this work,the size-dependent buckling of functionally graded(FG)Bernoulli-Euler beams under non-uniform temperature is analyzed based on the stressdriven nonlocal elasticity and nonlocal heat conduction.By utilizing the variational principle of virtual work,the governing equations and the associated standard boundary conditions are systematically extracted,and the thermal effect,equivalent to the induced thermal load,is explicitly assessed by using the nonlocal heat conduction law.The stressdriven constitutive integral equation is equivalently transformed into a differential form with two non-standard constitutive boundary conditions.By employing the eigenvalue method,the critical buckling loads of the beams with different boundary conditions are obtained.The numerically predicted results reveal that the growth of the nonlocal parameter leads to a consistently strengthening effect on the dimensionless critical buckling loads for all boundary cases.Additionally,the effects of the influential factors pertinent to the nonlocal heat conduction on the buckling behavior are carefully examined.
基金This work was supported in part by National Nature Science Foundation of China(11371141 and 11871218)by Science and Technology Commission of Shanghai Municipality(18dz2271000).
文摘Transonic shocks play a pivotal role in designation of supersonic inlets and ramjets.For the three-dimensional steady non-isentropic compressible Euler system with frictions,we constructe a family of transonic shock solutions in rectilinear ducts with square cross-sections.In this article,we are devoted to proving rigorously that a large class of these transonic shock solutions are stable,under multidimensional small perturbations of the upcoming supersonic flows and back pressures at the exits of ducts in suitable function spaces.This manifests that frictions have a stabilization effect on transonic shocks in ducts,in consideration of previous works which shown that transonic shocks in purely steady Euler flows are not stable in such ducts.Except its implications to applications,because frictions lead to a stronger coupling between the elliptic and hyperbolic parts of the three-dimensional steady subsonic Euler system,we develop the framework established in previous works to study more complex and interesting Venttsel problems of nonlocal elliptic equations.
文摘The nonlocal singularly perturbed nonlinear problem is considered. Underthe appropriate assumptions the author proved that there exists a solution andthe estimation of solution is obtained using the method of differential inequalities and a class of boundary layer functions.
文摘We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics,a nonlocal formulation of continuum mechanics.We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation.A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero.We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible.We fnd that,for polynomial degrees greater than or equal to two,there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible.We verify these fndings through extensive numerical experiments.
基金Zhao was supported by a scholarship from the China Scholarship Council,Li was partially supported by NSF of China(11731005)Cao was partially supported by NSF of China(11901264).
文摘This paper is concerned with the spatial propagation of an SIR epidemic model with nonlocal diffusion and free boundaries describing the evolution of a disease.This model can be viewed as a nonlocal version of the free boundary problem studied by Kim et al.(An SIR epidemic model with free boundary.Nonlinear Anal RWA,2013,14:1992-2001).We first prove that this problem has a unique solution defined for all time,and then we give sufficient conditions for the disease vanishing and spreading.Our result shows that the disease will not spread if the basic reproduction number R_(0)<1,or the initial infected area h_(0),expanding ability μ and the initial datum S_(0) are all small enough when 1<R_(0)<1+d/μ_(2)+α.Furthermore,we show that if 1<R_(0)<1+d/μ_(2)+α,the disease will spread when h_(0) is large enough or h_(0) is small but μ is large enough.It is expected that the disease will always spread when R_(0)≥1+d/μ_(2)+α which is different from the local model.
基金We gratefully acknowledge the support from the Natural Science Foundation of Chongqing(cstc2020jcyj-msxmX0037)State Key Laboratory of Mechanics and Control of Mechanical Structures(Nanjing University of Aeronautics and astronautics)under Grants MCMS-E-0120G01National Natural Science Foundation of China under Grants Nos.11925205 and 51921003,and the Fundamental Research Funds for the Central Universities of China.
文摘Free vibration of elastically constrained rectangular single-layered M0S_(2)is investigated by using a nonlocal Kirchhoff plate model with an initial stress.The variationally consistent elastically constrained boundary conditions are obtained by using the weighted residual method,while the governing equations of the nonlocal Kirchhoff plate model are known.A modified Fourier series method is applied to study the vibrational behaviors of elastically constrained nonlocal Kirchhoff plate models.The convergence and reliability of the modified Fourier series method is verified via comparison with the finite element method.A comprehensive parametric study is performed to show the influences of the boundary elastic constant,nonlocal parameter and initial stress on the vibrational behaviors of single-layered M0S2.The results should be good for the design of nanoresonators.
文摘In this paper,we are concerned with the numerical solutions for the parabolic and hyperbolic partial differential equations with nonlocal boundary conditions.Thus,we presented a new iterative algorithm based on the Restarted Adomian Decomposition Method(RADM)for solving the two equations of different types involving dissimilar boundary and nonlocal conditions.The algorithm presented transforms the given nonlocal initial boundary value problem to a local Dirichlet one and then employs the RADM for the numerical treatment.Numerical comparisons were made between our proposed method and the Adomian Decomposition Method(ADM)to demonstrate the efficiency and performance of the proposed method.
基金The author thanks the reviewers for useful suggestions. This work is supported by the National Natural Science Foundation of China （grant 10626002）.
文摘In this article, we will investigate the viscous Burgers equation with boundary feedback. The existence of the solution is proved by constructing a convergence sequence inductively. Moreover, the decay property of the solution is shown based on the maximum principle for nonlinear parabolic equations.
