This study introduces a pre-orthogonal adaptive Fourier decomposition(POAFD)to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli an...This study introduces a pre-orthogonal adaptive Fourier decomposition(POAFD)to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre(generalized Poisson equation).As a first step,the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence,and,as a second step,makes use of the semigroup or the reproducing kernel property of each of the expanding entries.Experiments show the effectiveness and efficiency of the proposed series solutions.展开更多
The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent ...The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.展开更多
In this paper,we establish the integration by parts formula for the solution of fractional noise driven stochastic heat equations using the method of coupling.As an application,we also obtain the shift Harnack inequal...In this paper,we establish the integration by parts formula for the solution of fractional noise driven stochastic heat equations using the method of coupling.As an application,we also obtain the shift Harnack inequalities.展开更多
We investigate the fractional heat equation with fractional translation in both time and position with different fractional orders.As examples,we consider a rod and anα-disk with an initial constant temperature and d...We investigate the fractional heat equation with fractional translation in both time and position with different fractional orders.As examples,we consider a rod and anα-disk with an initial constant temperature and discuss their cooling processes in the examined formalism.展开更多
In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 bou...In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 boundary data.We have a non-existence result,which is the justification for taking into account the restricted boundary data.There is a smooth positive boundary datum that precludes the existence of the positive classical solution.展开更多
In this paper,we prove Talagrand’s T2 transportation cost-information inequality for the law of stochastic heat equation driven by Gaussian noise,which is fractional for a time variable with the Hurst index H∈(1/2,1...In this paper,we prove Talagrand’s T2 transportation cost-information inequality for the law of stochastic heat equation driven by Gaussian noise,which is fractional for a time variable with the Hurst index H∈(1/2,1),and is correlated for the spatial variable.The Girsanov theorem for fractional-colored Gaussian noise plays an important role in the proof.展开更多
Let u(t,x)be the solution to the one-dimensional nonlinear stochastic heat equation driven by space-time white noise with u(0,x)=1 for all x∈R.In this paper,we prove the law of the iterated logarithm(LIL for short)an...Let u(t,x)be the solution to the one-dimensional nonlinear stochastic heat equation driven by space-time white noise with u(0,x)=1 for all x∈R.In this paper,we prove the law of the iterated logarithm(LIL for short)and the functional LIL for a linear additive functional of the form∫[0,R]u(t,x)dx and the nonlinear additive functionals of the form∫[0,R]g(u(t,x))dx,where g:R→R is nonrandom and Lipschitz continuous,as R→∞for fixed t>0,using the localization argument.展开更多
In this paper, an actuator fault diagnosis scheme based on the backstepping method is proposed for a class of nonlinear heat equations. The fault diagnosis scheme includes fault detection, fault estimation and time to...In this paper, an actuator fault diagnosis scheme based on the backstepping method is proposed for a class of nonlinear heat equations. The fault diagnosis scheme includes fault detection, fault estimation and time to failure (TTF) prediction. Firstly, we achieve fault detection by comparing the detection residual with a predetermined threshold, where the detection residual is defined as the difference between the observer output and the system measurement output. Then, we estimate the fault function through the fault parameter update law and calculate the TTF using only limited measurements. Finally, the numerical simulation is performed on a nonlinear heat equation to verify the effectiveness of the proposed fault diagnosis scheme.展开更多
The traditional differential quadrature method was improved by using theupwind difference scheme for the convective terms to solve the coupled two-dimensionalincompressible Navier-stokes equations and heat equation. T...The traditional differential quadrature method was improved by using theupwind difference scheme for the convective terms to solve the coupled two-dimensionalincompressible Navier-stokes equations and heat equation. The new method was compared with theconventional differential quadrature method in the aspects of convergence and accuracy. The resultsshow that the new method is more accurate, and has better convergence than the conventionaldifferential quadrature method for numerically computing the steady-state solution.展开更多
In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equ...In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two differen t met hods, respectively, based on variance compu tations and on pat h-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.展开更多
The purpose of this paper is to investigate the stability and asymptotic behavior of the time-dependent solutions to a linear parabolic equation with nonlinear boundary condition in relation to their corresponding ste...The purpose of this paper is to investigate the stability and asymptotic behavior of the time-dependent solutions to a linear parabolic equation with nonlinear boundary condition in relation to their corresponding steady state solutions. Then, the above results are extended to a semilinear parabolic equation with nonlinear boundary condition by analyzing the corresponding eigenvalue problem and using the method of upper and lower solutions.展开更多
This paper deals with blow-up of positive solution of the nonlinear heat equation with absorption subject to a nonlinear boundary condition.The conditions under which the solutions may exist globally or blow-up are ob...This paper deals with blow-up of positive solution of the nonlinear heat equation with absorption subject to a nonlinear boundary condition.The conditions under which the solutions may exist globally or blow-up are obtained by the comparison principles.展开更多
This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covarianc...This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.展开更多
This paper is concerned with some nonlinear heat equations with initial condition and anti-periodic boundary condition. Also some two-point value nonlinear heat equations with anti-periodic boundary condition are disc...This paper is concerned with some nonlinear heat equations with initial condition and anti-periodic boundary condition. Also some two-point value nonlinear heat equations with anti-periodic boundary condition are discussed. The existence and uniqueness of the solutions are given. Some asymptotic behaviors of the solutions are studied.展开更多
In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the...In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming (SQP) algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.展开更多
In this paper,we propose a new numerical method which is a least squares approximaton based on pseudospectral method for the Forward-Backward heat equation. The existence and uniqueness of the solution of the least sq...In this paper,we propose a new numerical method which is a least squares approximaton based on pseudospectral method for the Forward-Backward heat equation. The existence and uniqueness of the solution of the least squares approximation are proved. Error estimates for this approximation are given,which show that tile order of convergence depends only on the regularity of tile solution and the right hand of the Forward-Backward heat equation.展开更多
This paper deals with the blow-up properties of solutions to semilinear heat equation ut-uxx= up in (0, 1) × (0, T) with the Neumann boundary condition ux(0, t) = 0, u:x1, t) = 1 on [0, T). The necessary and suff...This paper deals with the blow-up properties of solutions to semilinear heat equation ut-uxx= up in (0, 1) × (0, T) with the Neumann boundary condition ux(0, t) = 0, u:x1, t) = 1 on [0, T). The necessary and sufficient conditions under which all solutions to have a finite time blow-up and the exact blow-up rates are established. It is proved that the blow-up will occur only at the boundary x = 1. The asymptotic behavior near the blow-up time is also studied.展开更多
Some nonclassical potential symmetry generators and group-invariant solutions of heat equation and wave equation were determined. It is shown that some new explicit solutions of partial differential equations in conse...Some nonclassical potential symmetry generators and group-invariant solutions of heat equation and wave equation were determined. It is shown that some new explicit solutions of partial differential equations in conserved form can he constructed by using the nonclassical potential symmetry generators which are derived from their adjoint system. These explicit solutions cannot he obtained by using the Lie or Lie-Baeicklund symmetry group generators of differential equations.展开更多
On a complete non-compact gradient shrinking Ricci soliton,we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space variable.This growth condition is sh...On a complete non-compact gradient shrinking Ricci soliton,we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space variable.This growth condition is sharp.As an application,we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with quadratic exponential growth on shrinkers.展开更多
A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-or...A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis.展开更多
基金supported by the Science and Technology Development Fund of Macao SAR(FDCT0128/2022/A,0020/2023/RIB1,0111/2023/AFJ,005/2022/ALC)the Shandong Natural Science Foundation of China(ZR2020MA004)+2 种基金the National Natural Science Foundation of China(12071272)the MYRG 2018-00168-FSTZhejiang Provincial Natural Science Foundation of China(LQ23A010014).
文摘This study introduces a pre-orthogonal adaptive Fourier decomposition(POAFD)to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre(generalized Poisson equation).As a first step,the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence,and,as a second step,makes use of the semigroup or the reproducing kernel property of each of the expanding entries.Experiments show the effectiveness and efficiency of the proposed series solutions.
文摘The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.
基金supported by the Natural Science Foundation of China(11901005,12071003)the Natural Science Foundation of Anhui Province(2008085QA20)。
文摘In this paper,we establish the integration by parts formula for the solution of fractional noise driven stochastic heat equations using the method of coupling.As an application,we also obtain the shift Harnack inequalities.
基金supported by the Internal Project of Excellent Research of the Faculty of Science of Hradec KrálovéUniversity(Grant No.2022/2218)。
文摘We investigate the fractional heat equation with fractional translation in both time and position with different fractional orders.As examples,we consider a rod and anα-disk with an initial constant temperature and discuss their cooling processes in the examined formalism.
基金supported by the National NaturalScience Foundation of China(11971069 and 12126307)。
文摘In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 boundary data.We have a non-existence result,which is the justification for taking into account the restricted boundary data.There is a smooth positive boundary datum that precludes the existence of the positive classical solution.
基金supported by the Shanghai Sailing Program (21YF1415300)the Natural Science Foundation of China (12101392)supported by the Natural Science Foundation of China (11871382,11771161).
文摘In this paper,we prove Talagrand’s T2 transportation cost-information inequality for the law of stochastic heat equation driven by Gaussian noise,which is fractional for a time variable with the Hurst index H∈(1/2,1),and is correlated for the spatial variable.The Girsanov theorem for fractional-colored Gaussian noise plays an important role in the proof.
基金supported by the National Natural Science Foundation of China(11771178 and 12171198)the Science and Technology Development Program of Jilin Province(20210101467JC)+1 种基金the Science and Technology Program of Jilin Educational Department during the“13th Five-Year”Plan Period(JJKH20200951KJ)the Fundamental Research Funds for the Central Universities。
文摘Let u(t,x)be the solution to the one-dimensional nonlinear stochastic heat equation driven by space-time white noise with u(0,x)=1 for all x∈R.In this paper,we prove the law of the iterated logarithm(LIL for short)and the functional LIL for a linear additive functional of the form∫[0,R]u(t,x)dx and the nonlinear additive functionals of the form∫[0,R]g(u(t,x))dx,where g:R→R is nonrandom and Lipschitz continuous,as R→∞for fixed t>0,using the localization argument.
文摘In this paper, an actuator fault diagnosis scheme based on the backstepping method is proposed for a class of nonlinear heat equations. The fault diagnosis scheme includes fault detection, fault estimation and time to failure (TTF) prediction. Firstly, we achieve fault detection by comparing the detection residual with a predetermined threshold, where the detection residual is defined as the difference between the observer output and the system measurement output. Then, we estimate the fault function through the fault parameter update law and calculate the TTF using only limited measurements. Finally, the numerical simulation is performed on a nonlinear heat equation to verify the effectiveness of the proposed fault diagnosis scheme.
文摘The traditional differential quadrature method was improved by using theupwind difference scheme for the convective terms to solve the coupled two-dimensionalincompressible Navier-stokes equations and heat equation. The new method was compared with theconventional differential quadrature method in the aspects of convergence and accuracy. The resultsshow that the new method is more accurate, and has better convergence than the conventionaldifferential quadrature method for numerically computing the steady-state solution.
基金supported by an NSERC granta startup fund of University of Albertasupported by the NSF grant DMS1613163
文摘In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two differen t met hods, respectively, based on variance compu tations and on pat h-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.
基金The project is supported by National Natural Science Foundation of China (10071026)
文摘The purpose of this paper is to investigate the stability and asymptotic behavior of the time-dependent solutions to a linear parabolic equation with nonlinear boundary condition in relation to their corresponding steady state solutions. Then, the above results are extended to a semilinear parabolic equation with nonlinear boundary condition by analyzing the corresponding eigenvalue problem and using the method of upper and lower solutions.
基金Project supported by the Education Department of Zhejiang Province (Grant No.Y200805137)the Zhejiang Ocean University (Grant No.X08Z04)
文摘This paper deals with blow-up of positive solution of the nonlinear heat equation with absorption subject to a nonlinear boundary condition.The conditions under which the solutions may exist globally or blow-up are obtained by the comparison principles.
基金supported by an NSERC granta startup fund of University of Alberta
文摘This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.
文摘This paper is concerned with some nonlinear heat equations with initial condition and anti-periodic boundary condition. Also some two-point value nonlinear heat equations with anti-periodic boundary condition are discussed. The existence and uniqueness of the solutions are given. Some asymptotic behaviors of the solutions are studied.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61374096 and 61104048)the Natural Science Foundation of Zhejiang Province of China(Grant No.Y6110751)
文摘In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming (SQP) algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.
文摘In this paper,we propose a new numerical method which is a least squares approximaton based on pseudospectral method for the Forward-Backward heat equation. The existence and uniqueness of the solution of the least squares approximation are proved. Error estimates for this approximation are given,which show that tile order of convergence depends only on the regularity of tile solution and the right hand of the Forward-Backward heat equation.
文摘This paper deals with the blow-up properties of solutions to semilinear heat equation ut-uxx= up in (0, 1) × (0, T) with the Neumann boundary condition ux(0, t) = 0, u:x1, t) = 1 on [0, T). The necessary and sufficient conditions under which all solutions to have a finite time blow-up and the exact blow-up rates are established. It is proved that the blow-up will occur only at the boundary x = 1. The asymptotic behavior near the blow-up time is also studied.
基金Project supported by the National Natural Sciences Foundation of China (No.10272021) and the Doctoral Program Foundation of Education Ministry of China (No.20040007022)
文摘Some nonclassical potential symmetry generators and group-invariant solutions of heat equation and wave equation were determined. It is shown that some new explicit solutions of partial differential equations in conserved form can he constructed by using the nonclassical potential symmetry generators which are derived from their adjoint system. These explicit solutions cannot he obtained by using the Lie or Lie-Baeicklund symmetry group generators of differential equations.
基金partially supported by the National Natural Science Foundation of China(11671141)the Natural Science Foundation of Shanghai(17ZR1412800)。
文摘On a complete non-compact gradient shrinking Ricci soliton,we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space variable.This growth condition is sharp.As an application,we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with quadratic exponential growth on shrinkers.
基金Supported by National Science Foundation of China(Grant 10871179)the National Basic Research Programme of China(Grant 2008CB717806)the Department of Education of Zhejiang Province(GrantY200803559).
文摘A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis.