THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS

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.

AN INTEGRATION BY PARTS FORMULA FOR STOCHASTIC HEAT EQUATIONS WITH FRACTIONAL NOISE

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.

Actuator Fault Diagnosis for a Class of One-Dimensional Nonlinear Heat Equations via Backstepping Method

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.

SOME RECENT PROGRESS ON STOCHASTIC HEAT EQUATIONS

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.

Functional Separable Solutions of Nonlinear Heat Equations in Non-Newtonian Fluids

We study the functional separation of variables to the nonlinear heat equation： ut = （A（x）D（u）ux^n）x＋ B（x）Q（u）, Ax≠0. Such equation arises from non-Newtonian fluids. Its functional separation of variables is studied by using the group foliation method. A classification of the equation which admits the functional separable solutions is performed. As a consequence, some solutions to the resulting equations are obtained.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

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.

THE BLOW-UP PROPERLIES OF SOLUTIONS TO SEMILINEAR HEAT EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS

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.

The cost of approximate controllability for semilinear heat equations

We consider the semilinear heat equation with globally Lipschitz non-linearity involving gradient terms in a bounded domain of R^n. In this paper, we obtain explicit bounds of the cost of approximate controllability, i.e., of the minimal norm of a control needed to control the system approximately. The methods we used combine global Carleman estimates, the variational approach to approximate controllability and Schauder＇s fixed point theorem.

Compact ADI Method for Solving Heat Equations in Multi-dimension

A compact alternating direction implicit(ADI) method has been developed for solving multi-dimensional heat equations by introducing the differential operators and the truncation error is O(τ 2 + h 4 ). It is shown by the discrete Fourier analysis that this new ADI scheme is unconditionally stable and the truncation error O(τ 3 + h 6 ) is gained with once Richardson's extrapolation. Some numerical examples are presented to demonstrate the efficiency and accuracy of the new scheme.

The Blow-up Rate for a System of Heat Equations with Neumann Boundary Conditions

This paper deals with the blow-up properties of solutions to a system of heat equations u_t=Δ_u,v_t=Δv in B_R×(0,T) with the Neumann boundary conditions u/η=e^v,v/η=e^u on S_R×[0,T).The exact blow-up rates are established.It is also proved that the blow-up will occur only on the boundary.

Artificial Boundary Conditions for Nonlocal Heat Equations on Unbounded Domain

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.

Point Sources Identification Problems for Heat Equations

We considered the point source identification problems for heat equations from noisy observation data taken at the minimum number of spatially fixed measurement points.We aim to identify the unknown number of sources and their locations along with their strengths.In our previous work,we proved that minimum measurement points needed under the noise-free setting.In this paper,we extend the proof to cover the noisy cases over a border class of source functions.We show that if the regularization parameter is chosen properly,the problem can be transformed into a poles identification problem.A reconstruction scheme is proposed on the basis of the developed theoretical results.Numerical demonstrations in 2D and 3D conclude the paper.

Large Deviation Principle for the Fourth-order Stochastic Heat Equations with Fractional Noises

In this paper, we shall study a fourth-order stochastic heat equation driven by a fractional noise, which is fractional in time and white in space. We will discuss the existence and uniqueness of the solution to the equation. Furthermore, the regularity of the solution will be obtained. On the other hand, the large deviation principle for the equation with a small perturbation will be established through developing a classical method.

ON CRITICAL EXPONENTS FOR SEMILINEAR HEAT EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

This paper deals with the blow up properties of solutions to semilinear heat equation u t- Δ u=u p in R N +×(0,T) with the nonlinear boundary condition -ο u ο x 1 = u q for x 1=0,t∈(0,T) .It has been proved that if max( p,q) ≤1,every nonnegative solution is global.When min (p,q) >1 by letting α=1p-1 and β=12(q-1) it follows that if max (α,β)≥N2 ,all nontrivial nonnegative solutions are nonglobal,whereas if max (α,β)<N2 ,there exist both global and nonglobal solutions.Moreover,the exact blow up rates are established.

Source-type solutions of heat equations with convection in several variables space

In this paper we study the source-type solution for the heat equation with convection： ut = △u ＋ b·▽un for （x,t） ∈ ST→ RN × （0,T] and u（x,0） = δ（x） for x ∈ RN, where δ（x） denotes Dirac measure in = RN,N 2,n 0 and b = （b1,...,bN） ∈ RN is a vector. It is shown that there exists a critical number pc = N＋2 such that the source-type solution to the above problem exists and is unique if 0 N n 〈 pc and there exists a unique similarity source-type solution in the case n = N＋1 , while such a solution does not exist N if n 〉 pc. Moreover, the asymptotic behavior of the solution near the origin is studied. It is shown that when 0 〈 n 〈 N＋1 the convection is too weak and the short time behavior of the source-type solution near the origin N is the same as that for the heat equation without convection.

Quenching rates for heat equations with coupled singular nonlinear boundary flux

We study finite time quenching for heat equations coupled via singular nonlinear boundary flux. A criterion is proposed to identify the simultaneous and non-simultaneous quenchings. In particular, three kinds of simultaneous quenching rates are obtained for different nonlinear exponent regions and appropriate initial data. This extends an original work by Pablo, Quirós and Rossi for a heat system with coupled inner absorption terms subject to homogeneous Neumann boundary conditions.

UPWIND LOCAL DIFFERENTIAL QUADRATURE METHOD FOR SOLVING COUPLED VISCOUS FLOW AND HEAT TRANSFER EQUATIONS

The differential quadrature method (DQM) has been applied successfully to solve numerically many problems in the fluid mechanics. But it is only limited to the flow problems in regular regions. At the same time, here is no upwind mechanism to deal with the convective property of the fluid flow in traditional DQ method. A local differential quadrature method owning upwind mechanism (ULDQM) was given to solve the coupled problem of incompressible viscous flow and heat transfer in an irregular region. For the problem of flow past a contraction channel whose boundary does not parallel to coordinate direction, the satisfactory numerical solutions were obtained by using ULDQM with a few grid points. The numerical results show that the ULDQM possesses advantages including well convergence, less computational workload and storage as compared with the low-order finite difference method.

A NONLOCAL NONLINEAR BOUNDARYVALUE PROBLEM FOR THE HEAT EQUATIONS

The existence and limit behaviour of the solution for a kind of nonlocal nonlinear boundaryvalue condition on a part of the boundary is studied for the heat equation, which physicallymeans that the potential is the function of the total flux. When this part of boundary shrinksto a point in a certain wayt this condition either results in a Dirac measure or simply disappearsin the corresponding problem.

Optimal Control of Non-well-posed Heat Equations

This work is concerned with Pontryagin＇s maximum principle of optimal control problems governed by some non-well-posed semilinear heat equations. A type of approach to the non-well-posed optimal control problem is given.

Null Controllability for Some Systems of Two Backward Stochastic Heat Equations with One Control Force

Abstract The authors establish the null controllability for some systems coupled by two backward stochastic heat equations. The desired controllability result is obtained by means of proving a suitable observability estimate for the dual system of the controlled system.