带非局部边界条件的热方程的LEGENDRE配置法

对带非局部边界条件的热方程的初边值问题提出了LEGENDRE配置法,并给出其半离散逼近和全离散逼近的稳定性和收敛性分析.数值试验验证了方法的有效性.     

Unique solvability of the inverse problem for the heat equation with nonlocal boundary conditions

We consider the inverse problem of source identification for the heat conduction problem. The neoclassical formulation of the direct problem with integral boundary condition is used. Conditions for unique solvability of the inverse problem are obtained. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 36–50, 2006.     

On the determination of unknown source in the heat equation with nonlocal boundary conditions

We consider the inverse problem for the heat equation with unknown source. The existence and uniqueness conditions are established in the case where the boundary and overdetermination conditions are nonlocal conditions of general form.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1438–1441, October, 1995.     

The heat equation with stochastic non-linear boundary conditions

In this paper we examine the problem of the heat equation with non-linear boundary conditions of stochastic type.     

The heat equation with nonlinear generalized Robin boundary conditions

Let Ω⊂RN be a bounded domain and let μ be an admissible measure on ∂Ω. We show in the first part that if Ω has the H¹-extension property, then a realization of the Laplace operator with generalized nonlinear Robin boundary conditions, formally given by on ∂Ω, generates a strongly continuous nonlinear submarkovian semigroup SB=(SB(t))_t?0 on L²(Ω). We also obtain that this semigroup is ultracontractive in the sense that for every u,v∈Lp(Ω), p?2 and every t>0, one has

     

Inverse problems for the heat-conduction equation with nonlocal boundary conditions

Conditions under which the time-dependent temperature conductivity coefficient is determined uniquely are established in the case where the boundary conditions and the overdetermination conditions are non local.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1066–1071, August, 1993.     

A nonlocal nonlinear diffusion equation with blowing up boundary conditions

We deal with boundary value problems (prescribing Dirichlet or Neumann boundary conditions) for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation. First, we prove existence, uniqueness and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

A nonlocal p-Laplacian evolution equation with Neumann boundary conditions

In this paper we study the nonlocal p-Laplacian type diffusion equation,

If p>1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation u_t=div(|u|^p−2u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L^∞(0,T;L^p(Ω)) to the solution of the p-Laplacian with homogeneous Neumann boundary conditions. The extreme case p=1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behavior of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition.     

An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions

This paper investigates the inverse problem of finding a time-dependent coefficient in a heat equation with nonlocal boundary and integral overdetermination conditions. Under some regularity and consistency conditions on the input data, the existence, uniqueness and continuous dependence upon the data of the solution are shown by using the generalized Fourier method.     

Bounded perturbations of two-dimensional diffusion processes with nonlocal conditions near the boundary

We study the existence of Feller semigroups arising in the theory of multidimensional diffusion processes. We study bounded perturbations of elliptic operators with boundary conditions containing an integral over the closure of the domain with respect to a nonnegative Borel measure without assuming that the measure is small. We state sufficient conditions on the measure guaranteeing that the corresponding nonlocal operator is the generator of a Feller semigroup.     

Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition

In this paper, we consider a semilinear heat equation ut=Δu+c(x,t)up for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition and nonnegative initial data where p>0 and l>0. We prove global existence theorem for max(p,l)?1. Some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given.     

Solution of Poisson's equation with global,local and nonlocal boundary conditions

Boundary value problems (BVPs) for partial differential equations are common in mathematical physics. The differential equation is often considered in simple and symmetric regions, such as a circle, cube, cylinder, etc., with global and separable boundary conditions. In this paper and other works of the authors, a general method is used for the investigation of BVPs which is more powerful than existing methods, so that BVPs investigated by the method can be considered in anti-symmetric and arbitrary regions surrounded by smooth curves and surfaces. Moreover boundary conditions can be local, non-local and global. The BVP is expanded in a convex and bounded region D in a plane. First, by generalized solution of the adjoint of the Poisson equation, the necessary boundary conditions are obtained. The BVP is then reduced to the second kind of Fredholm integral equation with regularized singularities.     

Mixed problems for the string vibration equation with homogeneous boundary conditions and inhomogeneous nonlocal conditions

We consider four mixed problems for the string vibration equation with homogeneous boundary and inhomogeneous nonlocal conditions of the first or second kind and with zero initial conditions. By using recursion relations, we find generalized solutions of the abovementioned problems.     

Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition

In this paper we consider a semilinear parabolic equation u_t=Δu−c(x,t)u^p for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition u∣_{∂Ω×(0,∞)}=∫_Ωk(x,y,t)u^ldy and nonnegative initial data where p>0 and l>0. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given.