Roles of weight functions in a nonlinear nonlocal parabolic system

This paper deals with a semilinear parabolic system with coupled nonlinear nonlocal sources subject to weighted nonlocal Dirichlet boundary conditions. We establish the conditions for global and non-global solutions. It is interesting to observe that the weight functions for the nonlocal Dirichlet boundary conditions play substantial roles in determining not only whether the solutions are global or non-global, but also whether (for the non-global solutions) the blowing up occurs for any positive initial data or just for large ones.     

Solvability of the mixed initial boundary-value problem for a parabolic equation with a nonlocal condition of first kind

The spectral method is applied to solve the mixed initial boundary-value problem for a parabolic equation with nonhomogeneous boundary conditions, one of which is nonlocal. We prove existence and uniqueness of the generalized solution of this problem in the Sobolev class W ₂ ^1,0 and represent it as a biorthogonal series. We also consider optimal control by the right-hand side of the equation, which is constructed as a biorthogonal series in the root functions of the spectral problem.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 209–220, 2002.     

On a mixed problem with integral conditions for one-dimensional parabolic equation

We consider an initial-boundary value problem for a one-dimensional parabolic equation with nonlocal boundary conditions. These nonlocal conditions are given in terms of integrals. Based on solution of the Dirichlet problem for the parabolic equation, we constructively establish the well-posedness for the nonlocal problem.     

The use of cubic B‐spline scaling functions for solving the one‐dimensional hyperbolic equation with a nonlocal conservation condition

Problems for parabolic partial differential equations with nonlocal boundary conditions have been studied in many articles, but boundary value problems for hyperbolic partial differential equations have so far remained nearly uninvestigated. In this article a numerical technique is presented for the solution of a nonclassical problem for the one‐dimensional wave equation. This method uses the cubic B‐spline scaling functions. Some numerical results are reported to support our study. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass

Some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications. In this paper, a numerical method which employs the Bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions. The properties of Bernstein polynomials, and the operational matrices for integration, differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

Results on Nonlocal Boundary Value Problems

In this article, we provide a variational theory for nonlocal problems where nonlocality arises due to the interaction in a given horizon. With this theory, we prove well-posedness results for the weak formulation of nonlocal boundary value problems with Dirichlet, Neumann, and mixed boundary conditions for a class of kernel functions. The motivating application for nonlocal boundary value problems is the scalar stationary peridynamics equation of motion. The well-posedness results support practical kernel functions used in the peridynamics setting.

We also prove a spectral equivalence estimate which leads to a mesh size independent upper bound for the condition number of an underlying discretized operator. This is a fundamental conditioning result that would guide preconditioner construction for nonlocal problems. The estimate is a consequence of a nonlocal Poincaré-type inequality that reveals a horizon size quantification. We provide an example that establishes the sharpness of the upper bound in the spectral equivalence.     

Single and multi‐interval Legendre spectral methods in time for parabolic equations

In this article, we take the parabolic equation with Dirichlet boundary conditions as a model to present the Legendre spectral methods both in spatial and in time. Error analysis for the single/multi‐interval schemes in time is given. For the single interval spectral method in time, we obtain a better error estimate in L²‐norm. For the multi‐interval spectral method in time, we obtain the L²‐optimal error estimate in spatial. By choosing approximate trial and test functions, the methods result in algebraic systems with sparse forms. A parallel algorithm is constructed for the multi‐interval scheme in time. Numerical results show the efficiency of the methods. The methods are also applied to parabolic equations with Neumann boundary conditions, Robin boundary conditions and some nonlinear PDEs. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A spectral method for parabolic differential equations

We present a spectral method for parabolic partial differential equations with zero Dirichlet boundary conditions. The region Ω for the problem is assumed to be simply-connected and bounded, and its boundary is assumed to be a smooth surface. An error analysis is given, showing that spectral convergence is obtained for sufficiently smooth solution functions. Numerical examples are given in both ?² and ?³.     

具有非局部边界的退化抛物方程组的爆破解

本文研究了具有非局部边界条件和非局部源的退化抛物方程组的弱解问题.利用基于比较原理的上下解的方法,在权函数和初始条件的假设下,获得了该方程组问题的爆破临界指标.此外,还获得了同时爆破解趋于爆破时间时的渐近行为,推广了已有的结果.     

非局部初值问题解的存在性定理

近来非局部问题的研究日见增多,但涉及带非线性边界条件的初值问题文献 较少.本文目的在于证明一个半线性方程的齐次边值问题和一个非线性边界条件问题 解的存在性.主要使用半群,分数次幂函数空间,广义Poincare算子等工具.     

On the uniform convergence of spectral expansions for a spectral problem with boundary conditions of the third kind one of which contains the spectral parameter

We study the uniform convergence, on a closed interval, of spectral expansions of Hölder functions in a given complete and minimal system of eigenfunctions corresponding to a spectral problem with spectral parameter in a boundary condition. We consider boundary conditions of the third kind and subject the function to be expanded to a condition of nonlocal type ensuring the uniform convergence. We prove a theorem stating that expansions in the entire system of eigenfunctions of the problem are possible without any additional conditions.     

Global boundedness and blow-up for a parabolic system with positive Dirichlet boundary value

This paper deals with a parabolic system coupled via nonlocal sources, subjecting to positive Dirichlet boundary value conditions. By using the super-, sub-solution methods and techniques, and piecewise functions, the blow-up criteria and global boundedness of nonnegative solutions are determined. The results show the positive boundary value ε ₀ plays an important role in the case of blow-up.     

非局部初边值条件下的抛物型偏微分方程

本文讨论在非局部初边值条件下的抛物型偏微分方程,在更为宽松的边界假设条件下讨论所构造的迭代序列的收敛速度问题.并且对非局部初值条件为离散形式的情况做了相应的讨论.     

Boundary value problem for second-order differential operators with integral conditions

In this article, we study a second-order differential operator with mixed nonlocal boundary conditions combined weighting integral boundary condition with another two-point boundary condition. Under certain conditions on the weighting functions and on the coefficients in the boundary conditions, called regular and nonregular cases, we prove that the resolvent decreases with respect to the spectral parameter in L ^p ?(0,?1), but there is no maximal decreasing at infinity for p?>?1. Furthermore, the studied operator generates in L ^p ?(0,?1) an analytic semigroup for p?=?1 in regular case, and an analytic semigroup with singularities for p?>?1 in both cases, and for p?=?1 in the nonregular case only. The obtained results are then used to show the correct solvability of a mixed problem for a parabolic partial differential equation with nonregular boundary conditions.     

具有加权非局部边界条件的多孔介质方程组的临界指标问题

本文研究了具有非局部边界条件和非线性内部源的多孔介质抛物型方程组问题。利用比较原理,获得了权函数和系数对整体解和爆破解的影响,并得到了解的爆破临界指标,推广了先前的研究结果。     

Green's functions for elliptic and parabolic systems with Robin-type boundary conditions

The aim of this paper is to investigate Green's function for parabolic and elliptic systems satisfying a possibly nonlocal Robin-type boundary condition. We construct Green's function for parabolic systems with time-dependent coefficients satisfying a possibly nonlocal Robin-type boundary condition assuming that weak solutions of the system are locally Hölder continuous in the interior of the domain, and as a corollary we construct Green's function for elliptic system with a Robin-type condition. Also, we obtain Gaussian bound for Robin Green's function under an additional assumption that weak solutions of Robin problem are locally bounded up to the boundary. We provide some examples satisfying such a local boundedness property, and thus have Gaussian bounds for their Green's functions.     

A degenerate parabolic system with localized sources and nonlocal boundary condition

This paper deals with the blow-up properties of the positive solutions to a degenerate parabolic system with localized sources and nonlocal boundary conditions. We investigate the influence of the reaction terms, the weight functions, local terms and localized source on the blow-up properties. We will show that the weight functions play the substantial roles in determining whether the solutions will blow-up or not, and obtain the blow-up conditions and its blow-up rate estimate.     

Blow-up of solutions to a parabolic system with nonlocal source

This paper deals with a parabolic system with different diffusion coefficients and coupled nonlocal sources, subject to homogeneous Dirichlet boundary conditions. The conditions on global existence, simultaneous or non-simultaneous blow-up, blow-up set, uniform blow-up profiles and boundary layer are got using comparison principle and asymptotic analysis methods.     

GLOBAL BLOW-UP FOR A DEGENERATE AND SINGULAR NONLOCAL PARABOLIC EQUATION WITH WEIGHTED NONLOCAL BOUNDARY CONDITIONS

This paper deals with the blow-up properties of positive solutions to a degenerate and singular nonlocal parabolic equation with weighted nonlocal boundary conditions.Under appropriate hypotheses, the global existence and finite time blow-up of positive solutions are obtained. Furthermore, by using the properties of Green's function, we find that the blow-up set of the blow-up solution is the whole domain(0, a), and this differs from parabolic equations with local sources case.     

Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditions

This paper investigates the inverse problem of finding a time-dependent heat source in a parabolic equation with nonlocal boundary and integral overdetermination conditions. Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown by using the generalized Fourier method. Numerical tests using the Crank-Nicolson finite difference scheme combined with an iterative method are presented and discussed.