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In this article, the one-dimensional parabolic equation with three types of integral nonlocal boundary conditions is approximated by the implicit Euler finite difference scheme. Stability analysis is done in the maximum norm and it is proved that the radius of the stability region and the stiffness of the discrete scheme depends on the signs of coefficients in the nonlocal boundary condition. The known stability results are improved. In the case of a plain integral boundary condition, the conditional convergence rate is proved and the regularization relation between discrete time and space steps is proposed. The accuracy of the obtained estimates is illustrated by results of numerical experiments. 相似文献
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Matthieu Felsinger 《偏微分方程通讯》2013,38(9):1539-1573
Weak solutions to parabolic integro-differential operators of order α ∈ (α0, 2) are studied. Local a priori estimates of Hölder norms and a weak Harnack inequality are proved. These results are robust with respect to α↗2. In this sense, the presentation is an extension of Moser's result from [20]. 相似文献
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陈玉娟 《数学物理学报(A辑)》2006,26(5):731-740
该文采用弱上下解方法以及正则化的技巧,研究了一类非局部的退化的抛物型方程组的解的爆破和整体存在性,给出了方程组的解的爆破指标pc=(p1+p2)(q1+q2)-mn,证得当pc<0时,对任意的初值,方程组的解整体存在;当pc>0时,对充分大的初值,解在有限时刻爆破,对充分小的初值,解整体存在;当pc=0时,若区域充分小,则方程组存在非负整体解,若区域包含了一个充分大的球, 则解在有限时刻爆破. 相似文献
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具有非局部源的退化半线性抛物型方程组解的爆破 总被引:4,自引:0,他引:4
本文讨论具有非局部源退化半线性抛物型方程组的初边值问题 .证明了局部解的存在唯一性并且得到当初值充分大时解在有限时刻爆破 . 相似文献
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陈玉娟 《数学物理学报(A辑)》2010,30(2):386-396
该文采用弱上下解方法和正则化技巧,研究了一类非局部退化抛物型方程组解的爆破和整体存在性,给出了爆破指标,并对非退化情形m=n=1,p_1=q_1=0,p_2q_21给出了一致爆破速率. 相似文献
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带非局部源的退化奇异半线性抛物方程的爆破 总被引:7,自引:0,他引:7
本文研究带齐次Dirichlet边界条件的非局部退化奇异半线性抛物方程ut-(xαux)x=∫0af(u)dx在(0,a)×(0,T)内正解的爆破性质,建立了古典解的局部存在性与唯一性.在适当的假设条件下,得到了正解的整体存在性与有限时刻爆破的结论.本文还证明了爆破点集是整个区域,这与局部源情形不同.进而,对于特殊情形:f(u)=up,p>1及,f(u)=eu,精确地确定了爆破的速率. 相似文献
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Zheng-Su Wan & Zhi-Zhong Sun 《计算数学(英文版)》2001,19(5):449-458
In paper [4] (J. Comput. Appl. Math.,76 (1996), 137-146), a difference scheme for a class of nonlocal parabolic equations with natural boundary conditions was derived by the method of reduction of order and the unique solvability and second order convergence in $L_2$-norm are proved. In this paper, we prove that the scheme is second order convergent in $L_\infty$ norm and then obtain fourth order accuracy approximation in $L_\infty$ norm by extrapolation method. At last, one numerical example is presented. 相似文献
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In this paper we start to develop the regularity theory of general two-phase free boundary problems for parabolic equations. In particular we consider uniformly parabolic operators in nondivergence form and we are mainly concerned with the optimal regularity of the viscosity solutions. We prove that under suitable nondegenerate conditions the solution is Lipschitz across the free boundary. 相似文献
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本文在Hackbush于1979年提出的对时间周期抛物型问题的快速解法的基础上,结合外推算法,得到了一种快速高精度算法——MGFE算法.并从理论上证明了这一算法的收敛性. 相似文献
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主要讨论具有非局部源与非局部边界条件的退化抛物型方程组,借助于上解与下解的技术,给出了该系统整体解的存在与有限时刻爆破的条件.此结果不仅扩充了已有的结论~([8]),而且表明,系数a,b和边界条件中的权重函数g_1(x,y),g_2(x,y),以及常数l_1,l_2在决定系统解的爆破与否中起着关键的作用. 相似文献
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In this article, we study the convergence analysis for the initial and boundary value problem of parabolic equations on a disk with singular solutions. It is assumed that the exact solution performs singular properties that its derivatives go to infinity at the boundary of the disk. We propose a fully implicit time-stepping numerical scheme. A stretching polynomial-like function with a parameter is used to construct a local grid refinement. Over the nonuniform partition, we combine the Swartztrauber-Sweet scheme and the backward Euler method in spatial and temporal discretization, respectively. We carry out convergence analysis and analyze the effects of the parameter. It is shown that our numerical scheme is of first order accuracy for temporal discretization and of almost second order accuracy for spatial discretization. Numerical experiments are performed to illustrate our analysis results and show that there exists an optimal value for the parameter to obtain a best approximate solution. 相似文献
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Nikolai Yu. Bakaev 《BIT Numerical Mathematics》2000,40(1):1-23
We examine single step time discrete approximations to an abstract Cauchy problem considered in a pair of Banach spaces, of which one (space of solutions) is densely embedded in the other (space of initial data). The stability and error analysis of such discretizations is carried out. Our approach is applicable to the analysis of approximations to parabolic PDE problems in various pairs of function spaces arising from practical needs. In the final part of the paper, we present a possible application of our results for studying a semi-discrete version of a model initial-boundary value problem of heat conduction. 相似文献
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A. Tudorascu 《偏微分方程通讯》2013,38(8):1426-1454
In this paper we prove that the steepest descent of certain porous-medium type functionals with respect to the quadratic Wasserstein distance over a constrained (but not weakly closed) manifold gives rise to a nonlinear, nonlocal parabolic partial differential equation connected to the study of the asymptotic behavior of solutions for filtration problems. The result by Carlen and Gangbo on constrained optimization for steepest descent of the negative Boltzmann entropy in the Wasserstein space is generalized to porous-medium type functionals. An interesting feature of the resulting Fokker-Planck equation is the nonlocality of its drift term occurring at the same time as its nonlinearity. 相似文献
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Tianliang Hou & Yanping Chen 《计算数学(英文版)》2015,33(2):158-178
In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time discretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We derive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori $L^2(0, T ;L^2(Ω))$ error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates. 相似文献