Iterative Solutions of the Dirichlet Problem for $\Delta u = u^2 $
Iterative Solutions of the Dirichlet Problem for $\Delta u = u^2 $
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$Delta u = u^2 $ 狄利克雷问题的迭代解
DOI：
10.1137/0107038
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发表时间：
1959
期刊：
影响因子：
--
通讯作者：
C. L. Perry
中科院分区：
文献类型：
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作者：
C. M. Ablow;C. L. Perry
(1)Au bu在给定区域中，区域边界上的u值已知。与潜在的化学问题相一致，给定u的边界值和常数裸非负as是所需的解。利用次调和函数的极大值原理证明了非负解的唯一性。通过讨论迭代法的收敛性证明了解的存在性，并给出了三个产生解的迭代过程。第一个是Kntorovich [8]和Zagadskii [10]的Newton方法的推广在本方程中的应用。它的特别优点是对比目前更复杂的问题有明显的适应性。
(1) Au bu in given region, the value of u on the boundary of the region being known. In consonance with the underlying chemical problem, the given boundary vlues of u and the constant bare nonnegative as is the desired solution. The uniqueness of the nonnegative solution is established through use of the maximum principle for subharmonic functions. The existence of the solution is demonstrated by the convergence of the iterative methods dis-cussed.Three itemtive processes yielding the solution are presented. The first is n pplication to the present equation of generalization of Newton’s method due to Kntorovich [8] and Zagadskii [10]. It hs the particular merit of being clearly upplicble to more complex problems thn the present one.