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Solving the 3D Cauchy problems of nonlinear elliptic equations by the superposition of a family of 3D homogenization functions

机译:通过叠加3D均质函数叠加的非线性椭圆方程的3D Cauchy问题

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摘要

We solve the 3D Cauchy problem of a nonlinear elliptic equation in a cuboid, using the derived family of 3D homogenization functions of different orders. When the solution is expressed by the weight superposition of a family of 3D homogenization functions, the unknown boundary data and the solution can be recovered quickly by solving a small scale linear system. It deserves to note that the superposition of homogenization functions method (SHFM) does not need to solve nonlinear equations and regularization, and is quite accurate to find the solution in the whole domain with the errors smaller than the level of noise being imposed on the overspecified Neumann data. Another advantage of the SHFM is that it can solve the Cauchy problem in a large size of the cuboid.
机译:通过不同订单的衍生3D均质函数的衍生3D均质函数,解决了长方体中非线性椭圆方程的3D Cauchy问题。当通过求解小刻度线性系统,通过叠加3D均匀化函数的重量叠加来表示溶液的重量叠加,可以快速地恢复未知的边界数据和解决方案。值得注意的是,均匀化函数方法(SHFM)的叠加不需要解决非线性方程和正规化,并且非常准确地找到整个域中的解决方案,错误的误差小于噪声的噪声级别上的噪音neumann数据。 SHFM的另一个优点是它可以在大尺寸的长方体中解决Cauchy问题。

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