A fast numerical algorithm for solving the Cauchy problem for elliptic equations with variable coefficients in standard calculation domains (rectangles, circles, or rings) is proposed. The algorithm is designed to calculate the heat flux at the inaccessible boundary. It is based on the separation of variables method. This approach employs a finite difference approximation and allows obtaining a solution to a discrete problem in arithmetic operations of the order of N ⁢ ln ⁡ N N\operatorname{ln}N , where 𝑁 is the number of grid points. As a rule, iterative procedures are needed to solve the Cauchy problem for elliptic equations. The currently available direct algorithms for solving the Cauchy problem have been developed only for (Laplace, Helmholtz) operators with constant coefficients and for use of analytical solutions for problems with such operators. A novel feature of the results of the present paper is that the direct algorithm can be used for an elliptic operator with variable coefficients (of a special form). It is important that in this case no analytical solution to the problem can be obtained. The algorithm significantly increases the range of problems that can be solved. It can be used to create devices for determining in real time heat fluxes on the parts of inhomogeneous constructions that cannot be measured. For example, to determine the heat flux on the inner radius of a pipe made of different materials.

中文翻译：

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用于计算不可到达边界处的热通量的直接数值算法

提出了一种快速数值算法，用于求解标准计算域（矩形、圆形或环）中变系数椭圆方程的柯西问题。该算法旨在计算不可到达边界处的热通量。它基于变量分离方法。该方法采用有限差分近似，并允许在算术运算中获得离散问题的解 氮 ⁢ 因 ⁡ 氮 N\操作符名称{ln}N ，其中 𝑁 是网格点数。通常，需要迭代过程来求解椭圆方程的柯西问题。目前可用的用于解决柯西问题的直接算法仅针对具有常数系数的（拉普拉斯、亥姆霍兹）算子而开发，并且用于使用此类算子的问题的解析解。本文结果的一个新颖特征是直接算法可以用于具有可变系数（特殊形式）的椭圆算子。重要的是，在这种情况下无法获得问题的解析解。该算法显着增加了可以解决的问题范围。它可用于创建用于实时确定无法测量的不均匀结构部件上的热通量的设备。例如，确定不同材料制成的管道内半径上的热通量。