Direct numerical algorithm for calculating the heat flux at an inaccessible boundary

Abstract 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.