SPECIAL HYPERBOLIC TYPE APPROXIMATION AND FOURIER METHOD FOR SOLVING 3-D STATIONARY DIFFUSION PROBLEM (ENVIRONMENT. TECHNOLOGY. RESOURCES. Proceedings of the International Scientific and Practical Conference, 2025)
In this paper, we explore the conservative averaging methods (CAM) used to solve three-dimensional boundary-value problems of second order. We investigate various interpretations of CAM to address these problems. We consider a special type of hyperbolic approximation - spline interpolation, that estimates middle integral values of piecewise smooth functions. By employing these splines, along with parabolic-type splines, we can reduce multidimensional mathematical physics problems in three dimensions to problems involving two dimensions for one coordinate. This approach also enables us to further simplify the two-dimensional problems into one-dimensional problems, where analytical solutions can be obtained. Additionally, we numerically solve the corresponding problem with homogeneous boundary conditions of the first kind using the Fourier series method, and we compare these numerical results with the analytical solutions.