Method for determining eigen numbers in heat conduction problems for a cylinder

Abstract

RELEVANCE. Due to the difficulties of finding eigenvalues and proper functions for bodies with axial (cylinder) and central (ball) symmetries defined in classical methods from the edge Sturm-Liouville problems, including Bessel equations whose exact analytical solutions are not obtained (known only numerical solutions, described by approximation formulas), there is a need to develop analytical methods of their solution.THE PURPOSE. Using orthogonal methods of weighted residuals, an approximate analytical method for determining eigenfunctions and eigenvalues in boundary value problems with axial and central symmetry (cylinder, ball) has been developed.METHODS. The method is based on the use of orthogonal systems of coordinate functions and additional boundary conditions. Latter are in such a form that their fulfillment by the desired solution is equivalent to the fulfillment of the differential equation of the boundary value problem at the boundary points of the region, leading to its fulfillment inside the considered region. Moreover, the accuracy of the equation depends on the number of approximations, which, in turn, depends on the number of additional boundary conditions used. Using the orthogonality property of trigonometric coordinate functions included in a series representing eigenfunctions makes it possible to increase the accuracy of the fulfillment of the differential equation of the Sturm-Liouville boundary value problem and the accuracy of determining the eigenvalues. To satisfy the initial condition, its residual is compiled and the condition of its orthogonality to all coordinate functions is required. The orthogonality of trigonometric systems of coordinate functions with respect to the unknown constants of integration leads to a system of algebraic linear equations, the number of which is equal to the number of approximations. As a result, the fulfillment of the initial condition is simplified and the accuracy of its fulfillment is increased.RESULTS. The advantage of the method is that the resulting solution contains only simple algebraic expressions, excluding special functions (Bessel function, Legendre function, gamma function).CONCLUSION. Thus, bypassing direct integration over a spatial variable, the use of additional boundary conditions makes it possible to find a solution of any complexity of the equations of the Sturm-Liouville boundary value problem, which reduces to the definition of simple integrals.