Solving an Unsteady-State and Non-Uniform Heat Conduction Transfer Problem Using Discrete-Analytical Method

摘要

The paper presents one of the methods to determine heat spread patterns in objects. Mathematical model of the process is a differential equation of the second order with initial and boundary conditions, which can be solved by only one function U(x, y, z, t). In this paper the problem of unsteady-state and non-uniform heat conduction transfer for 2 dimensions, with imposed initial and boundary conditions of the first, second and third kind, is solving using discrete-analytical method. The main idea of this method is to combine discrete and analytical method. In this case, initial problem is divided to 2 stages: in the first stage a discrete technique along ones directions will be applied; in the second stage an analytical method along other directions will be applied. The result will be a discrete set of analytical functions. For "discrete stage" is used a well-known method of finite differences, and for analytical stage is applied the virtue of the matrix exponent. In the general case, the problem can be submitted in operator form with non-orthogonal quadrangular mesh which is topologically equivalent to square mesh.