Application of the Maximum Principle for Differential Equations in Combination with the Finite Difference Method to Find Transient Approximate Solutions of Heat Equations and Error Analysis

摘要

This article tries to combine the maximum principle for differential equations with the finite-difference method that has been widely used to find upper and lower solutions of the exact solutions of transient heat conduction problems. As verified by some examples, the technique proposed in this article is successful in obtaining lower and upper approximate solutions of the exact solution correctly by using the finite-difference method, which originally does not apply to the maximum principle for differential equations. The lower and upper approximate solutions obtained with such a method not only can indicate the range where the exact solutions exist, but also can be used to further analyze the error between mean approximate solutions and unknown exact solutions under the most unfavorable condition.