论文标题
用可变的导热率求解热方程
Solving the heat equation with variable thermal conductivity
论文作者
论文摘要
我们考虑具有空间可变的导热率和均匀的dirichlet边界条件的热方程。使用FOKAS或统一变换方法的方法，我们将解决方案表示作为恒定界面问题的解决方案的极限，在该界面问题中，子域和接口数量无限。这会产生解决方案的明确表示，我们可以从中计算解决方案并确定其属性。使用此溶液表达式，我们可以找到相应的可变特征值问题的特征值作为先验函数的根。我们可以根据特征值明确地写出本征函数。热方程是可以使用此方法解决的更通用可变二阶初始值问题的第一个示例。
We consider the heat equation with spatially variable thermal conductivity and homogeneous Dirichlet boundary conditions. Using the Method of Fokas or Unified Transform Method, we derive solution representations as the limit of solutions of constant-coefficient interface problems where the number of subdomains and interfaces becomes unbounded. This produces an explicit representation of the solution, from which we can compute the solution and determine its properties. Using this solution expression, we can find the eigenvalues of the corresponding variable-coefficient eigenvalue problem as roots of a transcendental function. We can write the eigenfunctions explicitly in terms of the eigenvalues. The heat equation is the first example of more general variable-coefficient second-order initial-boundary value problems that can be solved using this approach.
