m维各向异性热传导方程的奇异内边界问题研究

Study of Singular Internal Boundary Problems for m Dimensional Anisotropic Heat Conduction Equations

摘要:

本文首先建立m维无穷区域 中各向异性的热传导方程的数学模型I；寻求向量函数 使所求问题的解函数w(x,t) 在任意时刻t∈(0,T) 在该向量函数上取区域R^m中的正的最大值，即 称向量函数x=x(t) ,0<tw(x,t);x(t)} ，使其满足 (I)

Abstract:

其中：δ(x-x(t))为m维狄拉克δ-函数；热传导方程的系数矩阵A=(a_kj) _m×m为m阶实对称非负矩阵。本文应用矩阵理论和广义特征函数法，在条件A为m阶实对称正定矩阵，A=BB^T ( B∈R^m×m为正线下三角矩阵)下，获得了数学模型I的充分光滑的精确解{w(x,t);x(t)} ，其中奇异内边界的表达式是 。同时获得了m维各向异性齐次热传导方程的自由边界问题IIa和问题IIb的充分光滑的精确解，且两者的自由边界都是相同的m维向量函数表达式 。

In this paper, first of all, the mathematical model I is established on anisotropic heat conduction equation in m dimension infinite domain. Mathematical model I: seek {w(x,t);x(t)} , make it satisfy to

The free term of the equation is Υ(t)δ(x-x(t)) , among them, δ(x-x(t)) is m dimensional Diracfunction, Υ(t) is strength function of singular source. The exact solution {w(x,t);x(t)} of the mathematical model I is obtained by the matrix theory and the generalized eigenfunction method, under the condition that the matrix A=(a_kj) _m×m is real symmetric positive definite matrix and A=BB^T . The matrix B is a lower triangular matrix, whose main diagonal element is positive. The singular internal boundary is demonstrated as . We establish the free boundary problem IIa and problem IIb on homogeneous heat conduction equation. The problem IIa is free boundary problem in region E_(t). The problem IIb is free boundary problem in region E₊(t) . It is also solves these two questions of problem IIa and IIb. These two free boundaries are the same .