A spectral method for solving the sideways heat equation
A spectral method for solving the sideways heat equation
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DOI：
10.1088/0266-5611/15/4/305
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发表时间：
1999
期刊：
影响因子：
2.1
通讯作者：
Fredrik Berntsson
中科院分区：
文献类型：
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作者：
Fredrik Berntsson
我们考虑一个逆热传导问题，侧热方程，这是一个问题的模型，其中一个要确定的温度在一个身体的表面上，使用内部测量。在数学上，它可以表述为热方程的柯西问题，其中数据沿沿着x = 1给出，并且在0 x<1的区间内寻求解。这个问题是不适定的，在这个意义上，解决方案并不连续依赖于数据。连续依赖的数据被恢复取代的时间导数的热方程与有界的基于光谱的近似。谱近似中的截止能级作为正则化参数。正则化的解决方案的误差估计，推导出一个程序，选择一个适当的正则化参数。离散化问题是空间变量中的常微分方程的初始值问题，其可以使用标准数值方法（例如，龙格-库塔方法）来求解。作为测试问题，我们采取方程的常数和可变系数。
We consider an inverse heat conduction problem, the sideways heat equation, which is the model of a problem where one wants to determine the temperature on the surface of a body, using interior measurements. Mathematically it can be formulated as a Cauchy problem for the heat equation, where the data are given along the line x = 1, and a solution is sought in the interval 0 x<1. The problem is ill-posed, in the sense that the solution does not depend continuously on the data. Continuous dependence of the data is restored by replacing the time derivative in the heat equation with a bounded spectral-based approximation. The cut-off level in the spectral approximation acts as a regularization parameter. Error estimates for the regularized solution are derived and a procedure for selecting an appropriate regularization parameter is given. The discretized problem is an initial value problem for an ordinary differential equation in the space variable, which can be solved using standard numerical methods, for example a Runge-Kutta method. As test problems we take equations with constant and variable coefficients.