In the paper, we solve a non-homogeneous heat conduction equation with non-homogeneous boundary conditions in a 2D rectangle. First, we derive the domain/boundary integral equations for both the forward and backward heat conduction problems. Then, by using the technique of homogenization, inserting the adjoint Trefftz test functions into the derived integral equations and expanding the solutions in terms of eigenfunctions, we can obtain the expansion coefficients in closed form. Hence, the analytic series solutions of forward heat conduction problems (FHCPs) and backward heat conduction problems (BHCPs) are available. For the FHCPs, only a few terms in the series render very high-order accurate solutions at any time, with errors of the order . For the BHCPs, we require to modify the closed-form series solutions via a new spring-damping regularization technique. Numerical tests for the BHCPs in a large space-time domain reveal that the present analytic series solution is very accurate to recover the initial temperature with an error of the order , although the measured final time temperature is very small when is large up to 100 and is even polluted by a large relative noise up to the level .

中文翻译：

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矩形中二维前向和后向热传导问题的解析级数解和一种新的正则化

在本文中，我们求解了二维矩形中具有非齐次边界条件的非齐次热传导方程。首先，我们推导出正向和反向热传导问题的域/边界积分方程。然后，利用均质化技术，将伴随的 Trefftz 检验函数插入到导出的积分方程中，并根据本征函数展开解，我们可以得到封闭形式的展开系数。因此，前向热传导问题（FHCPs）和后向热传导问题（BHCPs）的解析级数解是可用的。对于 FHCP，该系列中只有少数项在任何时候都呈现非常高阶的精确解，阶次误差为 。对于 BHCP，我们需要通过一种新的弹簧阻尼正则化技术来修改封闭形式的级数解。BHCPs在大时空域的数值试验表明，目前的解析级数解非常准确地恢复了初始温度，误差为 ，尽管测量的最终时间温度非常小，当大到 100 和甚至被一个大的相对噪音污染到了这个水平。