In the paper, we point out a drawback of the Fourier sine series method to represent a given odd function, where the boundary Gibbs phenomena would occur when the boundary values of the function are non-zero. We modify the Fourier sine series method by considering the consistent conditions on the boundaries, which can improve the accuracy near the boundaries. The modifications are extended to the Fourier cosine series and the Fourier series. Then, novel boundary consistent methods are developed to solve the 1D and 2D heat equations. Numerical examples confirm the accuracy of the boundary consistent methods, accounting for the non-zeros of the source terms and considering the consistency of heat equations on the boundaries, which can not only overcome the near boundary errors but also improve the accuracy of solution about four orders in the entire domain, upon comparing to the conventional Fourier sine series method and Duhamel’s principle.

中文翻译：

---

用边界一致方法减少非均匀热方程的近边界误差

在本文中，我们指出了用傅立叶正弦级数方法表示给定奇数函数的缺点，其中当函数的边界值非零时会出现边界吉布斯现象。我们通过考虑边界上的一致条件来修改傅里叶正弦级数方法，这可以提高边界附近的精度。修改扩展到傅立叶余弦级数和傅立叶级数。然后，开发了新颖的边界一致方法来求解一维和二维热方程。数值示例证实了边界一致性方法的准确性，考虑了源项的非零值并考虑了边界上热方程的一致性，