A new transform method I: domain-dependent fundamental solutions and integral representations
A new transform method I: domain-dependent fundamental solutions and integral representations
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DOI：
10.1098/rspa.2009.0512
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发表时间：
2010-08
期刊：
影响因子：
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通讯作者：
Euan A. Spence;A. Fokas
中科院分区：
文献类型：
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作者：
Euan A. Spence;A. Fokas
20世纪90年代末，作者提出了求解线性和某些非线性偏微分方程边值问题的一种新方法。对于线性偏微分方程，该方法构建了在傅里叶（变换）空间中表示的新颖的积分表示（IRs）。本文给出了椭圆偏微分方程的一种简化表示方法；也就是说，我们引入了一种算法来构造相关基本解的特定的、域相关的IRs，然后将其替换为Green的IRs。进一步，我们将该方法从多边形中的bvp扩展到极坐标中的bvp。在本文的续文中，这些结果被用于求解特定的bvp，这说明了该方法比经典变换方法具有实质性的优势。
A new method for solving boundary-value problems (BVPs) for linear and certain nonlinear PDEs was introduced by one of the authors in the late 1990s. For linear PDEs, this method constructs novel integral representations (IRs) that are formulated in the Fourier (transform) space. In this paper, we present a simplified way of obtaining these representations for elliptic PDEs; namely, we introduce an algorithm for constructing particular, domain-dependent, IRs of the associated fundamental solutions, which are then substituted into Green's IRs. Furthermore, we extend this new method from BVPs in polygons to BVPs in polar coordinates. In the sequel to this paper, these results are used to solve particular BVPs, which elucidate the fact that this method has substantial advantages over the classical transform method.