The Crane equation : The general explicit solution and a case study of Chebyshev polynomial series for functions with weak endpoint singularities

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• 作者：John P. Boyd ^{jpboyd@umich.edu jpboyd@engin.umich.edu}
• 关键词：Pseudospectral ; Chebyshev polynomials
• 刊名：Applied Mathematics and Computation
• 出版年：2017
• 出版时间：15 May 2017
• 年：2017
• 卷：301
• 期：Complete
• 页码：214-223
• 全文大小：657 K
• 卷排序：301

文摘

The boundary value problem uuxx=−2uuxx=−2 appears in Crane’s theory of laminar convection from a point source. We show that the solution is real only when |x|≤π/2. On this interval, denoting the constants of integration by A and s  , the general solution is AV([x−s]/A)AV([x−s]/A) where the “Crane function” V   is the parameter-free function V=exp(−{erfinv(−[2/π])x}2) and erfinv(z) is the inverse of the error function. V(x) is weakly singular at both endpoints; its Chebyshev polynomial coefficients an decrease proportionally to 1/n3. Exponential convergence can be restored by writing V(x)=∑n=0a2nT2n(z[x])V(x)=∑n=0a2nT2n(z[x]) where the mapping is z=arctanh(x/℧)L2+(arctanh(x/℧))2,℧=π/2. Another option is singular basis functions. V≈(1−x2/℧2){1−0.216log(1−x2/℧2)}V≈(1−x2/℧2){1−0.216log(1−x2/℧2)} has a maximum pointwise error that is less 1/2000 of the maximum of the Crane function.