Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5775945 | Applied Mathematics and Computation | 2017 | 10 Pages |
Abstract
The boundary value problem uuxx=â2 appears in Crane's theory of laminar convection from a point source. We show that the solution is real only when |x|â¤Ï/2. On this interval, denoting the constants of integration by A and s, the general solution is AV([xâs]/A) where the “Crane function” V is the parameter-free function V=exp(â{erfinv(â[2/Ï])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)=ân=0a2nT2n(z[x]) where the mapping is z=arctanh(x/â§)L2+(arctanh(x/â§))2,â§=Ï/2. Another option is singular basis functions. Vâ(1âx2/â§2){1â0.216log(1âx2/â§2)} has a maximum pointwise error that is less 1/2000 of the maximum of the Crane function.
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
John P. Boyd,