The Crane equation uuxx=−2: The general explicit solution and a case study of Chebyshev polynomial series for functions with weak endpoint singularities

The boundary value problem uuxx=−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) 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]) 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)} has a maximum pointwise error that is less 1/2000 of the maximum of the Crane function.