The Kidder Equation : uxx + 2 xux / √ 1 − α u = 0

The Kidder problem is uxx + 2x(1 − αu)−1/2ux = 0 with u(0) = 1 and u(∞) = 0 where α ∈ [0, 1]. This looks challenging because of the square root singularity. We prove, however, that |u(x ;α) − erfc(x)| ≤ 0.046 for all x, α. Other very simple but very accurate curve fits and bounds are given in the text; |u(x ;α) − erfc(x + 0.15076x/(1 + 1.55607x2))| ≤ 0.0019. Maple code for a rational Chebyshev pseudospectral method is given as a table. Convergence is geometric until the coefficients are O(10−12) when the coefficients an ∼ constant/n−6. An initial-value problem is obtained if ux (0, α) is known; the slope Chebyshev series has only a fourth-order rate of convergence until a simple change-of-coordinate restores a geometric rate of convergence, empirically proportional to exp(−n/8). Kidder’s perturbation theory (in powers of α) is much inferior to a delta-expansion given here for the first time. A quadratic-over-quadratic Padé approximant in the exponentially mapped coordinate z = erf(z) predicts the slope at the origin very accurately up to about α ≈ 0.8. Finally, it is shown that the singular case u(x ;α = 1) can be expressed in terms of the solution to the Blasius equation.