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The Kidder Equation: u x x + 2 x u x / 1 − α u = 0
Author(s) -
Iacono Roberto,
Boyd John P.
Publication year - 2015
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12073
Subject(s) - mathematics , mathematical analysis , rate of convergence , singularity , quadratic equation , polar coordinate system , convergence (economics) , geometry , channel (broadcasting) , electrical engineering , engineering , economics , economic growth
The Kidder problem isu x x + 2 x ( 1 − α u ) − 1 / 2u x = 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 + 0.15076 x / ( 1 + 1.55607 x 2 ) ) | ≤ 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 coefficientsa n ∼ constant / n − 6. An initial‐value problem is obtained ifu x ( 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.