Volume 1, Number 2, Spring 1993
AN EXACT SHOCK SOLUTION
G.R. NICKLASON AND M. HEGGIE
Abstract. A nonlinear, second order differential equation
arising from the integrodifferential equation 1  f^{2}(α) =
(d/dα) ∫_^{α}_{∞} V(αβ)f(β)dβ
for the relaxation function V(α) =
A_{1e}^{α/λ1} +A_{2e}^{α/λ2}
is solved for certain values of the input
parameters {A_{1}, A_{2}, λ_{1}, λ_{2}}. The solution is seen to depend
upon a class of functions known as generalized binomial functions.
These functions satisfy a particular type of functional
equation and their occurrence, while not previously noted in
this context, may be rather widespread. It is observed that
the solution of the corresponding first order problem arising
from the choice V(α) = A_{e}^{α/λ} also depends upon a particular
generalized binomial function.
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