On a Two-Point Boundary-Value Problem with Spurious Solutions

The Carrier–Pearson equation εü + u² = 1 with boundary conditions u(−1) = u(1) = 0 is studied from a rigorous point of view. Known solutions obtained from the method of matched asymptotics are shown to approximate true solutions within an exponentially small error estimate. The so-called spurious solutions turn out to be approximations of true solutions, when the locations of their “spikes” are properly assigned. An estimate is also given for the maximum number of spikes that these solutions can have.