He is known for his groundbreaking work in mathematical applications to aerodynamics and transonic flow, and in non-linear equations more generally.
[1][2] Cole earned an undergraduate degree in engineering from Cornell, after which he entered Caltech as a graduate student.
He worked with Hans Liepmann and Paco Lagerstrom, the latter his advisor, submitting a dissertation on transonic flow in 1949.
Lagerstrom and Cole continued their work, having formed a small research group at GALCIT to better understand the mathematics of fluid flow.
Cole took sabbatical in 1963–1964 at Harvard, where he wrote a book on this body of work: Perturbation Methods in Applied Mathematics.