O'Malley's current research emphasizes the relationship between singular perturbation theory and various regularization methods for differential-algebraic systems, geometric approaches to understanding the limiting solutions to singularly perturbed boundary value problems, the motion of shock layers and other interfaces, the interplay between asymptotic and numerical methods, and tough problems of asymptotic matching. He continues to collaborate with an international collection of interesting characters, and receives support for his scholarly work from the National Science Foundation.
O'Malley is known for several pioneering contributions to singular perturbation theory and applications.[4]
O'Malley has been especially active as a member of SIAM, the Society for Industrial and Applied Mathematics. He was president of SIAM in 1991 and 1992, and has been vice-president in charge of their publication program which encompasses ten journals and several book series. Among his other positions with SIAM has been that of program chair for several meetings including ICIAM `91, the second international industrial and applied mathematics conference, which drew 2,200 participants.
On the motion of viscous shocks and the supersensitivity of their steady-state limits, Methods Applics. Analysis, 1, (1994), (with J.G.L. Laforgue).
Shock layer movement for Burger's equation, SIAM J. Appl. Math. 54: (1994) (with J.G.L. Laforgue).
On regularizing differential-algebraic equations, Trends and Developments in Ordinary Differential Equations, Y. Alavi and P.F. Hsieh, editors, World Scientific, 1994 (with L.V. Kalachev).
The regularization of nonlinear differential-algebraic equations. SIAM J. Math Anal., 25: pp. 615–629, 1994 (with L.V. Kalachev).
Super sensitive boundary value problems, "Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters", H.G. Kaper and M. Garbey, editors, Kluwer, 1993, pp.215–223 (with J.G.L. Laforgue).
Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991.