Ivan Remizov, a senior researcher at Higher School of Economics, unveiled the approach in a study published in the Vladikavkaz Mathematical Journal, according to the university and Russian news agency TASS.

Second-order differential equations are a mathematical workhorse, used to model how systems change over time. They describe everything from the motion of a swinging pendulum to signals in power grids and other dynamic processes studied in physics and economics.

For nearly two centuries, mathematicians have known that there is no universal closed-form formula, similar to the quadratic formula, that can solve these equations when their coefficients vary. Instead, researchers have relied on numerical simulations or special-case methods, accepting that a general analytic expression was out of reach.

Remizov’s work does not overturn that classical limitation. Rather, it offers a new way to represent solutions using modern tools from operator theory.

His method is based on Chernoff approximation theory, which breaks a complex, continuously changing process into a large number of simple steps. Each step produces an approximation, and as the number of steps increases, the sequence converges toward the exact solution. Remizov and his colleagues have also shown how quickly this convergence occurs.

The study further demonstrates that applying the Laplace transform to these approximations leads precisely to the resolvent operator, a central concept in the theory of differential equations. This provides a constructive procedure for finding solutions, even if they cannot be written as a finite expression using elementary functions.

In practical terms, the approach allows mathematicians to substitute the coefficients a, b, cand g into a standard second-order equation, ay′′+by′+cy=gand obtain the solution function y through a well-defined limiting process.

Remizov, who earned his PhD from Moscow State University in 2018, works at both the Higher School of Economics and the Institute for Problems in Mathematical Transmission of Information of the Russian Academy of Sciences.

His research focuses on approximation methods for operator semigroups, an area with deep connections to mathematical physics.