Mathematician Devised a Formula for 19th-Century Problems
A Mathematical Breakthrough
Ivan Remizov, a Senior Research Fellow at HSE University and the IPPI RAS, has derived a universal formula for solving second-order differential equations that were considered analytically unsolvable for over 190 years. This achievement opens up novel avenues in applied mathematics, physics, and economics.
Specifics of the Formula
The equations in question are of the form $ay” + by’ + cy = g$, which describe intricate phenomena, ranging from pendulum oscillations to planetary motion. Before Remizov’s work, it was assumed that a general formula for these was unattainable. The mathematician expanded established methodologies by incorporating the computation of a sequence limit, enabling the direct input of coefficients to obtain the solution.
Application and Methodology
The approach builds upon Chernoff’s approximation theory and employs the Laplace transform. This framework permits the specification of specialized functions, such as Mathieu and Hill functions, which are vital for calculating satellite trajectories and particle movement in colliders. The resulting solution bears a resemblance to Feynman integrals in quantum mechanics, thus bridging classical mathematics with contemporary physics.
Significance for Science
The findings demonstrate that numerous 19th-century differential equations previously deemed intractable can now be examined directly. This heralds fresh prospects for scientific inquiry and simultaneously challenges existing models of astronomical and biological processes, including considerations regarding the probability of life arising.
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