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DOI: 10.46698/a3908-1212-5385-q
Chernoff Approximations of the Solution of Linear ODE with Variable Coefficients
Remizov, I. D.
Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 4.
Abstract: The Chernoff approximation method is a powerful and flexible tool of functional analysis, which allows in many cases to express \(\exp(tL)\) in terms of variable coefficients of a linear differential operator \(L\). In this paper, we prove a theorem that allows us to apply this method to find the resolvent of \(L\). Our theorem states that the Laplace transforms of Chernoff approximations of a \(C_0\)-semigroup converge to the resolvent of the generator of this semigroup. We demonstrate the proposed method on a second-order differential operator with variable coefficients. As a consequence, we obtain a new representation of the solution of a~nonhomogeneous linear ordinary differential equation of the second order in terms of functions that are coefficients of this equation, playing the role of parameters of the problem. For the Chernoff function, based on the shift operator, we give an estimate for the rate of convergence of approximations to the solution.
Keywords: operator semigroups, resolvent operator, linear ODE with variable coefficients, representation of the solution, Chernoff approximations
For citation: Remizov, I. D. Chernoff Approximations of the Solution of Linear ODE with Variable Coefficients, Vladikavkaz Math. J., 2025, vol. 27, no. 4, pp. 124-135. DOI 10.46698/a3908-1212-5385-q
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