Fourier Transformation and Hyers–Ulam Stability of Linear Differential Equations in L^2(R)
DOI:
https://doi.org/10.62054/ijdm/0303.18Abstract
This paper investigates the Hyers–Ulam stability of linear differential equations with constant coefficients in L^2(R) using Fourier transform methods. By converting differential operators into algebraic multipliers in the frequency domain, we obtain sufficient conditions for the existence of solutions and for stability under perturbations. Plancherel’s theorem provides the key norm estimate linking the frequency and time domains. Under a bounded inverse multiplier condition, we prove Hyers–Ulam stability for both second-order and general nth-order equations. Examples involving a Gaussian input, an RLC circuit, and a resonant oscillator illustrate the scope of the theory and show where stability fails.
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