Fourier-initial matching conjecture for the linear non-autonomous delay differential equation

Determine whether the following holds for the delay differential equation dX/dt + a t X(t) = b X(t − τ) with a > 0: If the initial function φ(t) on the interval t ∈ [−τ, 0] is set equal to X_f(t), where X_f is defined as the inverse Fourier transform of ĤX(ω) = C exp(−ω^2/(2a) + (b/(τ a)) e^{i ω τ}), then the forward-time solution X_s(t) for t ∈ [0, ∞) coincides identically with X_f(t) on that interval.

Background

The paper studies the delay differential equation dX/dt + a t X(t) = b X(t − τ) and shows that for a > 0, its Fourier transform leads to an explicit expression ĤX(ω) = C exp(−ω2/(2a) + (b/(τ a)) e{i ω τ}). By numerically inverting the Fourier transform, the authors construct a function X_f(t) defined for all t.

They then propose selecting the initial function φ(t) on [−τ, 0] to match X_f(t) on that interval, and conjecture that the resulting solution X_s(t) of the delay differential equation equals X_f(t) for all t ≥ 0. Numerical simulations presented in the paper support this assertion, but a formal proof is not provided.

References

However, the earlier findings suggest the following conjecture:

If the initial function $\phi(t)$ is represented by the solution ${X_f}(t)$ derived in equation (\ref{solx2}) for $t \in [-\tau, 0]$, then the solution ${X_s}(t)$ of equation (\ref{ddr}) is expressed by ${X_f}(t)$ for $t \in [0, \infty)$.

In other words, if a specific initial function $\phi(t)$ is carefully selected such that it is the same as ${X_f}(t)$ for $t \in [-\tau, 0]$, then the solution ${X_s}(t)$ for $t \in [0, \infty)$ is accurately given by ${X_f}(t)$ for $t \in [0, \infty)$.

— Solving a Delay Differential Equation through Fourier Transform  (2401.02027 - Ohira et al., 2024) in Section 3, Comparison with Numerical Solutions