Applying Laguerre’s functions for approximate calculation of Green’s function of a second-order differential equation

We consider the equation ẍ(t) = Ax(t) + f(t), t ∈ $\mathrm{ℝ}$. with the matrix coefficient A. This equation has a unique solution x, which is bounded on $\mathrm{ℝ}$, for any continuous bounded inhomogeneity f if and only if the spectrum of the matrix A does not intersect the semi-axis ${\mathrm{ℝ}}_{-}$ = {z ∈ $\mathrm{ℝ}$ : z ≤ 0}.

In this case, the solution x is defined by the formula

$x\left(t\right)=\underset{-\infty }{\overset{+\infty }{\int }}G(t-s)f\left(s\right)ds{,}_{}^{}G\left(t\right)=-\frac{1}{2}{e}^{-\sqrt{A\left|t\right|}}{\left(\sqrt{A}\right)}^{-1}$.

We discuss the problem of approximate calculation of Green’s function G(t) using its expansion into Laguerre’s series. The scale parameter τ in Laguerre’s polynomials is chosen to ensure the highest accuracy.