Inverse Laplace transform and spectral method based on Laguerre-Sobolev-type polynomials

In this contribution, given an analytic function defined in a suitable region of the complex plane, we established the existence of its inverse Laplace transform and showed that it admits a series representation of the form

$$e^{-bx}\sum _{n=0}^{\mathrm{\infty }}{c}_n{S}_n(2bx),x\ge 0,b>0,$$

where $\{S_n(2bx){\}}_{n\ge 0}$ denotes the sequence of Laguerre-Sobolev-type polynomials orthogonal with respect to the inner product

$$⟨f,g{⟩}_{s,b}={\int }_0^{\mathrm{\infty }}f(x)g(x)e^{-2bx}dx+\frac{\lambda }{2b}f(0)g(0),\lambda \ge 0,$$

defined on a suitable weighted Sobolev space. Furthermore, we proved that the series converges absolutely and uniformly on compact subsets of ${\mathbb{R}}^{+}$. As a consequence, the inverse Laplace transform could be efficiently approximated by truncated series. We also proved that the corresponding approximation error exhibits exponential decay, and that parameter $\lambda$ improves the approximation at $x=0$ without increasing the degree of the truncated series. Finally, we proposed a spectral method based on these Laguerre-Sobolev-type polynomials for the numerical solution of differential equations whose Laplace transform can be determined explicitly.