@article{Fuentes2026, 
author = {Edinson Fuentes and Lida E. Riscanevo and Manuel A. Silva},
title = {Inverse Laplace transform and spectral method based on Laguerre-Sobolev-type polynomials},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {7},
pages = {20364-20388},
keywords = {Laguerre-Sobolev-type polynomials, Laplace transform, spectral method},
url = {https://www.sciopen.com/article/10.3934/math.2026827},
doi = {10.3934/math.2026827},
abstract = {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          −      b      x            ∑          n      =      0              ∞            c    n        S    n    (  2  b  x  )  ,      x  ≥  0  ,      b  >  0  ,where    {      S    n    (  2  b  x  )      }          n      ≥      0       denotes the sequence of Laguerre-Sobolev-type polynomials orthogonal with respect to the inner product     ⟨  f  ,  g      ⟩          s      ,      b        =      ∫          0              ∞        f  (  x  )  g  (  x  )            e              −        2        b        x              d    x    +      λ          2      b        f  (  0  )  g  (  0  )  ,      λ  ≥  0  ,defined on a suitable weighted Sobolev space. Furthermore, we proved that the series converges absolutely and uniformly on compact subsets of              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    λ 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.}
}