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
$ \begin{equation*} e^{-bx}\sum\limits_{n = 0}^{\infty}c_n S_n(2bx), \ \ x\geq 0, \ \ b >0, \end{equation*} $
where $ \{S_n(2bx) \}_{n\geq 0} $ denotes the sequence of Laguerre-Sobolev-type polynomials orthogonal with respect to the inner product
$ \begin{equation*} \langle f, g \rangle_{s, b} = \int_{0}^{\infty} f(x) g(x){e^{-2bx}dx}+\frac{\lambda}{2b} f(0) g(0), \ \ \lambda \geq 0, \end{equation*} $
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.
Citation: Edinson Fuentes, Lida E. Riscanevo, Manuel A. Silva. Inverse Laplace transform and spectral method based on Laguerre-Sobolev-type polynomials[J]. AIMS Mathematics, 2026, 11(7): 20364-20388. doi: 10.3934/math.2026827
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
$ \begin{equation*} e^{-bx}\sum\limits_{n = 0}^{\infty}c_n S_n(2bx), \ \ x\geq 0, \ \ b >0, \end{equation*} $
where $ \{S_n(2bx) \}_{n\geq 0} $ denotes the sequence of Laguerre-Sobolev-type polynomials orthogonal with respect to the inner product
$ \begin{equation*} \langle f, g \rangle_{s, b} = \int_{0}^{\infty} f(x) g(x){e^{-2bx}dx}+\frac{\lambda}{2b} f(0) g(0), \ \ \lambda \geq 0, \end{equation*} $
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.
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Edinson Fuentes, Lida E. Riscanevo, Manuel A. Silva. Inverse Laplace transform and spectral method based on Laguerre-Sobolev-type polynomials[J]. AIMS Mathematics, 2026, 11(7): 20364-20388. doi: 10.3934/math.2026827
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