Computation of the Bell-Laplace transforms

Abstract

since it converts a function of a real variable t (often representing the time) to a function of a complex variable s (complex frequency). This transform is used for solving differential equations, since it transforms differential into algebraic equations and convolution into multiplication. It can be applied to local integrable functions on [0,+∞) and it converges in each half plane Re(s)> a, the constant a, also known as the convergece abscissa, depends on the growth behavior of f (t). A large number of LT can be found in the literature and, together with the respective antitransforms, are usually used in the solution of the most diverse differential problems. The numerical computation of the link between transforms and antitransforms was considered, for example, by F.G. Tricomi [22, 23], which highlighted the link with the series expansions in Laguerre polynomials. These results have been extended to more general expansions [17], however the results of Tricomi have been proven numerically more convenient by the point of view of numerical complexity. Recently, extensions of the LT have been considered in [16] and in [6] the numerical computation was carried out by approximating the respective kernels by means of expansions in a general Dirichlet series. Extensions of LT (called Laguerre-Laplace transforms) have been obtained in the first place [6] by replacing the exponential with the Laguerre-type exponentials, introduced in [8] and previously studied in [10], [11], [13]. Subsequently the kernel was replaced by an expansion whose coefficients are combination of Bell polynomials, exploiting a transformation, introduced in [16], which uses the Blissard formula, a typical tool of the umbral calculus [19, 20]. In order to validate the computational methods, some examples were taken into consideration, by the first author, with the aid of the computer algebra program Mathematica©. The rapid decay of the considered kernels allows to extend the integration interval to a right neighborhood of the origin. However, the higher computational complexity using the theoretical approach based on the generating function of the generalized Lucas polynomials suggested to approximate the original kernel by a truncation of a general Dirichlet’s series, and to use the matrix pencil method for evaluating the best coefficients. The results obtained confirm the correctness of the procedure introduced.