Abstract
This paper is devoted to the computation of the spectrum of the finite Laplace transform (FLT) and its applications. For this purpose, we give two different practical methods. The first one uses a discretization of the FLT. The second one is based on the Gaussian quadrature method. The spectrum of the FLT is then used to invert the Laplace transform of time limited functions as well as the Laplace transform of essentially time limited functions. Several numerical results are given to illustrate the results of this work.
Highlights
The finite Laplace transform (FLT) defined as bL0,bf (x) = e−xyf (y)dy, b > 0, plays an important role in solving boundary value problems for ordinary and partial differential equations, see [7]
The finite Laplace transform was studied first by Debnath and Thomas in [8]. They have given some properties of the FLT
We suppose that a = 0, we study the spectrum of the operator L0,b defined from L2[0, b] into itself and we give some of its applications
Summary
It is used to solve a weakly singular integral equation in transfer theory, see [18] Several applications of this transform in linear control problems have been studied in [15]. The finite Laplace transform was studied first by Debnath and Thomas in [8] They have given some properties of the FLT. The first one is based on the discretization of L0,b following a suitable set of orthogonal polynomials while the second method is based on the use of a Gaussian quadrature method We use such eigenfunctions and eigenvalues to invert the finite Laplace transform as well as the Laplace transform over the set of essentially time limited functions.
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