Real Inversion Formulas of the Laplace Transform
As stated in the preface, one of our strong motivations for writing this book is given by the historical success of the numerical and real inversion formulas of the Laplace transform which is a famous typical ill-posed and very difficult problem. In this
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Real Inversion Formulas of the Laplace Transform
As stated in the preface, one of our strong motivations for writing this book is given by the historical success of the numerical and real inversion formulas of the Laplace transform which is a famous typical ill-posed and very difficult problem. In this chapter, we will see their mathematical theory and formulas, as a clear evidence of the definite power of the theory of reproducing kernels when combined with the Tikhonov regularization.
4.1 Real Inversion Formulas of the Laplace Transform 4.1.1 Problem and Orientation We will consider the real inversion formulas of the Laplace transform Z L f .p/ D F.p/ D
1 0
ept f .t/ dt;
p>0
(4.1)
on certain function spaces that are applicable in numerical analysis. This integral transform is fundamental in mathematical science and engineering. The inversion of the Laplace transform is, in general, given by a complex form; however, we are still interested in its real inversion, which is the problem to find the original function f .t/ from a given image function F.p/; p 0 ; this is required in various practical problems. The real inversion is unstable in usual settings, thus the real inversion is ill-posed in the sense of Hadamard, and numerical real inversion methods have not been established [108, 241]. In other words, the image functions of the Laplace transform are analytic on a half complex plane, and the real inversion will be complicated. One is led to think that its real inversion is essentially involved, because we need to grasp analyticity from the real and discrete data. © Springer Science+Business Media Singapore 2016 S. Saitoh, Y. Sawano, Theory of Reproducing Kernels and Applications, Developments in Mathematics 44, DOI 10.1007/978-981-10-0530-5_4
197
198
4 Real Inversion Formulas of the Laplace Transform
In Sect. 4.1, we give our new approach to the numerical real inversion of the Laplace transform based on the compactness of the Laplace transform on some good reproducing kernel Hilbert spaces. We use Tikhonov regularization in combination with the theory of reproducing kernels above; however, in order to obtain good numerical results, we will also need some powerful numerical algorithm and computer system, in which we refer to the details later. We will give here mathematical background for the practical applications.
4.1.2 Known Real Inversion Formulas of the Laplace Transform In order to know the situation on the real inversion formula of the Laplace transform, we first recall well-known inversion formulas. The most popular formulas are .1/n n nC1 .n/ n D F.t/ f n!1 nŠ t t lim
(4.2)
(see Post [361, Section 16] and Widder [482, p. 61 Corollary]), and n Y t d h n n i f D F.t/; 1C n!1 k dt t t kD1 lim
(4.3)
(see Widder [482]). The analytical real inversion formula for the Laplace transform after the long steps in the spirit and logic in Chap. 2 was obtained in [388]: Z f .p/ D
1
ept F.t/dt
(4.4)
0
for p > 0, where F W .0; 1/ ! C is a measurable function satisfying Z
1 0
jF.
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