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Quadrature formulas of highest degree of precision for inversion of the Laplace transformation. (English) Zbl 0773.65090
Optimal recovery, Proc. 2nd Int. Symp. Optim. Control, Varna/Bulg. 1989, 273-280 (1992).
[For the entire collection see Zbl 0755.00006.]
The inversion problem for the Laplace transformation is a problem of finding the solution of the equation \(\int_ 0^ \infty e^{- pt}f(t)dt=F(p)\), where \(F(p)\) is a given mapping and \(f(t)\) is the wanted original function. The author considers the inversion formula for the Laplace transformation in the form of the Riemann-Mellin integral \[ f(t)={1\over 2\pi i}\int_{c-i\infty}^{c+i\infty}e^{pt}F(p) dp,\quad c>0, \tag{1} \] and discusses the construction of quadrature formulas for computation of the integral in (1). Several results about quadrature formulas of highest degree of precision are presented and their rate of convergence and error estimates are given.
65R10 Numerical methods for integral transforms
65D32 Numerical quadrature and cubature formulas
65R30 Numerical methods for ill-posed problems for integral equations
41A55 Approximate quadratures
44A10 Laplace transform