A problem of numerical inversion of implicitly defined Laplace transforms

A procedure is proposed for numerical inversion of Laplace transforms x(s) implicitly defined from the functional equation x(s) = g(a(s) − Cx(s)) where a(•) and g(•) are known functions, C is a known constant. This equation is encountered in queueing, inventory, and insurance problems. The procedure constructs coefficients of the Laguerre series for the original in the Laguerre series inversion method and a sequence of approximants in the Post-Widder inversion method. Scalar and matrix cases are treated in the same fashion. The numerical results are compared with those attainable with the Fourier series method.