In order to use the well known representation of the Mellin transform as a combination of two Laplace transforms, the inverse function $g\left(r\right)$ is represented as an expansion of Laguerre polynomials with respect to the variable $t=lnr$. The Mellin transform of the series can be written as a Laurent series. Consequently, the coefficients of the numerical inversion procedure can be estimated. The discrete least squares approximation gives another determination of the coefficients of the series expansion. The last...

Operation principle of the engineering systems occupies an important role in the reliability theory. In most of the studies, the reliability function of the system is obtained analytically according to the structure of the system. Also in such studies the mean operating time of the system is calculated. However, the reliability function of some systems, such as repairable system, cannot be easily obtained analytically. In this case, forming Laplace-Stieltjes transform of the system can provide a...

Error estimates in L∞(0,T;L2(Ω)), L∞(0,T;L2(Ω)2), L∞(0,T;L∞(Ω)), L∞(0,T;L∞(Ω)2), Ω in ${ℝ}^2$, are derived for a mixed finite element method for the initial-boundary value problem for integro-differential equation $$u_t=\mathrm{div}\{a▽u+{\int }_0^t{b}_1▽u\mathrm{d}\tau +{\int }_0^t𝐜u\mathrm{d}\tau \}+f$$ based on the Raviart-Thomas space Vh x Wh ⊂ H(div;Ω) x L2(Ω). Optimal order estimates are obtained for the approximation of u,ut in L∞(0,T;L2(Ω)) and the associated velocity p in L∞(0,T;L2(Ω)2), divp in L∞(0,T;L2(Ω)). Quasi-optimal order estimates are obtained for the approximation...