<![CDATA[Purpose: This study develops a practical framework that combines Laplace transforms, Runge-Kutta (RK) integration, error assessment and summability-based convergence acceleration for computing transient reliability and availability.Methodology: The methods are treated as complementary. Laplace transforms reduce linear time-invariant state equations, RK methods approximate dynamics when closed forms are unavailable, and Cesàro averaging or Richardson extrapolation is used only when the sequence has a defensible asymptotic structure. A two-state failure-repair model supplies exact benchmarks, and a smooth quadrature problem illustrates extrapolation.Results: Classical RK4 reproduced the exact availability with an observed order of about 4.06, and successive error ratios approached 16 as the step size was halved, while Euler and midpoint RK2 gave observed orders of 1.01 and 2.10. Richardson extrapolation raised the trapezoidal rule from second to fourth order on the smooth benchmark.Conclusions: Acceleration cannot repair an unstable or biased numerical solution, so the model, base solver and error estimator must be validated before any summability procedure is applied.Limitations: The worked model has two states and constant rates, and all results are analytical illustrations rather than field data.Contributions: The study offers a global error bound under Lipschitz assumptions, an exactly solvable benchmark, a verification checklist and decision rules for reproducible reliability computation.]]>
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