This study presents a comprehensive comparison of five analytical-numerical algorithms designed to solve boundary value problems involving bodies coated with functionally graded materials (FGM). The investigation focuses on an axisymmetric heat conduction problem about local heating of the surface of a coated half-space. All proposed algorithms leverage the Hankel integral transformation to handle the problem's radial symmetry efficiently. Three of the methods approximate the continuously varying properties of the FGM coating by discretizing it into a finite number of homogeneous or inhomogeneous layers, for which analytical solutions are derived individually. The change in the thermal conductivity coefficient of the inhomogeneous layer along its thickness is described by a linear or exponential function. The fourth approach transforms the original boundary value problem into the Hankel transform domain, where derivatives are approximated using established finite-difference schemes, enabling a numerical solution. The fifth algorithm uses the approximation of the solution in the functionally graded coating using modeling functions, providing an alternative to layer discretization. The performance, accuracy, and computational efficiency of these five algorithms are assessed and discussed to identify their suitability for practical engineering applications involving FGMs.