Abstract <p>We derive the time fractional heat conduction equation (FHCE) for a functionally graded elliptical annulus plate using non-Fourier heat conduction principles that account for memory effects rather than instantaneous responses, influenced by a moving laser heat source. Thermal conductivity, heat capacity, and density are presumed to vary axially and depend on temperature. The upper and lower plates are at zero, while Biot criteria and the Kirpichev reference number for convection energy transfer boundaries regulate the geometrically curved areas. Kirchhoff’s variable transformation linearises the FHCE governing the given conditions. The heat equation is resolved through the application of the Laplace transform, modified Mathieu transform, and Taylor series, followed by their inversions. The inverse of Kirchhoff’s Laplace domain transformation establishes the temperature distribution. Numerical analyses of titanium carbide and nickel properties yielded graphs depicting temperature, motion, and stress variations during laser pulse duration and velocity.</p>