Effect of a Laser‐Moving Heat Source on a Functionally Graded Elliptical Plate's Thermal Behavior Using a Fractional‐Order Mechanism

ABSTRACT This study investigates the thermal behavior of a thick elliptical plate subjected to a moving laser heat‐source using a novel application of the time‐fractional heat conduction equation with the Caputo‐fractional derivative. By incorporating a long‐tail power kernel to account for the memory‐dependent heat flow, this approach provides a more accurate representation of heat transfer dynamics in functionally graded materials (FGMs) under transient conditions. The material properties of the elliptical annulus plate, including thermal conductivity, heat capacities, and Poisson's ratio, are axially graded to model ceramic–metal–based FGMs, specifically titanium carbide and nickel. Analytical solutions for the displacement and temperature distributions are derived using the extended Mathieu function, Laplace transform, and perturbation methods, while numerical methods are employed for stress computation. The study introduces a comprehensive framework to evaluate the influence of time‐fractional parameters on thermal, deformation, and stress profiles, offering enhanced predictive capabilities for high‐temperature applications in advanced material design. The novelty lies in the integration of fractional‐order modeling and a moving laser source, which provides deeper insights into the thermal–mechanical responses of FGMs, paving the way for optimized designs in thermal industries. The merit of this study lies in its innovative modeling approach, novel application to FGMs with elliptical geometry, and relevance to high‐temperature industrial applications. The integration of fractional calculus and moving heat sources represents a significant advancement in understanding and optimizing thermal behavior in advanced materials.