Fractional Order Three-Dimensional Generalized Boundary Value Problem for Rectangular Plate Moving with Heat Source

Abstract

This paper investigates a three-dimensional generalized boundary value problem (BVP) for a rectangular plate subjected to a moving heat source within the framework of fractional-order thermoelasticity. The study employs a combination of Laplace and double Fourier transform techniques to obtain analytical solutions in the transform domain. These are subsequently inverted using numerical methods, including Fourier expansion and Riemann-sum approximation, to derive physical field quantities such as temperature increment, stress, strain, and displacement. The model accounts for the memory and non-local effects characteristic of fractional calculus, providing a more accurate description of thermal and mechanical responses. Numerical results are presented for various fractional orders and heat source velocities, demonstrating the significant influence of these parameters on the physical behaviour of the plate. The findings have direct implications for materials and processes involving transient thermal loading, such as in aerospace, electronics cooling, and advanced manufacturing systems.

Introduction: In recent years, fractional-order thermoelasticity has emerged as a powerful framework for modelling materials exhibiting memory and non-local effects, where traditional integer-order theories fail to capture the true dynamic response. The present study extends the generalized thermoelastic theory to a three-dimensional rectangular plate subjected to a moving heat source, using fractional calculus. Earlier works by Ezzat, Youssef, and Gaikwad established the groundwork for fractional thermoelastic behaviour in simpler geometries. This research aims to advance that understanding by solving a fractional-order boundary value problem (BVP) that accounts for transient thermal and mechanical interactions in a continuously moving thermal environment. The results provide valuable insights for engineering systems involving rapid thermal loading, such as in aerospace components, electronic cooling, and high-precision manufacturing.

Objectives:

• To formulate and analyse a three-dimensional generalized fractional-order boundary value problem for a rectangular plate under a moving heat source.


• To apply Laplace and double Fourier transform techniques to derive analytical expressions for temperature, stress, strain, and displacement fields.


• To evaluate the influence of fractional order (α), heat source velocity (ν), and time (t) on thermoelastic field quantities.


• To validate the significance of fractional-order parameters in accurately describing memory-dependent and non-local thermoelastic effects.

Methods: The model is developed under the assumptions of homogeneous and isotropic material behaviour using the generalized thermoelasticity theory with one relaxation time. The governing equations of motion, heat conduction, and constitutive relations are expressed in fractional differential form. The analytical solution is obtained through the Laplace and double Fourier transforms for dimensional reduction and solution in the transform domain. Also the numerical inversion of the transforms using Fourier expansion and Riemann-sum approximation techniques to obtain results in the physical domain. Copper is used as the reference material, and its standard thermophysical constants are employed for numerical simulation. Field variables such as temperature increment, stress, strain, and displacement are evaluated for varying heat source speeds and fractional orders.

Results: The numerical analysis demonstrates that the fractional-order parameter, heat source velocity, and time significantly influence the thermoelastic behaviour of the rectangular plate. The temperature increment is found to depend strongly on the speed of the moving heat source. As the heat source velocity increases, the temperature initially rises rapidly and then decreases once the source moves beyond a particular region, indicating the transient nature of the thermal field. For lower fractional orders, the temperature distribution exhibits sharper gradients and slower diffusion, which highlights the nonlocal and memory-dependent characteristics of fractional thermoelasticity. The stress distribution shows a dual-phase response: it decreases initially with an increase in heat source velocity due to reduced thermal gradients, but later rises because of rapid cooling and thermal mismatch effects. Similarly, the displacement component along the x-axis decreases with increasing source velocity, showing that higher velocities allow less time for thermal expansion, resulting in smaller mechanical deformation. Spatially, the effects are more prominent near the centre (y = z = 0) due to direct exposure to the heat source, while off-centre regions (y = z = 0.5) experience attenuated responses. These results collectively confirm that fractional-order modelling provides deeper insight into the coupled thermal and mechanical behaviour of materials under moving heat loads.

Conclusions: The present investigation establishes that the fractional-order generalized thermoelastic model effectively captures the nonlocal and memory-dependent behavior of materials subjected to a moving heat source. The study concludes that the fractional-order parameter (α), heat source velocity, and time play vital roles in determining temperature, displacement, and stress distributions in the rectangular plate. As the fractional order increases toward unity, the system behavior gradually approaches that predicted by classical thermoelastic theory, while smaller values of α produce stronger thermal gradients and higher stress magnitudes due to enhanced memory effects. The heat source velocity influences both the magnitude and distribution of thermal and mechanical responses-higher speeds result in lower displacements and reduced peak temperatures. These findings demonstrate that the fractional-order approach provides a more accurate and generalized framework than traditional models, particularly for materials and processes that involve transient heat transfer and time-dependent deformation. The proposed model can therefore be effectively applied to engineering systems in aerospace, electronics cooling, and precision manufacturing, where rapid and localized thermal loading plays a significant role in material performance.