Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3528 https://internationalpubls.com Fractional Order Three-Dimensional Generalized Boundary Value Problem for Rectangular Plate Moving with Heat Source Changdev B. Kothule1, Tarachand L. Holambe2, Satish G. Khavale1*, Bhausaheb R. Sontakke3 1,3 Department of Mathematics, Pratishthan Mahavidyalaya Paithan, Chhatrapati Sambhajinagar (M.S.) India 2 Department of Mathematics, Kai. Shankarrao Gutte ACS College, Dharmapuri, Beed, (M.S.) India 1Department of Mathematics, Statistics & Computer Science, Chetna’s Hazarimal Somani College of Commerce & Economics, Smt. Kusumtai Chaudhari College of Arts, Bandra(E), Mumbai, (M.S.) India kothulepatil1995@gmail.com , tarachandholambe@gmail.com, khavalesatish8@gmail.com, satish.khavale@chetanacollege.in Article History: Received: 19-09-2024 Revised: 24-10-2024 Accepted: 11-11-2025 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 mailto:kothulepatil1995@gmail.com mailto:tarachandholambe@gmail.com mailto:khavalesatish8@gmail.com mailto:satish.khavale@chetanacollege.in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3529 https://internationalpubls.com 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. Keywords: Fractional-order, Thermoelasticity, Boundary Value Problem, Rectangular Plate, Moving Heat Source, Integral Transforms. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3530 https://internationalpubls.com Nomenclature λ, 𝜇 Lame’s parameters 𝜌 Density 𝐶𝐸 Specific heat of the material with constant strain t Time T Absolute temperature 𝑇0 Reference temperature 𝜃 = (T −𝑇0) Temperature increment |𝑇 − 𝑇0| / 𝑇0 ≪ 1. 𝜔𝑇 Linear thermal expansion coefficient 𝛾 = 𝜔𝑇(3𝜆 + 2𝜇) 𝜎𝑖𝑗 Stress tensor 𝑒𝑖𝑗 Strain tensor 𝑢𝑖 Displacement components K Thermal conductivity 𝜏0 Relaxation time 𝑐0 = √ 𝜆+2𝜇 𝜌 𝜂 = 𝜌𝐶𝐸 𝐾 휀 = 𝛾2𝑇0 𝜌𝐶𝐸(𝜆+2𝜇) 𝛽 = ( 𝜆+2𝜇 𝜇 ) 1/2 1. Introduction De Chant [1] introduced a numerical approach to model impulsive displacement in quasi-linear viscoelastic materials, a topic with broad implications in dynamic systems where viscoelastic effects are significant. Duan et al. [3] expanded on the fundamental principles of fluid-structure interaction under dynamic loading by examining the nonlinear stability of rarefaction waves in compressible fluids. These approaches are essential for understanding how complex material responses influence the design and performance of structures subjected to thermal and mechanical loads. Ezzat and Youssef focused on generalized thermoelastic models, extending classical theories to include more complex material behaviours. Youssef [4] presented a two-temperature generalized thermoelastic medium subjected to a moving heat source, emphasizing the roles of heat conduction and mechanical stress in the material. Ezzat and Youssef’s work [5] on three-dimensional thermal shock problems in generalized thermoelastic half-spaces provided deeper insights into the impact of rapid thermal loading on material responses. These studies are critical for applications in heat treatment processes, aerospace engineering, and industries involving rapid temperature variations. Gaikwad and Ghadle [28, 29] investigated the thermoelastic deformation of thin circular disks and plates with internal heat generation and non-uniform heat supply. Their studies are particularly relevant for structural health monitoring and the design of thermally stressed components in engineering applications. The practical applications of these theoretical studies are vast. For instance, Parnell et al. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3531 https://internationalpubls.com [8] addressed transient thermal mixed boundary value problems in half-spaces, applying advanced computational methods to solve these systems. Numerical solutions are essential for real-world problems where exact analytical solutions are unavailable. Tahouneh and Naei [9] examined the effect of multi-directional nanocomposite materials on the vibrational response of thick shell panels, highlighting the interaction between thermal and mechanical loading in structures with complex material properties. The work by Parnell et al. [8] on transient thermal problems in half-spaces and Youssef and Al-Lehaibi [12] on thermoelastic half-spaces subjected to moving heat sources are directly applicable to scenarios in aerospace engineering, electronics cooling, and geophysics. Gaikwad et.al. [13-26] focused on time-fractional heat conduction and thermoelastic stress analysis in various geometries, such as thin hollow circular disks and rectangular plates. Their studies demonstrate the application of fractional calculus to address complex boundary conditions and material responses. Fractional-order models provide a more accurate description of memory effects and non-local behaviour, which are not captured by traditional integer-order models. Khavale and Gaikwad [14, 18] explored transient thermoelastic problems, using fractional derivatives to analyse temperature distributions and thermal deflections in thin circular disks. These models are particularly useful in cases where materials exhibit anomalous diffusion or time-dependent behaviour. Youssef and Al-Lehaibi [12] studied generalized thermoelastic diffusion, where thermal and mass diffusion processes are coupled, applying this model to a thermoelastic half-space subjected to thermal pulses. This work is important for analysing materials undergoing both heat transfer and mass transport, such as semiconductors and porous materials. Similarly, Khavale and Gaikwad [21, 24] investigated magneto-thermoviscoelasticity, considering the interaction between thermal, mechanical, and magnetic fields in materials exhibiting fractional-order behaviour. These studies are vital for understanding the behaviour of materials in applications like electromagnetic heating and advanced materials science. The use of fractional-order models, as demonstrated in the studies by Khavale and Gaikwad [14, 25], can enhance predictions of material performance in systems where heat conduction does not follow classical diffusion laws. 2. The Basic Equations The system a homogeneous and isotropic Boundary Value Problem on the generalized thermoelasticity with one relaxation time and without any external body forces in undefined coordinates {𝑖, 𝑗, 𝑘 = 1, 2, 3} takes the (Ezzat, Youssef [5]; Youssef, Al-Lehaibi [12]): The equations of motion are 𝜇𝑢𝑖,𝑗𝑗 + (𝜇 + 𝜆)𝑢𝑖,𝑗𝑗 − (3𝜆 − 2𝜇)𝛼𝜃𝑖 = 𝜌𝑢𝑖̈ (1) The heat equation is, 𝐾𝜃𝑖𝑖 = ( 𝜕𝛼 𝜕𝑡𝛼 + 𝜏0 𝜕𝛼+1 𝜕𝑡𝛼+1 ) [𝜌𝐶𝑒𝜃 + 𝛾𝑇0𝑒] − (1 + 𝜏0 𝜕𝛼 𝜕𝑡𝛼 ) 𝑄 (2) The constitutive relations are 𝜎𝑖𝑗 = 2𝜇𝑒𝑖𝑗 + 𝜆𝑒𝑘𝑘𝛿𝑖𝑗 − 𝛾𝜃𝛿𝑖𝑗 (3) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3532 https://internationalpubls.com The displacement relation with the strain takes the form 𝑒𝑖𝑗 = 1 2 (𝑢𝑖,𝑗 − 𝑢𝑗,𝑖). (4) 3. Problem Formulation Assume an isotropic, homogeneous boundary value problem in three dimensions occupies the space 𝜓 = {𝑥, 𝑦, 𝑧 ∶ 0 ≤ 𝑥 < ∞, −∞ < 𝑦 < ∞, −∞ < 𝑧 < ∞} where the body is quiescent initially and is loaded thermally by a moving heat source with constant speed 𝜐, which starts from the bounding plane of the surface 𝑥 = 0 and moves along the x-axis when these surfaces are traction-free as in Fig. 1 Fig.1. Graphical representation of problem The equations of motion are (𝜆 + 2𝜇) 𝜕2𝑢 𝜕𝑥2 + 𝜇 ( 𝜕2𝑢 𝜕𝑦2 + 𝜕2𝑢 𝜕𝑧2 ) + (𝜆 + 𝜇) ( 𝜕2𝑣 𝜕𝑥𝜕𝑦 + 𝜕2𝑤 𝜕𝑥𝜕𝑧 ) − 𝛾 𝜕𝜃 𝜕𝑥 = 𝜌𝑢 ̈ (5) (𝜆 + 2𝜇) 𝜕2𝑣 𝜕𝑦2 + 𝜇 ( 𝜕2𝑣 𝜕𝑥2 + 𝜕2𝑣 𝜕𝑧2 ) + (𝜆 + 𝜇) ( 𝜕2𝑢 𝜕𝑥𝜕𝑦 + 𝜕2𝑤 𝜕𝑧𝜕𝑦 ) − 𝛾 𝜕𝜃 𝜕𝑦 = 𝜌𝑣 ̈ (6) (𝜆 + 2𝜇) 𝜕2𝑤 𝜕𝑧2 + 𝜇 ( 𝜕2𝑤 𝜕𝑥2 + 𝜕2𝑤 𝜕𝑦2 ) + (𝜆 + 𝜇) ( 𝜕2𝑢 𝜕𝑧𝜕𝑥 + 𝜕2𝑣 𝜕𝑧𝜕𝑦 ) − 𝛾 𝜕𝜃 𝜕𝑧 = 𝜌𝑤 ̈ (7) The heat equation is 𝐾 ( 𝜕2𝜃 𝜕𝑥2 + 𝜕2𝜃 𝜕𝑦2 + 𝜕2𝜃 𝜕𝑧2 ) = 𝜌𝐶𝐸 ( 𝜕𝛼 𝜕𝑡𝛼 + 𝜏0 𝜕𝛼+1 𝜕𝑡𝛼+1 ) 𝜃 +𝛾𝑇0 ( 𝜕𝛼 𝜕𝑡𝛼 + 𝜏0 𝜕𝛼+1 𝜕𝑡𝛼+1 ) ( 𝜕𝑢 𝜕𝑥 + 𝜕𝑣 𝜕𝑦 + 𝜕𝑤 𝜕𝑧 ) − (1 + 𝜏0 𝜕𝛼 𝜕𝑡𝛼 ) 𝑄 (8) The stress-strain relations as: 𝜎𝑥𝑥 = 2𝜇𝑒𝑥𝑥 + 𝜆𝑒 − 𝛾𝜃, (9) 𝜎𝑦𝑦 = 2𝜇𝑒𝑦𝑦 + 𝜆𝑒 − 𝛾𝜃, (10) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3533 https://internationalpubls.com 𝜎𝑧𝑧 = 2𝜇𝑒𝑧𝑧 + 𝜆𝑒 − 𝛾𝜃, (11) 𝜎𝑥𝑦 = 2𝜇𝑒𝑥𝑦 (12) 𝜎𝑥𝑧 = 2𝜇𝑒𝑥𝑧 (13) 𝜎𝑦𝑧 = 2𝜇𝑒𝑦𝑧 (14) The strain components are: 𝜎𝑥𝑥 = 𝜕𝑢 𝜕𝑥 , 𝜎𝑦𝑦 = 𝜕𝑣 𝜕𝑦 , 𝜎𝑧𝑧 = 𝜕𝑤 𝜕𝑧 (15) 𝜎𝑥𝑦 = 1 2 ( 𝜕𝑢 𝜕𝑦 + 𝜕𝑣 𝜕𝑥 ), 𝜎𝑥𝑧 = 1 2 ( 𝜕𝑢 𝜕𝑧 + 𝜕𝑤 𝜕𝑥 ) , 𝜎𝑦𝑧 = 1 2 ( 𝜕𝑤 𝜕𝑦 + 𝜕𝑣 𝜕𝑧 ) (16) 𝜕𝑢 𝜕𝑥 + 𝜕𝑣 𝜕𝑦 + 𝜕𝑤 𝜕𝑧 = 𝜎𝑥𝑥 + 𝜎𝑦𝑦 + 𝜎𝑧𝑧 = 𝑒. (17) Equations (5)-(7) can be rewritten by using Eq. (17) as: 𝜇∇2 𝜕𝑢 𝜕𝑥 + (𝜆 + 𝜇) 𝜕2𝑒 𝜕𝑥2 − 𝛾 𝜕2𝜃 𝜕𝑥2 = 𝜌 𝜕𝑢 ̈ 𝜕𝑥 (18) 𝜇∇2 𝜕𝑣 𝜕𝑦 + (𝜆 + 𝜇) 𝜕2𝑒 𝜕𝑦2 − 𝛾 𝜕2𝜃 𝜕𝑦2 = 𝜌 𝜕𝑣 ̈ 𝜕𝑦 (19) 𝜇∇2 𝜕𝑤 𝜕𝑧 + (𝜆 + 𝜇) 𝜕2𝑒 𝜕𝑧2 − 𝛾 𝜕2𝜃 𝜕𝑧2 = 𝜌 𝜕𝑤 ̈ 𝜕𝑧 (20) where, ∇2= 𝜕2 𝜕𝑥2 + 𝜕2 𝜕𝑦2 + 𝜕2 𝜕𝑧2 . The heat conduction equation takes the form 𝐾∇2𝜃 = 𝜌𝐶𝐸 ( 𝜕𝛼 𝜕𝑡𝛼 + 𝜏0 𝜕𝛼+1 𝜕𝑡𝛼+1 ) 𝜃 + 𝛾𝑇0 ( 𝜕𝛼 𝜕𝑡𝛼 + 𝜏0 𝜕𝛼+1 𝜕𝑡𝛼+1 ) 𝑒 − (1 + 𝜏0 𝜕𝛼 𝜕𝑡𝛼 ) 𝑄 (21) Now, the following dimensionless variables will be applied (Ezzat, Youssef [5-6-10]: (𝑥∗, 𝑦∗, 𝑧∗) = 𝑐𝜂(𝑥, 𝑦, 𝑧), (𝑡∗, 𝜏0 ∗) = 𝑐2𝜂(𝑡, 𝜏0), 𝜃∗ = 𝛾𝜃 (𝛾 + 2𝜇) 𝑄∗ = 𝑄 𝐾𝑐2𝜂2𝑇0 , 𝜎𝑖𝑗 ∗ = 𝜎𝑖𝑗 𝛾 + 2𝜇 , 𝑐 = √ 𝛾 + 2𝜇 𝜌 , 𝜂 = 𝜌𝐶𝐸 𝐾 . Thus, we obtain Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3534 https://internationalpubls.com 𝛽∇2 𝜕𝑢 𝜕𝑥 + (1 − 𝛽) 𝜕2𝑒 𝜕𝑥2 − 𝜕2𝜃 𝜕𝑥2 = 𝜕𝑢 ̈ 𝜕𝑥 (22) 𝛽∇2 𝜕𝑣 𝜕𝑦 + (1 − 𝛽) 𝜕2𝑒 𝜕𝑦2 − 𝜕2𝜃 𝜕𝑦2 = 𝜕𝑣 ̈ 𝜕𝑦 (23) 𝛽∇2 𝜕𝑤 𝜕𝑧 + (1 − 𝛽) 𝜕2𝑒 𝜕𝑧2 − 𝜕2𝜃 𝜕𝑧2 = 𝜕𝑤 ̈ 𝜕𝑧 (24) ∇2𝜃 = ( 𝜕𝛼 𝜕𝑡𝛼 + 𝜏0 𝜕𝛼+1 𝜕𝑡𝛼+1 ) 𝜃 + 휀1 ( 𝜕𝛼 𝜕𝑡𝛼 + 𝜏0 𝜕𝛼+1 𝜕𝑡𝛼+1 ) 𝑒 − 휀2 (1 + 𝜏0 𝜕𝛼 𝜕𝑡𝛼 ) 𝑄 (25) 𝜎𝑥𝑥 = 2𝛽𝑒𝑥𝑥 + (1 − 2𝛽)𝑒 − 𝜃, (26) 𝜎𝑦𝑦 = 2𝛽𝑒𝑦𝑦 + (1 − 2𝛽)𝑒 − 𝜃, (27) 𝜎𝑧𝑧 = 2𝛽𝑒𝑧𝑧 + (1 − 2𝛽)𝑒 − 𝜃, (28) 𝜎𝑥𝑦 = 2𝛽𝑒𝑥𝑦 (29) 𝜎𝑥𝑧 = 2𝛽𝑒𝑥𝑧 (30) 𝜎𝑦𝑧 = 2𝛽𝑒𝑦𝑧 (31) where, 𝛽 = 𝜇 𝜆 + 2𝜇 , 휀1 = 𝛾2𝑇0 𝜌𝐶𝐸(𝜆 + 2𝜇) 휀2 = 𝛾𝑇0 (𝜆 + 2𝜇) . By taking the sum of the eq. (22)-(24) and substituting eq. (17), we get ∇2𝑒 − ∇2𝜃 = �̈� (32) Assume that the function 𝜎 is the invariant stress, as follows: 𝜎 = 𝜎𝑥𝑥 + 𝜎𝑦𝑦 + 𝜎𝑧𝑧 3 . (33) By using eq. (26)-(28), we obtain 𝜎 = 𝜔𝑒 − 𝜃 (34) where, 𝜔 = 3−4𝛽 3 . 4. Solution of The Problem Using Laplace transform, which is defined for any function G(t) as follows: �̅�(𝑠) = ∫ 𝐺(𝑡) ∞ 0 𝑒−𝑠𝑡𝑑𝑡. (35) Applying the transform to (35), we obtain: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3535 https://internationalpubls.com 𝛽∇2 𝜕�̅� 𝜕𝑥 + (1 − 𝛽) 𝜕2𝑒̅ 𝜕𝑥2 − 𝜕2�̅� 𝜕𝑥2 = 𝑠2 𝜕�̅� 𝜕𝑥 (36) 𝛽∇2 𝜕�̅� 𝜕𝑦 + (1 − 𝛽) 𝜕2𝑒̅ 𝜕𝑦2 − 𝜕2�̅� 𝜕𝑦2 = 𝑠2 𝜕�̅� 𝜕𝑦 (37) 𝛽∇2 𝜕�̅� 𝜕𝑧 + (1 − 𝛽) 𝜕2𝑒̅ 𝜕𝑧2 − 𝜕2�̅� 𝜕𝑧2 = 𝑠2 𝜕�̅� 𝜕𝑧 (38) ∇2𝑒̅ − ∇2�̅� = 𝑠2𝑒̅ (39) ∇2�̅� = (𝑠𝛼 + 𝜏0𝑠𝛼+1)�̅� + 휀1(𝑠𝛼 + 𝜏0𝑠𝛼+1)𝑒̅ − 휀2(1 + 𝜏0𝑠𝛼)�̅� (40) �̅�𝑥𝑥 = 2𝛽𝑒�̅�𝑥 + (1 − 2𝛽)𝑒̅ − �̅�, (41) �̅�𝑦𝑦 = 2𝛽𝑒�̅�𝑦 + (1 − 2𝛽)𝑒̅ − �̅�, (42) �̅�𝑧𝑧 = 2𝛽𝑒�̅�𝑧 + (1 − 2𝛽)𝑒̅ − �̅�, (43) �̅�𝑥𝑦 = 2𝛽𝑒�̅�𝑦 (44) �̅�𝑥𝑧 = 2𝛽𝑒�̅�𝑧 (45) and �̅�𝑦𝑧 = 2𝛽𝑒�̅�𝑧 (46) �̅� = 𝜔�̅� − �̅� (47) Using the Laplace transform, we applied the following initial conditions: 𝑢(𝑟, 𝑡)|𝑡=0 = 𝑣(𝑟, 𝑡)|𝑡=0 = 𝑤(𝑟, 𝑡)|𝑡=0 = 𝜃(𝑟, 𝑡)|𝑡=0 = 0 𝜕𝑢(𝑟, 𝑡) 𝜕𝑡 | 𝑡=0 = 𝜕𝑣(𝑟, 𝑡) 𝜕𝑡 | 𝑡=0 = 𝜕𝑤(𝑟, 𝑡) 𝜕𝑡 | 𝑡=0 = 𝜕𝜃(𝑟, 𝑡) 𝜕𝑡 | 𝑡=0 = 0 𝑟 = 𝑥, 𝑦, 𝑧 . (48) We assumed the medium is subjected to a rectangular heat source moving with constant speed and constant strength, releasing its energy continuously on a band of constant dimensions 2𝑎 × 2𝑏 centered on the y-axis and z-axis, respectively, while moving with constant speed υ along the x-axis and being zero elsewhere as in fig.1. Thus, the rectangular moving heat source is assumed to be of the following dimensionless form [4]: 𝑄 = 𝑄0𝛿(𝑥 − 𝜐𝑡)𝐻(𝑎 − |𝑦|)𝐻(𝑏 − |𝑧|), (49) Where a and b are constants, 𝑄0 is the heat source strength (constant), 𝛿 is the Dirac delta function, and H(t) is the Heaviside function. Applying the Laplace transform, we get 𝑄 = 𝑄0 υ 𝐻(𝑎 − |𝑦|)𝐻(𝑏 − |𝑧|)𝑒−( 𝑠 𝜐 )𝑥 . (50) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3536 https://internationalpubls.com Eliminating 𝑒̅ in eq. (39), (40) and (47), we get ∇2�̅� = 𝛼1�̅� + 𝛼2�̅� − 𝛾1𝐻(𝑎 − |𝑦|)𝐻(𝑏 − |𝑧|)𝑒 −( 𝑠 𝜐 )𝑥 . (51) and ∇2�̅� = 𝛼3�̅� + 𝛼4�̅� − 𝛾2𝐻(𝑎 − |𝑦|)𝐻(𝑏 − |𝑧|)𝑒−( 𝑠 𝜐 )𝑥 . (52) where, 𝛼1 = 𝑠2𝜔 − (𝑠𝛼 + 𝜏0𝑠𝛼+1)(1 − 𝜔)(𝜔 + 휀1) 𝜔 , 𝛼2 = 𝑠2𝜔 − 휀1(1 − 𝜔) 𝜔 , 𝛼3 = (𝑠𝛼 + 𝜏0𝑠𝛼+1)(𝛼 + 휀1) 𝜔 , 𝛼4 = 휀1(𝑠𝛼 + 𝜏0𝑠𝛼+1) 𝜔 , 𝛾1 = 휀2(1 + 𝜏0𝑠𝛼)(𝜔 − 1) 𝑄0 𝜐 , 𝛾2 = 휀2(1 + 𝜏0𝑠𝛼) 𝑄0 𝜐 . The double Fourier transform for any function 𝑓(𝑥, 𝑦, 𝑧) is defined as follows: 𝐹[𝑓 ̅(𝑥, 𝑦, 𝑧, 𝑠)] = 𝑓̅̃(𝑥, 𝑝, 𝑞, 𝑠) = 1 2𝜋 ∫ ∫ 𝑓 ̅ ∞ −∞ ∞ −∞ (𝑥, 𝑦, 𝑧, 𝑠)𝑒−𝑖(𝑝𝑦+𝑞𝑧)𝑑𝑦 𝑑𝑧, (53) Where the inverse of the double Fourier transform takes the form 𝐹−1 [𝑓̅̃(𝑥, 𝑝, 𝑞, 𝑠)] = 𝑓 ̅ (𝑥, 𝑦, 𝑧, 𝑠) = 1 2𝜋 ∫ ∫ 𝑓̅̃ ∞ −∞ ∞ −∞ (𝑥, 𝑝, 𝑞, 𝑠)𝑒−𝑖(𝑝𝑦+𝑞𝑧)𝑑𝑝 𝑑𝑞, (54) Thus, we have 𝐹[∇2𝑓(̅𝑥, 𝑦, 𝑧, 𝑠)] = ( 𝜕2 𝜕𝑥2 − 𝑞2 − 𝑝2) 𝑓̅̃ (𝑥, 𝑝, 𝑞, 𝑠) (55) Applying the transforms to (53) and (55), we get the following ordinary differential equations: 𝜕2�̃̅� 𝜕𝑥2 = 𝛼1𝜃 ̃̅ + 𝛽1�̃̅� − 𝛾3𝑒−( 𝑠 𝜐 )𝑥 (56) And 𝜕2�̃̅� 𝜕𝑥2 = 𝛽2𝜃 ̃̅ + 𝛼4�̃̅� − 𝛾4𝑒−( 𝑠 𝜐 )𝑥 (57) where 𝛽1 = 𝑝2 + 𝑞2 + 𝛼2 , 𝛽2 = 𝑝2 + 𝑞2 + 𝛼3 𝛾3 = 2 𝜋 𝛾1 sin(𝑞𝑎) sin (𝑝𝑎) 𝑞𝑝 𝑎𝑛𝑑 𝛾4 = 2 𝜋 𝛾2 sin(𝑞𝑏) sin (𝑝𝑏) 𝑞𝑝 Eliminating �̃̅� from eq. (56) and (57), we get Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3537 https://internationalpubls.com [ 𝜕4 𝜕𝑥4 − (𝛽1 + 𝛽2) 𝜕2 𝜕𝑥2 + (𝛽1𝛽2 − 𝛼1𝛼4)] 𝜃 ̃̅ = −𝛽5𝑒−( 𝑠 𝜐 )𝑥 (58) Similarly, eliminating 𝜃 ̃̅ from (56) and (57) we obtain [ 𝜕4 𝜕𝑥4 − (𝛽1 + 𝛽2) 𝜕2 𝜕𝑥2 + (𝛽1𝛽2 − 𝛼1𝛼4)] �̃̅� = −𝛽6𝑒−( 𝑠 𝜐 )𝑥 (59) where 𝛽5 = 𝛾4 ( 𝑠2 𝑣2 − 𝛽1) + 𝛼4𝛾3 , 𝛽6 = 𝛾3 ( 𝑠2 𝑣2 − 𝛽2) + 𝛾4𝛼1 The solution of eq. (58) takes the form 𝜃 ̃̅ = 𝐴1𝑒−𝑘1𝑥 + 𝐴2𝑒−𝑘2𝑥 − 𝐴3𝑒−( 𝑠 𝜐 )𝑥 (60) where 𝐴3 = 𝛽5 ( 𝑠2 𝑣2−𝑘1 2)( 𝑠2 𝑣2−𝑘2 2) . The solution of eq. (59) takes the form �̃̅� = 𝐵1𝑒−𝑘1𝑥 + 𝐵2𝑒−𝑘2𝑥 − 𝐵3𝑒−( 𝑠 𝜐 )𝑥 (61) Where 𝐵3 = 𝛽6 ( 𝑠2 𝑣2−𝑘1 2)( 𝑠2 𝑣2−𝑘2 2) With 𝐴1, 𝐴2, 𝐵1 and 𝐵2 being some parameters, and ±𝑘1 2 𝑎𝑛𝑑 ± 𝑘2 2 the roots of the characteristic equation: 𝑘4 − 𝐿𝑘2 + 𝑀 = 0 (62) Where 𝐿 = 𝛽1 + 𝛽2 and 𝑀 = 𝛽1𝛽2 − 𝛼1𝛼4, which satisfy the relations: 𝑘1 2 + 𝑘2 2 = 𝛽1 + 𝛽2, 𝑘1 2𝑘2 2 = 𝛽1𝛽2 − 𝛼1𝛼4 . (63) From eq. (60), (61) and (56), we get 𝐵1 = (𝑘1 2 − 𝛼1) 𝛽1 𝐴1 𝐵2 = (𝑘2 2 − 𝛼1) 𝛽1 𝐴2 (64) Hence, we have �̃̅� = (𝑘1 2 − 𝛼1)𝐴1 𝛽1 𝑒−𝑘1𝑥 + (𝑘2 2 − 𝛼1)𝐴2 𝛽1 𝑒−𝑘2𝑥 + 𝐵3𝑒−( 𝑠 𝜐 )𝑥 . (65) To finalize the solution, we have to obtain the parameters 𝐴1, 𝐴2 by applying certain boundary conditions. We consider that the bounding plane of the surface is traction-free and it has no external thermal loading which gives, by using all the above transformations, the following conditions: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3538 https://internationalpubls.com �̃̅�(0, 𝑞, 𝑝, 𝑠) = �̃̅�(0, 𝑦, 𝑧, 𝑡) = 0 (66) Substituting (66) into eq. (60) and (61), we get the following system: 𝐴1 + 𝐴2 = 𝐴3 (67) (𝑘1 2 − 𝛼1)𝐴1 + (𝑘2 2 − 𝛼1)𝐴2 = 𝛽1𝐵3 (68) The solution of the above system gives 𝐴1 = 𝐴3(𝑘2 2 − 𝛼1) − 𝛽1𝐵3 𝑘2 2 − 𝑘1 2 , 𝐴2 = − 𝐴3(𝑘1 2 − 𝛼1) − 𝛽1𝐵3 𝑘2 2 − 𝑘1 2 (69) Hence, we have the solutions in Fourier and Laplace transform domain as follows: �̃̅�(𝑥, 𝑝, 𝑞, 𝑠) = 𝐴3(𝑘2 2 − 𝛼1) − 𝛽1𝐵3 (𝑘2 2 − 𝑘1 2) 𝑒−𝑘1𝑥 − 𝐴3(𝑘1 2 − 𝛼1) − 𝛽1𝐵3 (𝑘2 2 − 𝑘1 2) 𝑒−𝑘2𝑥 − 𝐴3𝑒−( 𝑠 𝜐 )𝑥 (70) �̃̅�(𝑥, 𝑝, 𝑞, 𝑠) = (𝑘1 2 − 𝛼1)[𝐴3(𝑘2 2 − 𝛼1) − 𝛽1𝐵3] 𝛽1(𝑘2 2 − 𝑘1 2) 𝑒−𝑘1𝑥 − (𝑘2 2 − 𝛼1)[𝐴3(𝑘1 2 − 𝛼1) − 𝛽1𝐵3] 𝛽1(𝑘2 2 − 𝑘1 2) 𝑒−𝑘2𝑥 − 𝐵3𝑒 −( 𝑠 𝜐 )𝑥 (71) And 𝑒̅̃(𝑥, 𝑝, 𝑞, 𝑠) = 1 𝛼 [�̃̅�(𝑥, 𝑝, 𝑞, 𝑠) + �̃̅�(𝑥, 𝑝, 𝑞, 𝑠)] (72) 5. Inversion of Fourier and Laplace Transforms To get the final solution in its original variables, we should calculate the inverse of the double Fourier and Laplace transforms in eq. (70)-(72). These expressions may be formally written as functions of 𝑥, and all the parameters of the Fourier and Laplace transforms, namely 𝑝, 𝑞 and 𝑠 in the form 𝑓̅̃(𝑥, 𝑞, 𝑝, 𝑠) ( Ezzat, Youssef [5, 6, 10]). In the beginning we invert the double Fourier transforms using the formula in (54) which gives the expression 𝑓̅̃(𝑥, 𝑦, 𝑧, 𝑠) in the Laplace transform domain as follows (Ezzat, Youssef [5]): 𝑓̅̃(𝑥, 𝑦, 𝑧, 𝑠) = 𝐹−1[𝑓̅̃ (𝑥, 𝑝, 𝑞, 𝑠) = 1 2𝜋 ∫ ∫ 𝑓̅̃(𝑥, 𝑞, 𝑝, 𝑠) ∞ −∞ ∞ −∞ 𝑒𝑖(𝑝𝑦+𝑞𝑧)𝑑𝑝 𝑑𝑞 = 1 2𝜋 ∫ ∫ [cos (𝑝𝑦 + 𝑞𝑧)𝑓̅̃ 𝑒 ∞ −∞ ∞ −∞ + 𝑖𝑠𝑖𝑛(𝑝𝑦 + 𝑞𝑧)𝑓̅̃ 0] 𝑑𝑝 𝑑𝑞 (73) Where 𝑓̅̃ 0 𝑎𝑛𝑑 𝑓̅̃ 𝑒 denote the odd and even parts of the function 𝑓̅̃(𝑥, 𝑞, 𝑝, 𝑠), respectively. To invert the Laplace, transform, the technique of the Riemann-sum approximation will be applied as (Tzou [31]): Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3539 https://internationalpubls.com 𝑔(𝑡) = 𝑒𝑘𝑡 𝑡 [ 1 2 �̅�(𝑘) + 𝑅𝑒 ∑(−1)𝑛�̅� (𝐾 + 𝑖𝑛𝜋 𝑡 ) 𝑁 𝑛=1 ] (74) Here Re means the real part and 𝑖 = √−1. Almost all the computational experiments have shown that the value of k comes from the relation 𝐾𝑡 ≈ 4.7, which gives faster convergence [31]. 6. Numerical Result and Discussion Copper plate was taken for the numerical calculations and the constants of the material have been taken as follows [7]: Physical constant Value Thermal conductivity (K) 386 N/K sec Linear thermal expansion coefficient (𝜔𝑇) 1.78 × 10−5𝐾−1 Specific heat of the material with constant strain (𝐶𝐸) 383.1 𝑚2/K Reference temperature (𝑇0) 293 K Lame’s parameters (𝜇) 3.86×1010 N/m2 Lame’s parameters (λ) 7.79 ×1010 N/m2 Density (𝜌) 8954 kg/m2 Relaxation time (𝜏0) 0.33 × 10-15 sec 𝜏0 = 0.002 𝛽 = 0.25 𝛼 = 0.67 휀1 = 0.0168 휀2 = 0.010444 𝜂 = 8886.73 m/sec2 The study investigates the distribution of temperature increment, stress, strain and displacement (𝑢𝑥) components over a wide range of distance x (from 0 to 1) for varying heat source speeds 𝜈 (values 0.1, 0.3, 0.5, 0.7). specifically, the analysis forces on two distinct cases for the positions y and z where y = z = 0.0 and y = z = 0.5. Temperature Increment: -The temperature increment shows a significant dependence on the heat speed 𝜈. As the speed increases, the temperature increment values are initially large but decrease after reaching a critical position 𝑥 = 𝜈 𝑡 where t is time. For position 𝑥 < 𝜈 𝑡 , the heat source is still influencing the region, leading to rapid increases in temperature increment. For position 𝑥 ≥ 𝜈 𝑡 the heat source is no longer present which leads to rapid decrease in temperature increment. Stress Distribution: -Thermal stress initially decreases with increasing heat source speed, due to a lower thermal gradient. After the peak stress point, further increases in speed lead to higher stress, possibly because of rapid cooling and thermal mismatch. This behaviour illustrates a dual-phase response: lower stress due to milder gradients early on, but higher stress post-heating due to quicker relaxation/cooling. Displacement (𝑢ₓ): - Displacement decreases with increasing speed of the heat source. Higher speeds give the material less time to undergo thermal expansion, resulting in smaller net displacement. This Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3540 https://internationalpubls.com shows that rapid heating minimizes mechanical deformation, which could be important in precision thermal processing applications. Influence of Position (y = z = 0 vs y = z = 0.5): - At the canter (y = z = 0), effects are more pronounced in all aspect’s temperature, stress, and displacement due to direct exposure to the heat source. At off- centre positions (y = z = 0.5), responses are attenuated, confirming the localized nature of thermal effects in the analysed geometry. Temperature for different values of time Temp for different values of alpha Displacement for Different values of time Displacement of different values of alpha Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3541 https://internationalpubls.com Stresses for different values of time Stresses for different values of alpha 7. Concluding Remarks Effect of Time (t): 1. Temperature (θ): • Temperature rises sharply at early times due to active heating by the moving source. • Over time, the temperature profile shifts along the x-axis in the direction of heat source motion, indicating the transient nature of thermal propagation. • After the source passes, cooling is evident as temperature falls significantly. 2. Displacement (uₓ): • Displacement increases initially with time due to thermal expansion. • Later, it stabilizes or decreases as the material begins to cool. • Peak displacement shifts with time, indicating delayed mechanical response to thermal input. 3. Stress (σ): • Stress shows an initial increase, followed by a decrease over time, aligned with temperature and strain variations. • Peak stress shifts downstream over time, mimicking the thermal gradient progression. • Temporal evolution reflects thermoelastic coupling and relaxation behaviour. Effect of Fractional Order Parameter (α): 1. Temperature (θ): • As α increases (approaching classical behaviour), the temperature distribution becomes smoother and more diffused. • Lower α values (more fractional behaviour) produce sharper temperature gradients and delayed diffusion, highlighting memory effects. 2. Displacement (uₓ): • Displacement decreases with increasing α. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3542 https://internationalpubls.com • Fractional models (lower α) capture stronger nonlocal and history-dependent mechanical responses, leading to higher displacements. 3. Stress (σ): • Stress magnitude reduces with increasing α. • This suggests that fractional models (lower α) result in stronger stress responses due to enhanced memory and time-delayed effects. 3. General Observations: • Heat source speed, time, and fractional order α all significantly influence the physical behaviour. • Fractional order models provide more realistic and flexible descriptions of thermoelastic responses, especially in materials exhibiting non-Fourier heat conduction or memory effects. • These insights are critical for advanced materials and structures used in aerospace, electronics, and thermal management systems. 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