## 1 Introduction  
Many manufacturing processes are usually exposed to moving heat sources such as welding, cutting, surface laser treatment, and some tribological problems [1, 2]. Two important applications of a laser pulse beam, i.e., laser texturing and laser piercing processes, are illustrated in Figure 1. Thanks to its simplicity and effectiveness, \"beam theory\" is an essential tool in solid mechanics that enables the designers and engineers to evaluate the behavior of different structures and systems from macro- to micro-scale dimensions. Beam theories are commonly used in the pre-design stages because they offer a substantial understanding of the performance of structures. This is true even though more sophisticated theories for stress investigation of complex structural systems are now accessible. Several theories concerning beams have been developed, all of which are predicated on various assumptions and produce varying levels of precision. The Euler-Bernoulli beam model is widely recognized as one of the field's most fundamental and significant concepts. Microscale beams use microscale beams, many microstructure devices, systems like sensors and actuators, microelectromechanical systems (MEMS), and very thin electronic gadgets [3,4]. Understanding the thermal behavior of such structural components in the manufacturing processes is critical for predicting the performance of such sensitive devices. The response of beams that are on two-parameter elastic foundations, which have a variety of features in design and related industries, has been the subject of a significant amount of research in the past few years, and various theoretical works have been presented recently that have analyzed this behavior and attitude [5]. Many different structures and systems can be described as beam structures that foundations support. An essential area ofotechnical engineering is the study of structural foundations that are supported by soil media. It is both necessary and practical to idealize the soils' responses to external loads because the majority of naturally existing soils have complicated stress-strain properties. Geotechnical assessments of beams that have been resting on elastic soil layers or pile behaviors caused by horizontal stresses, etc., have all benefited from using the [PERSON] model (1867). Based on \"[PERSON]'s hypothesis\", which argues that the deflection at every point on the surface of an elastic continuum is proportional exclusively to the load that is being applied to the surface, this model was developed. An independent set of vertical linear springs is considered to compose the continuum in this mechanical model, which follows from the premise in question. Disadvantages arise, however, from the fact that the displacements that are in adjacent cells do not blend smoothly [6]. By introducing shear interaction between the nearby Winkler spring elements, [PERSON] (1954) revolutionized the field. By using the words \"Pasternak model\" to describe a generalized foundation model, [PERSON] (1964) viewed [PERSON]'s mechanical model as a shear layer on the Winkler framework. The [PERSON] model, referred to as PM from here on out, is applied in a similar way. For the past many decades, the approaches for obtaining the parameters have been addressed because the two-parameter models inevitably need both values for their use in practical applications. Despite not providing the parameters' exact values, [PERSON] himself recommended utilizing plate loading tests to evaluate the two parameters. The interaction between the elastic substrate and the main structures and their influence on the system's thermal behavior may be important in several engineering fields. Despite their significant structural design challenges, elastic-founded structures are frequently utilized in modern architecture, civil engineering, and electronic devices.