Higher-order time-differential heat transfer model with three-phase lag including memory-dependent derivatives

Abstract

Memory-dependent derivative (MDD) has become a new hot spot in many fields, including vibration mechanics, particle physics and thermoelasticity due to its simple formulation and high expressive force. In this study, a new generalized mathematical thermoelastic model that includes a high-order heat transfer law with different phase delays is developed based on the MD derivatives. The time delay in this new concept represents the duration of the memory influence, while the kernel function represents the dependent weight. A problem of an infinite thermoelastic half-space whose surface is stress-free and periodically heated by a laser-pulse is described in order to validate the proposed model. The system of equations is solved using Laplace transforms technique as well as its inversion is applied based on an appropriate numerical technique. Changes in thermophysical fields are obtained with several values of higher-order time derivatives. The effects of kernel function, time delay parameters and laser pulse duration are also investigated and explained by giving some graphs and tables. Moreover, certain comparisons are made taking into account the various unique features of the current model as well as when the memory effect is ignored. The results indicate that MDD is more suitable for modeling thermoelasticity.