analysis of the effects of time periods and fractal patterns on heat flow in porous materials

The proposed mathematical model applies non-integer derivative orders to study how porosity influences a generalized thermoelastic medium while utilizing improved multi-phase delays framework. the system consists of a porous homogenous isotropic domain where the volume percent shows two-dimensional variations. the modified couple stress theory enables transformation of controlling differential equations into ordinary differential equations. declining to the normal mode approach allows analysis of these equations' mathematical solutions. the calculation of precise solutions for displacement, temperature change, shear and perpendicular loads and volume fraction parameter depends on harmonic wave analysis. the research outcomes present graphical illustrations which present the influence of complex derivative orders together with material porosity levels. the paper demonstrates how models that use fractal differentiation differ from standard models through comparison studies. the research findings enhance thermoelastic behavior understanding in porous media while facilitating the creation of improved performance materials. the study provides significant ramifications for materials science and offers solutions to multi-physics problems as well as enables the development of high-precision industrial technologies.