Differential equations form the basis of consecutive equations in the field of engineering sciences, in particular in mechanics and material sciences. In fact, the use of rheological constitutive equations incorporating fractional derivatives has garnered significant attention over the past fifty years. Fractional derivative equations have demonstrated consistency with molecular theories [1]. In terms of thermodynamic compatibility, rheological models incorporating fractional derivatives offer a more precise fit with laboratory measurement results [2, 3]. Therefore, in this chapter, we demonstrate in a first part (Sect. 1) the interest of differential equations with fractional order in the description of the mechanical behaviour of materials. Due to its ability to absorb deformation energy, flexible polyurethane (PU) foam finds widespread usage in various comfort applications, including automotive seat cushions and mattresses. Designing a mechanical model to characterize its behaviour under different test conditions holds considerable interest for optimizing comfort of the personalized seating occupant system under static, quasi-static, and dynamic conditions. When it is subject to a significant compression deformation, PU exhibits highly nonlinear elasticity and viscoelastic behaviour. The theory of viscoelasticity holds significant importance in describing materials characterized by time-dependent mechanical behaviour. Constitutive models for small one-dimensional deformations in viscoelastic materials can be expressed in either integer or fractional differential equations. In the second part (Sect. 2) of this chapter, we focus on modelling the quasi-static behaviour of polyurethane foam using a memory integer and fractional models. The parameters of two memory models (fractional memory model and integer memory model) were determined through original minimization and identification methods. The identified parameters meet the thermodynamic conditions. The results show good agreement between experimental behaviour of polyurethane foam and those predicted by two memory models. Finally, a comparison between these models was conducted.