In the present study, we derive a Boussinesq–type nonlinear partial differential equation to describe solitary wave propagation in isotropic elastic materials. The mathematical formulation is based on the modified strain gradient elasticity (MSGE) framework, which accounts for micro–deformations arising from micro–structural effects as well as macro–scale deformation due to surface effects. The derivation is based on Hamilton's principle, which equates the variation of the strain energy functional to the virtual work done by external forces. The resulting mathematical model is formulated in tensor form to maintain generality and is subsequently specialized to the one–dimensional case to elucidate the nonlinear nature of solitary wave propagation and the influence of micro–structural effects on the material's dynamic response. A key result of this study is the demonstration that the type of wave propagation in the medium can be controlled by appropriately selecting the length–scale parameter associated with micro–inertia, as well as the material length–scale parameters. Three types of initial and boundary conditions are considered: (ⅰ) Constant initial and boundary conditions, (ⅱ) dynamic boundary conditions, and (ⅲ) static initial conditions, moreover; all physical quantities are plotted and discussed in detail.

Recent advances in fractional calculus have highlighted the role of distributed-order operators in modeling anomalous diffusion processes, particularly through bi-fractional diffusion equations of the natural type. In this work, we introduce a distributed-order fractional integral formulation that leads to a generalized bi-fractional Fourier law involving two Riemann–Liouville fractional integrals. The proposed constitutive relation captures a class of anomalous heat conduction characterized by decelerating thermal transport, wherein the effective thermal conductivity is relatively large in the short-time regime and diminishes in the long-time regime. For a quasi-static thermoelastic problem in an unbounded domain, exact analytical solutions for the temperature and displacement fields are derived and expressed in terms of the Fox H-function. Within the quasi-static framework, it is rigorously shown that the appropriate zero initial condition must be imposed on the normal stress rather than on the volumetric strain. A damped cosinusoidal boundary condition at infinity is incorporated and is shown to affect the elastic response due to the infinite propagation speed of mechanical disturbances under the quasi-static assumption. The coupled thermo-mechanical analysis reveals that thermal and mechanical fields exhibit analogous transitional behavior: Decelerating thermal conduction induces a corresponding retardation in the deformation of the medium.