A thermal shock problem in a semiconducting rod under modified Green-Lindsay theory of length l is discussed. The partial differential equations are simplified by applying the Laplace transformation technique. It is assumed that one end of the rod is subjected to a sudden heat source. The governing equations of a semiconducting medium are transformed using the Laplace transform, and analytical expressions for displacement, stress, carrier density, and temperature distribution are obtained. These transformed components are then inverted by the numerical inversion method and are presented graphically to show the results in different theories of thermoelasticity.

This paper presents a novel framework for modeling nonlinear fractional evolution control systems. This framework utilizes a power non-local fractional derivative (PFD), which is a generalized fractional derivative that unifies several well-known derivatives, including Caputo–Fabrizio, Atangana–Baleanu, and generalized Hattaf derivatives, as special cases. It uniquely features a tunable power parameter “p”, providing enhanced control over the representation of memory effects compared to traditional derivatives with fixed kernels. Utilizing the fixed-point theory, we rigorously establish the existence and uniqueness of solutions for these systems under appropriate conditions. Furthermore, we prove the Hyers–Ulam stability of the system, demonstrating its robustness against small perturbations. We complement this framework with a practical numerical scheme based on Lagrange interpolation polynomials, enabling efficient computation of solutions. Examples illustrating the model’s applicability, including symmetric cases, are supported by graphical representations to highlight the approach’s versatility. These findings address a significant gap in the literature and pave the way for further research in fractional calculus and its diverse applications.

Several industrial fields require mixing and mechanical agitation processes. This operation is mainly used to enhance heat and mass transfer inside stirred tank systems and improve the degree of homogeneity to obtain a high-quality final product. The main goal of this research paper is to analyze the thermal and hydrodynamic behavior of non-Newtonian nanofluid (Bingham–Papanastasiou–Al₂O₃) inside a symmetrically stirred tank. A 3D numerical study has been conducted for a stationary laminar flow inside a symmetric cylindrical vessel under influencing parameters, including the inertia parameter ($Re=1, 20, 100$) and the volume fraction of nanoparticles ($\mathsf{Ø}=0.02, 0.06, 0.1$) with different geometric configurations, has been introduced into the stirring system. According to the findings, with high inertia ($Re=100$), the heat transfer inside the stirred tank is enhanced. Furthermore, increasing the nanoparticle fraction volume had a significant impact on the acceleration of heat transfer along the stirred vessel. It has been also found that the geometric configuration of an anchor with added arm blade (Case 2) is more efficient compared with the rest of the anchor agitator.

This study explores the stochastic Benjamin-Bona-Mahony (BBM) equation with a beta derivative (BD), thereby incorporating multiplicative noise in the Itô sense. We derive various analytical soliton solutions for these equations utilizing two distinct expansion methods: the $ \frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}} $-expansion and the modified $ \frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}} $-expansion techniques, both within the framework of beta derivatives. A fractional multistep transformation is employed to convert the equations into nonlinear forms with respect to an independent variable. After performing an algebraic manipulation, the solutions are trigonometric and hyperbolic trigonometric functions. Our analysis demonstrates that the wave behavior is influenced by the fractional-order derivative in the proposed equations, thus providing deeper insights into the wave composition as the fractional order either increases or decreases. Additionally, we explore the effect of white noise on the propagation of the waves solutions. This study underscores the computational robustness and adaptability of the proposed approach to investigate various phenomena in the physical sciences and engineering.

This work examines the complex interaction between acoustic pressure, photothermal excitation, and magnetic fields in a nonlocal semiconductor material according to the dual phase lag thermoelastic model. An extensive theoretical framework is employed to investigate the dynamic behavior of the semiconductor system, encompassing a wide range of physical phenomena when the thermal conductivity is variable to modify the heat conduction equation. An investigation is carried out using theoretical analysis and numerical simulations to comprehend the complex interaction between these fundamental physical processes. By including the relevant equations in our simulations, we can examine the impact of the magnetic field on the thermal-mechanical characteristics of the medium. The magnetic field and acoustic pressure combine to generate complex variations in temperature, strain, and displacement. The harmonic wave approach is employed to get the analytical expression of the fundamental physical fields, including temperature, mechanical stress, displacement, carrier density, and acoustic pressure, by mathematical means. The theoretical findings are examined and visibly shown due to the powerful magnetic field of the nonlocal semiconductor medium according to changes in thermal conductivity.