The study scrutinizes the effects of thermal radiation, heat generation, and induced magnetic field on steady, fully developed hydromagnetic free convection flow of an incompressible viscous and electrically conducting couple stress fluid in a vertical channel. The channel walls are maintained at an isoflux-isothermal condition, such that the left channel wall is maintained at a constant heat flux. In contrast, the right channel wall is maintained at a constant temperature. The governing simultaneous equations are solved analytically utilizing the method of undetermined coefficient, and closed form solutions in dimensionless form have been acquired for the velocity field, the induced magnetic field, and the temperature field. The expression for the induced current density has been also obtained. A parametric study for the velocity, temperature, and induced magnetic field profiles, as well as for the skin-friction coefficient, Nusselt number, and induced current density, is conducted and discussed graphically.

This paper considers a general three-dimensional ABC fractional dynamical system formulated in a bounded locally compact Hausdorff space. The locally compact Hausdorff structure ensures that the compact-open topology on the space of continuous functions coincides with the topology induced by the supremum norm, providing an appropriate Banach space framework for the analysis of the system. Within this setting we study the continuity, boundedness, and Lipschitz properties of the nonlinear operators associated with the fractional model. Based on these properties, the existence of solutions is established using Schaefer fixed point theorem, while uniqueness is obtained through Banach contraction principle under suitable conditions. Furthermore, the Hyers–Ulam stability of the system is investigated, showing that small perturbations lead to small deviations in the corresponding solutions. Finally, the theoretical results are applied to the fractional Lorenz system and the two-dimensional fractional Euler system, illustrating the applicability of the proposed framework to models arising in chaotic dynamics and fluid mechanics.

This paper develops a functional operator-theoretic framework for nonlinear Erdelyi–Kober (EK) fractional dynamical systems formulated in Banach spaces endowed with the compact-open topology. Within this setting, sufficient conditions for existence, uniqueness, and Ulam–Hyers stability of solutions are established using the Banach and Schaefer fixed-point theorems. The continuity, boundedness, and Lipschitz properties of the associated nonlinear operators are analyzed to ensure well-posedness of the fractional system. As a constructive complement to the theoretical results, a power series iterative method (PSIM) is employed to obtain an explicit fractional series representation of the solution in the case $0<\alpha <1$. The applicability of the theoretical framework is illustrated through a nonlinear fractional dynamical Belousov–Zhabotinsky system (DBZS), where the assumptions of the main theorems are verified and the solution is constructed via the proposed series scheme. The results provide a coherent link between abstract fixed-point analysis and a constructive semi-analytical representation of solutions for EK fractional systems.

This paper offers a thorough examination of a unified class of Humbert’s polynomials in two variables, extending beyond well-known polynomial families such as Gegenbauer, Humbert, Legendre, Chebyshev, Pincherle, Horadam, Kinnsy, Horadam–Pethe, Djordjević, Gould, Milovanović, Djordjević, Pathan, and Khan polynomials. This study’s motivation stems from exploring polynomials that lack traditional nomenclature. This work presents various expansions for Humbert–Hermite polynomials, including those involving Hermite–Gegenbauer (or ultraspherical) polynomials and Hermite–Chebyshev polynomials. The proofs enhanced our understanding of the properties and interrelationships within this extended class of polynomials, offering valuable insights into their mathematical structure. This research consolidates existing knowledge while expanding the scope of Humbert’s polynomials, laying the groundwork for further investigation and applications in diverse mathematical fields.

In this article, we considered the one-dimensional swelling problem in porous elastic soils with microtemperatures effects in the case of fluid saturation. First, we showed that the system is well-posed in the sens of semigroup. Then, we constructed a suitable Lyapunov functional based on the energy method and we proved that the dissipation given only by the microtemperatures is strong enough to provoke an exponential stability for the solution irrespective of the wave speeds of the system.

This research aims to introduce and examine a new type of polynomial called the ${\Delta }_h$ Legendre–Appell polynomials. We use the monomiality principle and operational rules to define the ${\Delta }_h$ Legendre–Appell polynomials and explore their properties. We derive the generating function and recurrence relations for these polynomials and their explicit formulas, recurrence relations, and summation formulas. We also verify the monomiality principle for these polynomials and express them in determinant form. Additionally, we establish similar results for the ${\Delta }_h$ Legendre–Bernoulli, Euler, and Genocchi polynomials.