This paper studies the (3 + 1) extended Zakharov–Kuznetsov (EZK) dynamical equation that is utilized to describe nonlinear three-dimensional dust-ion-acoustic solitary waves in a magnetized two-ion-temperature dusty plasma. The bifurcation theory for planar dynamical systems is applied to the EZK equation. For a given restriction on the parameters, some new traveling wave solutions are characterized in terms of Jacobi-elliptic functions. These solutions are clarified in a graphical way. Physically, these wave solutions represent the related electric potential. Based on Maxwell equations, we calculate the corresponding electric magnetic fields, and we clarify them graphically.

A system of inhomogeneous second-order difference equations with linear parts given by noncommutative matrix coefficients are considered. The closed form of its solution is derived by means of a newly defined delayed matrix sine/cosine using the $\mathcal{Z}$ transform and determining function. This representation helps with analyzing iterative learning control by applying appropriate updating laws and ensuring sufficient conditions for achieving asymptotic convergence in tracking.

Mathematics and physics are deeply interconnected. In fact, physics relies on mathematical tools like calculus and differential equations. The aim of this article is to introduce tempered Riemann–Liouville (RL) fractional operators and their properties with applications in mathematical physics. The tempered RL substantial fractional derivative is also defined, and its properties are discussed. The Laplace transforms (LTs) of the introduced fractional operators are evaluated. The Hyers–Ulam stability and the existence of a novel tempered fractional differential equation are examined. Moreover, a fractional integro-differential kinetic equation is formulated, and the LT is used to find its solution. A growth model and its graphical representation are established, highlighting the role of novel fractional operators in modeling complex dynamical systems. The developed mathematical framework offers valuable insights into solving a range of scenarios in mathematical physics.

In this work, we investigate the theoretical properties of a generalized coupled system of finite-delay fractional differential equations involving Caputo derivatives. We establish rigorous criteria to ensure the existence and uniqueness of solutions under appropriate assumptions on the problem parameters and constituent functions, employing contraction mapping principles and Schauder’s fixed-point theorem. Then, we examine the Ulam–Hyers stability of the proposed system. To illustrate the main findings, three examples are provided. Moreover, we provide numerical solutions using the Adams–Bashforth–Moulton method. The practical significance of our results is demonstrated through illustrative examples, highlighting applications in predator–prey dynamics and control systems.