Advances in Mathematical Physics

To improve the complexity of the chaotic system and achieve the effective transmission of image information, in this paper, a five-dimensional memristive chaotic system with cubic nonlinear terms is constructed, which has four pairs of symmetric coordinates. First, the cubic nonlinear memristive chaotic system is analyzed using the Lyapunov exponential map, bifurcation map, and attractor phase diagram. The experimental results show that under four pairs of symmetric coordinates, the system exists not only parameter-dependent symmetric rotational coexisting attractor and transient chaotic phenomena but also exists super-multistationary with alternating chaotic cycles dependent on the initial value of the memristor. Then, it is proposed to add a constant term to the linear state variable to explore the effect of the offset increment of the linear state variable on the system in four pairs of symmetric coordinates, while circuit simulation of the five-dimensional chaotic system is carried out using Simulink to verify its existence and realisability. Finally, the synchronization of the dimensionality reduction system and the confidential transmission of the image are achieved, using the control voltage of the system to replace the internal state variables of the memristor to achieve the one-dimensional reduction process, and an adaptive synchronization controller is designed to synchronize the system before and after the dimensionality reduction. Based on the above, an image to be transmitted is modulated into a one-dimensional array and then subjected to the fractional and cyclic operations and combined with the linear encryption and decryption functions and the chaotic masking technique, the simple encryption and decryption of the image processes are realized.

In this paper, we consider the existence and iterative approximation of solutions for a class of nonlinear fourth-order integro-differential equations (IDEs) with Navier boundary conditions. We first prove the existence and uniqueness of analytical solutions for a linear fourth-order IDE, which has rich applications in engineering and physics, and then we establish a maximum principle for the corresponding operator. Based upon the maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to obtain iterative solutions for the nonlocal nonlinear problem under certain conditions. Some examples are presented to illustrate the main results.

The mechanical behavior of the fine-grained piezoelectric/substrate structure with multiple interface cracks under the electromechanical impact loading is investigated. Using the Laplace and Fourier integral transforms, the double-coupled singular integral equations and single-valued conditions of the problems are formulated. Both the singular integral equation and single-valued conditions are simplified into an algebraic equation through the Chebyshev point placement method and solved by numerical calculation. Then, the expression of the dynamic energy release rate is given with the help of the dynamic intensity factors of electric displacement and stress obtained. Finally, numerical results of the dynamic energy release rate with material parameters are demonstrated. The results show that the dynamic energy release rate depends on the size of the interface cracks, coating thickness, and the mechanical–electrical loading. Meanwhile, the fine-grained piezoelectric structures exhibit safer structural performance compared to normal one.

The extended conformable k-hypergeometric function finds various applications in physics due to its ability to describe complex mathematical relationships arising in different physical scenarios. Here are a few instances of its uses in physics, including nuclear physics, fluid dynamics, quantum mechanics, and astronomy. The main objectives of this paper are to introduce the extended conformable k-hypergeometric and confluent hypergeometric functions by utilizing the new definition of the -beta function and studying its important properties, like integral representation, summation formula, derivative formula, transform formula, and generating function. Also, introduce the extension of the Riemann–Liouville fractional derivative and establish some results related to the newly defined fractional operator, such as the Mellin transform and relations to extended -hypergeometric functions.

This paper addresses the unsteady hydrodynamic convective heat and mass transfer of three fluids namely air, water, and electrolyte solution past an impulsively started vertical surface with Newtonian heating in a porous medium under the influences of magnetic field and chemical reaction. Suitable dimensionless parameters are used to transform the flow equations and the approximate analytic method employed to solve the flow problem. The results are illustrated graphically for the velocity, temperature, and concentration profiles. Though, low Prandtl numbers produce high-thermal boundary layer thickness, however, as a novelty, the presence of the magnetic field delayed the convection motion hence, the thermal boundary layer thickness is greater for water with high P_r = 7.0 as compared to air with low P_r = 0.71 and electrolyte solution with low P_r = 1.0. Practically, water with a high-Prandtl number can effectively absorb and release heat. This makes water useful in applications such as geothermal heat pumps and solar thermal collectors, industrial processes such as chemical reactions, distillation, and drying, and in oceanography in predicting the movement and behavior of ocean currents, which in turn can impact weather patterns and climate. Another major observation from the study is that the rate of cooling associated with air, water, or electrolyte impacts differently on the product being cooled.

Thermal diffusion is a phenomenon where the concentration gradient or diffusive flux is created due to the temperature gradient. Thermal diffusion is induced because of the higher temperature and uneven distribution of the mixture. Formally, thermal diffusion is called the Soret effect, and it is a crucial factor in a number of natural occurrences like the separation of isotopes technique of purification. In this research paper, Maxwell fluid’s flow in the vicinage of a flat plate is discussed by considering the effect of the thermodiffusion subject to the first-order slip at the boundary with the application of a constant proportional Caputo (CPC) fractional derivative. The effect of heat generation and radiation is also taken into consideration, as well as the effect of a magnetic field of constant magnitude. The generalized heat and mass fluxes are considered, and this generalization of heat and mass fluxes is done by utilizing the CPC fractional derivative. After converting the current model’s governing equations into a dimensionless form, the temperature, concentration, and velocity fields’ analytical solutions are found. By drawing graphs of the temperature, concentration, and velocity fields for the parametric modifications, the results are graphically illustrated. It becomes clear from the results discussion that the outcomes produced by the constant proportional derivative are more decaying than those obtained with the classical differential operator of order one.