Vol. 16, No. 1 (2027): Regular Issue

The one-dimensional shallow water wave equations represent a fundamental model in fluid dynamics and is widely used to describe various practical problems. Existence of classical solutions to the wave equation provide valuable understanding and predictive power for phenomena observed in nature and engineering applications. This paper is focused on existence of one or more classical solutions to shallow water equations. To this end a novel way of integral representation of the solutions is introduced. The provided example supports the main findings in this paper.

In two-dimensional turbulent convection, the effects of roughness configurations on heat transport and flow reversal are examined in this work. The impact of five distinct rough models on the Nusselt number (Nu) as a function of Rayleigh number (Ra) is investigated and analysed. All the rough models show reduced heat transport at low Ra; the model with locally compact roughness elements shows the most significant reduction in heat transport. As Ra increases, the normalized Nu generally increases, with differences observed between models with sparsely distributed and locally compact roughness. Flow reversals in 2DRB convection are also explored, with the presence or absence of reversals categorized among the rough models. Flow reversal processes are identified using angular momentum analysis. The study reveals chaotic oscillations in the flow field and Nu for certain models, indicating the influence of roughness on the Large-Scale Circulation (LSC). Sparse models with widely spaced rough elements exhibit more active correlations between the cavity's fluid and LSC, leading to enhanced heat transfer. The scaling relationship between Nu and Ra is investigated, showing distinct scaling regimes for different Ra ranges. The distribution of roughness elements and the relative contributions of the majority of the surface and boundary-layer areas to thermal dissipation influence scalar behaviour. Machine learning techniques, including Convolutional Auto-encoders (CAEs) and Gated Recurrent Units (GRUs), are employed to compress and predict snapshots of turbulent convection data. These techniques offer a promising approach to analyse complex turbulence data and facilitate sequence analysis and prediction. Overall, this work delivers valuable insights into the role of roughness configurations in two-dimensional turbulent convection, shedding light on heat transport, flow reversals, and scaling relationships. The use of machine learning models enhances the understanding and prediction of complex turbulence behaviour.

This study examines a delayed predator-prey paradigm wherein a discrete time delay influences the evolution of the predator. The model incorporates nonlinear interaction variables, harvesting, and the fear effect. This study's originality is in the simultaneous examination of prey harvesting, fear-induced behavioral changes, and predator reproductive delays within a cohesive Leslie–Gower framework an area not previously investigated in the literature. The model effectively represents complex ecological dynamics by integrating delay-induced feedback in predator reproduction with a rational functional response. Our model offers novel insights into the interplay of prey harvesting, fear-induced behavioral modifications, and temporal delay factors typically examined in isolation on predator-prey dynamics in a hitherto underexplored area of ecological modelling. By demonstrating the boundedness and invariance of solutions, we ensure the biological relevance of the system. Equilibrium points are computed for both the non-delayed and delayed versions of the model, and their local stability is examined. The impact of delay on the stability of coexistence equilibrium is investigated by a comprehensive Hopf bifurcation examination. We ascertain the period, stability, and orientation of bifurcating periodic orbits by the use of centre manifold and normal form theory. The theoretical findings are corroborated by numerical simulations, encompassing bifurcation diagrams for harvesting rate and terror impact without delay. These findings underscore the significance of behavioral responses and harvesting strategies in affecting population dynamics in the presence of temporal delays.

Effective and sustainable inventory modeling is essential for organizations, with the shelf life of goods being a key consideration. Goods are subject to deterioration over time due to factors such as damage, waste, decay, and drying, which reduce their value. Holding costs, another critical aspect of sustainable inventory management, are influenced by a range of variables. Typically, holding costs are time-dependent, with a linear time-dependent model assuming a constant rate of change in carrying costs over time. However, such a model rarely reflects real-world market systems. Conversely, an exponentially time-dependent holding cost is also impractical, as it suggests an unrealistic rate of change. A more realistic approach is the parabolic time-dependent holding cost model, which provides a more feasible solution. In this study, we develop a two-warehouse model for inventory management. One warehouse has finite capacity and is owned, while the other has unlimited capacity and is rented. The model assumes a constant deterioration rate for both warehouses, with demand following a linear time-dependent function. Shortages are allowed, with full backordering assumed to enhance realism. The primary objective is to maximize total average profit. To illustrate the model's applicability, a numerical example is provided, along with a sensitivity analysis to evaluate the impact of key system parameters. Graphical representations of the results are generated using MATLAB software (version 2021b).

In this paper, recursive backstepping control method for tracking control and synchronization of five-dimensional hyperchaotic improved Sprott-C systems are investigated. Based on synchronized hyperchaotic improved Sprott-C systems, a new secure technique for encrypting audio signals is put forth. The efficiency of the suggested synchronization and control strategies is confirmed by numerical simulations. We demonstrate that there is no difference between the original and decrypted audio signals by quantitatively validating the suggested encryption scheme.

Leukemia is a hematological malignancy marked by uncontrolled proliferation of immature leukocytes, disrupting normal blood cells formation and limiting long-term therapeutic success. Despite advancements in treatment, achieving sustained remission remains a formidable challenge. To investigate therapeutic outcomes under chimeric antigen receptor (CAR) T-cell immunotherapy, a nonlinear ordinary differential equation model is developed that integrates susceptible, infected, cancer, and immune cell populations. The model incorporates immune-mediated cell clearance through a parameter $\gamma$, representing the elimination of infected cells by immune action. Analytical results establish positivity, boundedness, and the existence of biologically feasible equilibria. The basic reproduction number $\mathcal R_0$ is derived using the next-generation matrix approach, serving as the threshold for disease persistence. Local and endemic equilibria are analyzed through Routh-Hurwitz and Lyapunov criteria to determine asymptotic stability conditions. Numerical simulations illustrate how CAR T-cell infusion and immune stimulation feedback influence leukemia suppression and immune persistence. Sensitivity analysis highlights $\beta, \gamma, k_0$ as dominant parameters shaping therapeutic outcomes. The proposed framework offers a compact, mechanistic description of immune-tumor dynamics and supports optimization of immunotherapeutic dosing strategies.

This article delves into investigating the existence of solutions for fractional-order neural networks (FNN), alongside analyzing the finite-time stability (FTS) of fractional differential equations. FTS, which focuses on trajectories converging to equilibrium within a short time frame, is a central aspect of this exploration. Employing Gronwall's inequality as the primary analytical tool, the study derives conditions for FTS. The theoretical findings are then substantiated through rigorous numerical simulations.

The article provides a comprehensive review of advancements in the fractional Euler method, with a particular emphasis on its improved and modified versions. A detailed investigation of the method’s theoretical foundations is conducted, focusing on the analysis of error bounds. Notably, the study establishes two new error bounds and distinguishes one as superior to the other through a comparative theoretical analysis. To substantiate the theoretical findings, numerical examples are presented, offering a thorough comparison of the error bounds and reinforcing the validity of the obtained results.

This manuscript presents a systematic study on stochastic fractional neural networks with time delay, focusing on the establishment of existence and uniqueness of solutions in the phase space $\mathbb{M}^2([-\uptau,T];\mathbb{R}^n)$. Using the Picard iteration method, explicit sufficient conditions guaranteeing the well-posedness of the system are rigorously established. A key contribution of this work is the derivation of analytical upper bounds for the Picard approximation sequence, which provides insight into the convergence behavior and reliability of the obtained solutions. Numerical examples are presented to confirm the theoretical findings and demonstrate the effectiveness of the proposed approach.

This study focuses on determining the minimum energy control for fractional delay systems. The optimal energy performance index for such systems is derived using solutions obtained through the Laplace transform. The proposed control strategy ensures the system transitions from its initial state to the desired final state with minimal energy consumption. The approach is further extended to accommodate various control types. To support the theoretical findings, the numerical examples along with graphical simulations are provided and discussed.

In this paper, the dynamics of a double-well Duffing oscillator impacting with single displacement boundary is discussed. The flow switchability theory is used for developing the analytical conditions for the impact, grazing and stuck motions of the impact double-well Duffing oscillator at the displacement boundary. Through the discretization of a nonlinear dynamical system, the implicit mapping is developed for the solution of a double-well Duffing oscillator. The mapping structures for period-${n}$ motions with impact chatters are introduced for impact periodic motions of the double-well Duffing oscillator. The bifurcation trees of impact chatter periodic motions are developed. The corresponding saddle-node and period-doubling bifurcations of periodic motions are determined through the eigenvalue analysis, and the periodic motion switching are determined through the grazing bifurcations. The impact chattered periodic motions from period-1 to period-8 motions with/without grazing at the impact boundary is presented numerically and analytically for illustrations of motion complexity in the impact double-well Duffing oscillator. Such a study can help one understand impact dynamics and motion complexity of the double-well Duffing systems with a displacement boundary. The method presented in this paper can be applied to other discontinuous nonlinear dynamical systems.

The class of parabolic equations in time sclases with forward jum operators are investigated for existence of at least one solution, at least two nonnegative and at least three nonnegative solutions. For this aim, a new integral representation of the solutions is derived. The results are illustrated by using the Burgers equation.