Wafa Shammakh

This work develops and experimentally validates a Caputo fractional-order model for levofloxacin degradation and Chemical Oxygen Demand (COD) mineralization in Electro-Fenton wastewater treatment, capturing memory effects in long-time mineralization dynamics. Stability analysis using Lyapunov methods demonstrated asymptotic convergence to the equilibrium point. The system is solved using the Adams-Bashforth-Moulton (ABM) scheme, and parameters are calibrated via grid search over α with constrained kinetic optimization. The optimal orders indicate stronger memory in COD removal (α⋆ = 0.7) than in levofloxacin decay (α⋆ = 0.9), achieving high agreement with experiments (R2 = 0.9192 for Levofloxacin and R2 = 0.9974 for COD) across varying electrolyte, pH, catalyst, and biodegradability conditions. Relative to the integer-order model, the fractional model reduces residual error, and accurately predicts levofloxacin decay and COD mineralization under optimized conditions, highlighting the biodegradability potential of the treated wastewater. Finally, local sensitivity analysis identifies fractional order α and nonlinear reaction order n as the most influential parameters for predictive accuracy and process optimization.

Asthma is one of the most prevalent chronic respiratory diseases worldwide, and environmental pollutants such as desert dust play a significant role in triggering respiratory sensitization and aggravating asthma symptoms. In this study, a mathematical model is proposed to investigate the dynamics of dust-induced asthma progression. The population is divided into unaware susceptible individuals, aware susceptible individuals, dust-sensitized individuals, and asthmatic individuals, while two additional variables describe dust concentration in the human population and the environment. Fundamental qualitative properties of the model, including positivity, boundedness, existence of equilibria, and stability conditions, are established. It is shown that the asthma-free equilibrium is locally and globally asymptotically stable whenever the dust-induced asthma threshold is below unity, whereas a unique asthma-persistent equilibrium exists when the dust-induced asthma threshold exceeds unity. To reduce the burden of asthma, an optimal control problem is formulated by incorporating three time-dependent interventions: awareness campaigns, medical intervention for dust-sensitized individuals, and environmental dust reduction. Pontryagin’s Maximum Principle is employed to characterize the optimal controls and derive the corresponding adjoint system. Numerical simulations, performed using the forward–backward sweep method, demonstrate that the proposed interventions significantly reduce the number of asthmatic individuals compared with the uncontrolled case. Furthermore, a cost-effectiveness analysis based on the Average Cost-Effectiveness Ratio (ACER) and Incremental Cost-Effectiveness Ratio (ICER) is conducted to compare seven intervention strategies. The results indicate that the medical intervention strategy is the most economically attractive option, while the combined strategy involving all controls achieves the largest reduction in asthma burden.

This paper examines a family of operators that combine the features of fractional-order and classical operators. Our goal is to obtain results on their invertibility in function spaces, based on their inherent improving properties. The class of proportional operators we study is extensive and includes both fractional-order and classical operators. This leads to interesting function spaces in which we obtain the right- and left-handed properties of invertibility. Thus, we extend and unify results concerning fractional-order and proportional operators. To confirm the relevance of our results, we have supplemented the paper with a series of results on the equivalence of differential and integral forms for various problems, including terminal value problems.

Recently, a new type of derivative has been introduced, known as Caputo proportional derivatives. These are motivated by the applications of such derivatives (which are a generalization of Caputo’s standard fractional derivative) and the need to incorporate such calculus into the research on operators. The investigation therefore focuses on the equivalence of differential and integral problems for proportional calculus problems. The operators are always studied in the appropriate function spaces. Furthermore, the investigation extends these results to encompass the more general notion of Hilfer hybrid derivatives. The primary aim of this study is to preserve the maximal regularity of solutions for this class of problems. To this end, we consider such operators not only in spaces of absolutely continuous functions, but also in particular in little Hölder spaces. It is widely acknowledged that these spaces offer a natural framework for the study of classical Riemann–Liouville integral operators as inverse operators with derivatives of fractional order. This paper presents a comprehensive study of this problem for proportional derivatives and demonstrates the application of the obtained results to Langevin-type boundary problems.

In this contribution, we prove some results of the uniqueness of the common fixed point for quadruple mappings in a complete b ‐metric‐like space. Then, two nontrivial illustrative examples are provided to highlight the reliability of our basic findings. Furthermore, we apply this to the context of b ‐metric‐like spaces to prove the existence of a solution to a class of integral equations.

In this paper, we investigate the existence of solutions to a [ ]-Kirchhoff problem involving the [ ]-Hilfer fractional derivative. Specifically, we reformulate the problem by analyzing the associated functional energy. We then construct three disjoint sets and demonstrate that the functional energy has a critical point in each of these sets. Finally, based on the definitions of these sets, we show that in the first set the solution is positive, in the second set the solution is negative, and in the third set, the solution changes sign.

The problem of equivalence between differential and integral problems is absolutely crucial when applying solution methods based on operators and their properties in function spaces. In this paper, we complement the solution of this important problem by considering the case of general derivatives and integrals of fractional order for vector functions for weak topology. Even if a Caputo differential fractional order problem has a right-hand side that is weakly continuous, the equivalence between the differential and integral forms may be affected. In this paper, we present a complete solution to this problem using fractional order Pettis integrals and suitably defined pseudo-derivatives, taking care to construct appropriate Hölder-type spaces on which the operators under study are mutually inverse. In this paper, we prove, in a number of cases, the equivalence of differential and integral problems in Hölder spaces and, by means of appropriate counter-examples, investigate cases where this property of the problems is absent.