A generalized Chebyshev operational method for Volterra integro-partial differential equations with weakly singular kernels.

Volterra integro-partial differential equations with weakly singular kernels (VIPDEWSK) are utilized to model diverse physical phenomena. A matrix collocation method is proposed for determining the approximate solution of this functional equation category. The method employs shifted Chebyshev polynomials of the fifth kind (SCPFK) to construct two-dimensional pseudo-operational matrices of integration, avoiding the need for explicit integration and thereby speeding up computations. Error bounds are examined in a Chebyshev-weighted space, providing insights into approximation accuracy. The approach is applied to several experimental examples, and the results are compared with those obtained using the Bernoulli wavelets and Legendre wavelets methods.

A fractional-order model for nosocomial infection caused by Pseudomonas aeruginosa in Northern Cyprus.

Pseudomonas aeruginosa, a resilient gram-negative bacterium, poses a persistent threat as a leading cause of nosocomial infections, particularly in resource-constrained regions. Despite existing treatment and control measures, the bacterium continues to challenge healthcare systems, especially in developing nations. This paper introduces a fractional-order model to elucidate the dynamic behavior of nosocomial infections caused by P. aeruginosa and to compare the efficacy of carbapenems and aminoglycosides in treatment. The model's existence and uniqueness are established, and both global and local stability are confirmed. The effective reproduction number is computed, revealing an epidemic potential with a value of 1.02 in Northern Cyprus. Utilizing real-life data from a university hospital and employing numerical simulations, our results indicate that patients exhibit higher sensitivity and lower resistance to aminoglycoside treatment compared to carbapenems. Aminoglycosides consistently outperform carbapenems across key metrics, including the reduction of susceptible population, infection numbers, treatment efficacy, total infected population, hospital occupancy, and effective reproduction number. The fractional-order approach emerges as a suitable and insightful tool for studying the transmission dynamics of the disease and assessing treatment effectiveness. This research provides a robust foundation for refining treatment strategies against P. aeruginosa infections, contributing valuable insights for healthcare practitioners and policymakers alike.

A fractional-order mathematical model for lung cancer incorporating integrated therapeutic approaches.

This paper addresses the dynamics of lung cancer by employing a fractional-order mathematical model that investigates the combined therapy of surgery and immunotherapy. The significance of this study lies in its exploration of the effects of surgery and immunotherapy on tumor growth rate and the immune response to cancer cells. To optimize the treatment dosage based on tumor response, a feedback control system is designed using control theory, and Pontryagin's Maximum Principle is utilized to derive the necessary conditions for optimality. The results reveal that the reproduction number [Formula: see text] is 2.6, indicating that a lung cancer cell would generate 2.6 new cancer cells during its lifetime. The reproduction coefficient [Formula: see text] is 0.22, signifying that cancer cells divide at a rate that is 0.22 times that of normal cells. The simulations demonstrate that the combined therapy approach yields significantly improved patient outcomes compared to either treatment alone. Furthermore, the analysis highlights the sensitivity of the steady-state solution to variations in [Formula: see text] (the rate of division of cancer stem cells) and [Formula: see text] (the rate of differentiation of cancer stem cells into progenitor cells). This research offers clinicians a valuable tool for developing personalized treatment plans for lung cancer patients, incorporating individual patient factors and tumor characteristics. The novelty of this work lies in its integration of surgery, immunotherapy, and control theory, extending beyond previous efforts in the literature.