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The utilization of time-fractional PDEs in diverse fields within science and technology has attracted significant interest from researchers. This paper presents a relatively new numerical approach aimed at solving two-term time-fractional PDE models in two and three dimensions. We combined the Liouville–Caputo fractional derivative scheme with the Strang splitting algorithm for the temporal component and employed a meshless technique for spatial derivatives utilizing Lucas and Fibonacci polynomials. The rising demand for meshless methods stems from their inherent mesh-free nature and suitability for higher dimensions. Moreover, this approach demonstrates the effective approximation of solutions across both regular and irregular domains. Error norms were used to assess the accuracy of the methodology across both regular and irregular domains. A comparative analysis was conducted between the exact solution and alternative numerical methods found in the contemporary literature. The findings demonstrate that our proposed approach exhibited better performance while demanding fewer computational resources. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a ψ-Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here, we use the fiber method and the Nehari manifold to prove our results. [ABSTRACT FROM AUTHOR]

Infectious diseases can have a significant economic impact, both in terms of healthcare costs and lost productivity. This can be particularly significant in developing countries, where infectious diseases are more prevalent, and healthcare systems may be less equipped to handle them. It is recognized that the hepatitis B virus (HBV) infection remains a critical global public health issue. In this study, we develop a comprehensive model for HBV infection that includes vaccination and hospitalization through a fractional framework. It has been shown that the solutions of the recommended system of HBV infection are positive and bounded. We examine the steady states of the model and determine the basic reproduction number; denoted by R 0 . The qualitative and quantitative behavior of the model is demonstrated using mathematical skills and numerical techniques. It has been proved that the infection-free steady state of the system is locally asymptotically stable if R 0 < 1 and unstable otherwise. Furthermore, the Ulam–Hyers stability (UHS) of the recommended fractional models is investigated and the significant conditions are provided. We present an iterative technique to visualize the dynamical behavior of the system. We perform different simulations to illustrate the effect of different input factors on the solution pathways of the system of HBV infection to conceptualize the role of parameters in the control and prevention of the infection. [ABSTRACT FROM AUTHOR]

The economic burden of HIV extends beyond the individual level and affects communities and countries. HIV can lead to decreased economic growth due to lost productivity and increased healthcare costs. In some countries, the HIV epidemic has led to a reduction in life expectancy, which can impact the overall quality of life and economic prosperity. Therefore, it is significant to investigate the intricate dynamics of this viral infection to know how the virus interacts with the immune system. In the current research, we will formulate the dynamics of HIV infection in the host body to conceptualize the interaction of T-cells and the immune system. The recommended model of HIV infection is presented with the help of fractional calculus for more precious outcomes. We introduce numerical methods to demonstrate how the input parameters affect the output of the system. The dynamical behavior and chaotic nature of the system are visualized with the variation of different input factors. The system's tracking path has been numerically depicted and the impact of the viruses on T-cells has been demonstrated. In addition to this, the key factors of the system has been predicted through numerical findings. Our results predict that the strong non-linearity of the system is responsible for the chaos and oscillation, which are so closely related. The chaotic parameters of the system are highlighted and are recommended for the control of the chaos of the system. [ABSTRACT FROM AUTHOR]

It is renowned that the immune reaction in the tumour micro environment is a complex cellular process that requires additional research. Therefore, it is important to interrogate the tracking path behaviour of tumor-immune dynamics to alert policy makers about critical factors of the system. Here, we use fractional derivative to structure tumor-immune interactions. Furthermore, in our research, we concentrated on the qualitative investigation and time series analysis of tumor-immune cell interactions. The solution routes are examined using a new numerical technique to emphasis the impact of the factors on tumor-immune system. We focused on the behaviour of the system with fluctuation of different values. The most crucial components of the proposed system are identified and policymakers are advised. The outcomes of the present study are the strong predictor of clinical success and the in-out of immune cells in a tumour is also critical to treatment efficacy. As a result, studying the behaviour of tumor-immune cell interactions is important to predict crucial factors for the prevention and management to the health officials. [ABSTRACT FROM AUTHOR]