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Epidemiological models feature reliable and valuable insights into the prevention and transmission of life-threatening illnesses. In this study, a novel SIR mathematical model for COVID-19 is formulated and examined. The newly developed model has been thoroughly explored through theoretical analysis and computational methods, specifically the continuous Galerkin–Petrov (cGP) scheme. The next-generation matrix approach was used to calculate the reproduction number (R0)({R}_{0}). Both disease-free equilibrium (DFE) (E0)({E}^{0}) and the endemic equilibrium (E⁎)({E}^{\ast }) points are derived for the proposed model. The stability analysis of the equilibrium points reveals that (E0)({E}^{0}) is locally asymptotically stable when R01{R}_{0}\gt 1. We have examined the model’s local stability (LS) and global stability (GS) for endemic equilibrium \text{ }and DFE based on the number (R0)({R}_{0}). To ascertain the dominance of the parameters, we examined the sensitivity of the number (R0)({R}_{0}) to parameters and computed sensitivity indices. Additionally, using the fourth-order Runge–Kutta (RK4) and Runge–Kutta–Fehlberg (RK45) techniques implemented in MATLAB, we determined the numerical solutions. Furthermore, the model was solved using the continuous cGP time discretization technique. We implemented a variety of schemes like cGP(2), RK4, and RK45 for the COVID-19 model and presented the numerical and graphical solutions of the model. Furthermore, we compared the results obtained using the above-mentioned schemes and observed that all results overlap with each other. The significant properties of several physical parameters under consideration were discussed. In the end, the computational analysis shows a clear image of the rise and fall in the spread of this disease over time in a specific location.

In this study, a global solution is presented for an initial boundary value problem involving a wave equation with logarithmic nonlinear source terms and fractional boundary dissipation. The solution is obtained through the application of the energy method combined with the Faedo-Galarkin procedure. Additionally, a blow-up result for the solution is established, contingent upon the condition of non-positive initial energy. [ABSTRACT FROM AUTHOR]

In this paper, the existence of a weak positive solution for non-local magnetic fractional systems is studied in the fractional magnetic Sobolev space through a sub-supersolution method combined with iterative techniques. [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]

In this work, we establish a theorem concerning the extension of positive weak solutions for a stationary fractional Laplacian problem featuring weight functions that change sign. Additionally, we introduce novel conditions to ensure the existence of a positive solution for the given problem. These conditions are derived utilizing the approach of sub-super solutions, thereby extending and complementing existing results in the literature. [ABSTRACT FROM AUTHOR]

This work deals with the existence and multiplicity of solutions for a class of variable-exponent equations involving the Kirchhoff term in variable-exponent Sobolev spaces according to some conditions, where we used the sub-supersolutions method combined with the mountain pass theory. [ABSTRACT FROM AUTHOR]

In this study, we investigate the existence and multiplicity of solutions for a fractional discrete p−Laplacian equation on Z. Under suitable hypotheses on the potential function V and the nonlinearity f, with the aid of Ekeland’s variational principle, via mountain pass lemma, we obtain that this equation exists at least two nonnegative and nontrivial homoclinic solutions when the real parameter λ>0 is large enough.

Human immunodeficiency virus (HIV) infection is said to be a dangerous, extremely severe, and potentially fatal disease. After weakening a person's immune system, it impedes the body's ability to fend off future illnesses. In this study, we use the Caputo operator's fractional framework to describe how antiretroviral therapy (ART) affects the HIV/AIDS transmission process. Using fractional calculus, the basic characteristics of the formulated system are found. In addition, we investigated the steady states of the fractional system and determined the reproduction parameter $ \mathcal{R}_0 $. A fractional framework has been used for stability analysis. The optimal value of the Caputo fractional-order $ \alpha $ is around $ 9.437300e-01 $, and the system parameters are determined using the least-squares method, demonstrating that the system matches real HIV/AIDS data the best. Average and maximum absolute relative percentile errors have been used to draw comparisons between the classical and Caputo systems. The case summaries obtained demonstrate the Caputo operator's supremacy. A sensitivity test is run using the partial rank correlation coefficient (PRCC) technique to clearly see which parameters are crucial to the formulation of $ \mathcal{R}_0 $. Lastly, a powerful numerical technique is used to run simulations that show the HIV/AIDS time series in the Caputo fractional framework. [ABSTRACT FROM AUTHOR]

The decay rate of solutions to a Bresse system in thermoelasticity of type III with respect to the distributed delay term is the subject of this study. We demonstrate our major finding utilising the energy approach in the Fourier space. [ABSTRACT FROM AUTHOR]

The paper studies a system of two singular one-dimensional nonlinear equations that arise in generalized viscoelasticity with long-term memory, with general source terms and nonlocal boundary condition. We prove the existence of a global solution to the problem using the potential-well theory. Furthermore, we construct a Lyapunov functional and use it together with the perturbed energy method to prove a general decay result. [ABSTRACT FROM AUTHOR]