Our focus in this work is the proposition of a fractional order model based on Caputo fractional derivatives for the understanding of how coronavirus disease is transmitted in a community, using Nigeria as a case study. By using Laplace transform, we show that the state variables of the model are non-negative at all times and show the existence and uniqueness of solutions for the model. Thorough analysis of the model shows that the model is Ulam-Hyers-Rassias stable and that its disease-free equilibrium is locally and globally asymptotically stable whenever the reproduction number of the disease is less than unity. By gathering real-life data about the disease in Nigeria from accredited authority, Nigerian Centre for Disease Control (NCDC), we estimate parameters driving the spread of the disease by fitting this data to our model. By adopting these parameter estimates, using MATLAB, we perform the numerical simulation of the model with a view to validating results from qualitative analysis of the model. Numerical results show that plots for the model at different fractional orders have major determining influence on various compartments of the model as it varies. Various distinct results were observed for each of the compartments in different fractional orders, highlighting the importance of consideration of the fractional order in modelling the highly contagious COVID-19 disease. This work highlights the advantage of fractional order model over the classical integer order model in the sense that the solution obtained for the fractional order model possesses a higher degree of freedom that enables variation of the system so as to obtain as many preferable responses of the different classes as desired since variation of fractional order $ξ$ can be done at any preferable fractional rate 0.7, 0.4, 0.2 etc.

Diabetes is a chronic disease which many people suffer from seriously. This study introduces a novel approach called the Haar wavelet collocation method (HWCM) to study the analysis and numerical approximation of the fractional order diabetes mellitus model. We built the operational matrix of integration (OMI) using the Haar wavelet to solve the diabetes mellitus model, a system of fractional differential equations. First, we transform the diabetes mellitus model into a system of algebraic equations using the operational matrix of integration of the Haar wavelet. The obtained system is further considered using the Newton-Raphson technique to extract the unknown Haar coefficients. Here, we use the calculus of fractional derivatives of a mathematical model to study and investigate the dynamic behavior of diabetes. We find numerical results for the validation of fractional order derivatives. Using the model parameter values, these numerical results are seen from both mathematical and biological perspectives. Numerical tables and graphical representations provide a visual presentation of the obtained results. The results of the developed method, the RK4 method, and the ND solver solution are compared. The numerical results show how highly accurate and efficient HWCM is in solving the fractional order diabetes mellitus model. Further, we show the method’s efficacy and dynamics in various settings by performing simulations with parameter values. Mathematica, a mathematical software, has been utilized for numerical computations and implementation.

We consider a discrete nonlinear Schrödinger equation with long-range interactions and a memory effect on the infinite lattice $h\mathbb{Z}$ with mesh-size $h > 0.$ Such models are common in the study of charge and energy transport in biomolecules. Because the distance between base pairs is small, we consider the continuum limit: a sharp approximation of the system as $h → 0.$ In this limit, we prove that solutions to this discrete equation converge strongly in $L^2$ to the solution to a continuous NLS-type equation with a memory effect, and we compute the precise rate of convergence. In order to obtain these results, we generalize some recent ideas proposed by Hong and Yang in $L^2$-based spaces to classical functional settings in dispersive PDEs involving the smoothing effect and maximal function estimates, as originally introduced in the pioneering works of Kenig, Ponce and Vega. We believe that our approach may therefore be adapted to tackle continuum limits of more general dispersive equations.

In the present paper, we first give four post-quantum integrals for functions of two variables, denoted by $_{ac}T_{p,q},$ $^d_ aT_{p,q},$ $^b_cT_{p,q}$ and $^{bd}T_{p,q}.$ Afterwards, each of these newly defined integrals is illustrated. Moreover, some new Hermite-Hadamard inequalities are established based on these definitions. We also show the correctness of these inequalities with the aid of some numerical examples.

Monotonicity analysis is an important aspect of fractional mathematics. In this paper, we perform a monotonicity analysis for a generalized class of nabla discrete fractional proportional difference on the $h\mathbb{Z}$ scale of time. We first define the sums and differences of order $0 < α ≤ 1$ on the time scale for a general form of nabla fractional along with Riemann-Liouville $h$-fractional proportional sums and differences. We formulate the Caputo fractional proportional differences and present the relation between them and the fractional proportional differences. Afterward, we introduce and prove the monotonicity results for nabla and Caputo discrete $h$-fractional proportional differences. Finally, we provide two numerical examples to verify the theoretical results along with a proof for a new version of the fractional proportional difference of the mean value theorem on $h\mathbb{Z}$ as an application.

In this investigation, we introduce a novel approach for establishing Milne’s type inequalities in the context of quantum calculus for differentiable convex functions. First, we prove a quantum integral identity. We derive numerous new Milne’s rule inequalities for quantum differentiable convex functions. These inequalities are relevant in open Newton-Cotes formulas, as they facilitate the determination of bounds for Milne’s rule applicable to differentiable convex functions in both classical and $q$-calculus. In addition, we conduct a computational analysis of these inequalities for convex functions and provide mathematical examples to demonstrate the validity of the newly established results within the framework of $q$-calculus.

In this paper, we derive and analyze a semi-discrete Lasota-Waźewska model. First, the existence, uniqueness, and local dynamical properties of the positive fixed point are systematically investigated. Subsequently, we explore the existence of Neimark-Sacker bifurcation and the stability of the bifurcated invariant curve. Finally, numerical simulations are provided to illustrate the theoretical findings.

In our work, our objective is to prove the existence and uniqueness of a weak solution to a class of nonlinear degenerate elliptic $(p, q)$-Laplacian problem with Dirichlet-type boundary condition by giving $L^ ∞$ data. The principal technique utilized here is the variational method alongside the theory of Orlicz spaces and Minty-Browder theorem.

In recent decades, various methodologies have been proposed to model the complexities of challenging global problems across different domains. One such challenge involves understanding multi-step behaviors observed in certain situations. Newly proposed piecewise derivatives are known to address these issues. This study utilizes a mathematical model to examine the spread of a social issue of divorce among married couples resulting from extramarital affairs, using piecewise derivatives. Initially, we develop the model with Caputo fractional derivative and conduct some basic mathematical computations. Furthermore, the model is explored within the framework of the piecewise operator, incorporating both classical and Caputo operators. Within this framework, the study presents the existence and uniqueness of the solution using the fixed-point results. To analyze the behavior of the considered model, the Newton polynomial interpolation method is employed. The findings are subsequently illustrated through graphical representations, considering various values of fractional order.

In this work, we consider a class of eigenvalue elliptic systems involving the fractional $(p(x, .), q(x, .))$-Laplacian operators. Our main tools are based on Mountain Pass Theorem and Fountain Theorem.

This paper deals with the existence, asymptotics and computation of solutions of nonlinear Sturm-Liouville problems with general separated boundary conditions. The approach centers first on converting these problems into Hammerstein integral equations with modified argument, and then applying the Banach and Rothe fixed point theorems to solve them. This approach not only enabled us to prove existence theorems for these problems, but also to derive general and accurate asymptotic formulae for their solutions. Finally, an illustrative numerical example is presented using the Picard’s iteration method.

Acetaminophen (N-acetyl-para-aminophenol) is the most widely used painkiller in the world. Consuming acetaminophen involves a complex metabolic system, since it is converted into non-toxic and toxic metabolites called N-acetyl-p-benzoquinone imine (NAPQI). Acetaminophen is metabolized through a series of complex metabolic processes in the liver that involve enzymes as catalysts. This study presents a metabolic analysis of acetaminophen metabolism and its optimal regulation to reduce toxin formation. The metabolic study uses kinetic modeling combined with metabolic control analysis to identify key enzymes that can be modified to reduce hepatotoxicity due to excessive drug consumption. The sensitivity result shows that cytochrome (CYP) and uridine 5’-diphosphate-glucuronosyltransferase (UGT) are the two key enzymes that can be considered as internal control parameters. By inhibiting the reaction rate of CYP and UGT, the formation of N-acetyl-p-benzoquinone imine (NAPQI) can be reduced up to 69.9%, and the formation of Acetaminophen-Glucuronide (APAP-G) can be increased by approximately 0.49% such that Acetaminophen in the liver (APAP-L) can be excreted directly into the urine. Increasing the concentration of antioxidant GSH can also prevent hepatotoxicity by forming the complex NAPQI-GSH so that hepatotoxicity due to overconsumption of acetaminophen can be reduced.

In this paper, using the Mountain Pass Theorem, we present results on compactness and the existence of solutions for a class of local and nonlocal $p$-Laplacian equations involving Robin boundary conditions, with critical nonlinearity and a small perturbation.

By applying the Mountain Pass Theorem, we establish the existence of a weak solution for a class of nonlinear elliptic problem involving an $α(z)$-biharmonic operator and with an $l(z)$-hardy term in a bounded domain of $\mathbb{R}^N.$ Provided that certain additional assumptions are made regarding the nonlinearities, the corresponding functional will satisfy the Palais-Smale condition.

In this paper, we establish the existence, uniqueness and construct fourth-order numerical method for solving fully third-order nonlinear differential equation with integral boundary conditions. The method is based on the discretization of an iterative method on continuous level with the use of the trapezoidal quadrature formulas with corrections. Some examples demonstrate the applicability of the theoretical results of existence and uniqueness of solution and the fourth-order convergence of the proposed numerical method. The approach used for the third-order nonlinear differential equation with integral boundary conditions can be applied to differential equations of any order.

In this study, we propose a modified fractional order derivative in the scope of Caputo to investigate the dynamics of Dengue fever transmission. The existence, uniqueness and boundedness of the fractional model were established using a fixed-point approach. The stability analysis of the model was done with respect to the reproduction number which was found to be stable locally and globally at infection free and endemic state respectively. The fractional order (DHF) model was estimated using the fractional Adams-Bashforth predictor-corrector technique. Additionally, the numerical validation was done to ascertain the impact of various parameters on the dynamics as a whole, as well as the effect of vaccines on the model. The graphical solutions show that the fractional order $(α)$ and vaccinations affect the dynamics of the model when they are varied within the model. The findings indicate that saturation of infectious individuals in the system helps to flatten out the infection transmission.

This study introduces an advanced analytical technique for solving the fourth-order Extended Fisher-Kolmogorov (EFK) equation, focusing on the application of Adomian decomposition methods (ADM) and modified Adomian decomposition methods (MADM) . The research outlines a systematic approach to deriving numerical solutions, facilitating the characterization and understanding of complex dynamics associated with the EFK equation. Additionally, the technique is generalized to higher-order extensions, enhancing its applicability in modeling various physical phenomena. The results illustrate the effectiveness of the proposed methods in achieving accurate solutions while addressing challenges inherent in higher-order differential equations.

Alzheimer’s disease (AD) is a progressive neurodegenerative disorder characterized by the accumulation of amyloid-beta $({\rm A}\beta)$ plaques and tau neurofibrillary tangles (NFTs), as well as by chronic neuroinflammation and blood-brain barrier (BBB) dysfunction. Although these pathological features are well known, their complex interactions remain poorly understood. This study proposes a comprehensive multiscale coupled reaction-diffusion model composed of 13 partial differential equations to simulate the spatio-temporal dynamics of ${\rm A}β$ and tau pathology, neuroinflammatory responses, BBB integrity, and neuronal degeneration. The model captures biochemical reaction kinetics, diffusion-driven propagation, and regulatory feedback among key cellular components, including microglia, astrocytes, and cytokines. Furthermore, the effects of therapeutic interventions, such as anti-amyloid drugs and dietary modifications, are incorporated to assess their influence on disease progression. Numerical simulations using finite difference methods provide insights into how these factors contribute to or mitigate AD pathogenesis. The results support the potential of mathematical modeling as a tool to understand disease mechanisms and evaluate treatment strategies.

This study focuses on examining the controllability of nonlinear fractional differential systems in finite-dimensional spaces, considering multiple delays in both state and control. For linear systems, the necessary and sufficient conditions for relative controllability are established through the definition and application of the Gramian matrix. For nonlinear systems, controllability conditions are derived using Schauder’s fixed point theorem.

The aim of this paper is to characterize the simplest three-dimensional polynomial differential system having an equilibrium and a 2-dimensional orientable smooth compact manifold with genus $g ≤ 1$ in $\mathbb{R}^3 ,$ where the 2-dimensional orientable smooth compact manifold is sphere $\mathbb{E}^ 2$ or torus $\mathbb{T}^2 $ We first look for the smallest degree of polynomial differential systems with both an equilibrium and an isolated compact invariant algebraic surface $\mathbb{E}^2$ or $\mathbb{T}^2.$ It is shown that the smallest degree of the system depends on the relative position between the equilibrium and the compact invariant algebraic surface in $\mathbb{R}^3.$ Furthermore, the sufficient and necessary algebraic conditions are given for the smallest order three-dimensional polynomial differential system having both an equilibrium and an isolated compact invariant algebraic surface. Lastly, we discuss the influence of the coexistence of an isolated compact invariant algebraic surface and an equilibrium on dynamics of the three-dimensional polynomial differential system.