Journal of Nonlinear Modeling and Analysis
The Journal of Nonlinear Modeling and Analysis (JNMA), started in 2019, aims to publish original research papers, communications as well as survey papers on theory and application of nonlinear analysis related to differential equations, dynamical systems and other areas in mathematics and mathematical sciences, including mathematical biology. The journal was published quarterly in March, June, September and December between 2019 and 2024. From 2025 on, it will be published bimonthly with its issues released in February, April, June, August, October and December.
The Second-Order Melnikov Function and Chaotic Tangles on Toral van der Pol Equation
The van der Pol equation on the torus is considered. This equation contains a heteroclinic cycle consisting of four symmetric heteroclinic orbits. By high order Melnikov method, the periodic forced van der Pol equation is investigated and chaotic dynamics is obtained. An explicit formula of the second-order Melnikov function is derived for the splitting heteroclinic connection. It is used to detect chaotic dynamics when the first-order(classical) Melnikov function is degenerate. By the second-order Melnikov function, we deduce chaotic heteroclinic tangles with rigorously theoretical analysis as well as numerical simulations.
Copyright (c) 2025 Journal of Nonlinear Modeling and Analysis
Application of the Functional Variable Method to Some Nonlinear Evolution Equations
The functional variable method is a highly effective approach for deriving exact solutions to nonlinear evolution equations. It offers broad applicability in addressing nonlinear wave equations. In this paper, the functional variable method is employed to obtain soliton solutions for the Radhakrishnan-Kundu-Lakshmanan (RKL) equation and the Landau-Ginzburg-Higgs equation. Exact solutions play a crucial role in uncovering the internal mechanisms of physical phenomena. Graphical representations of the obtained optical soliton solutions are provided to illustrate some of their physical parameters.
Bifurcation Analysis of Predator-Prey Mathematical Model with Resource-Limited Growth and Predator Switching Behavior
Ecology examines the interactions between organisms and their environment, with a particular focus on population dynamics, resource availability, and predator-prey relationships. This study presents a mathematical model designed to investigate the interactions between two prey populations, one in an unprotected region and the other in a protected region, along with a predator population and shared resource availability. The model employs nonlinear differential equations to capture processes such as prey growth, predation, and resource utilization. By identifying equilibrium points and conducting eigenvalue analysis, we assess the system’s stability. Numerical simulations demonstrate a range of outcomes, including stable states, cyclic behavior, and population collapse, depending on ecological conditions (parametric values). Biologically, predator-prey coexistence equilibria may lose stability, shifting to extinction or dominance scenarios. This makes bifurcation point. Bifurcation analysis reveals how competition and predation impact stability, with critical points marking transitions between coexistence, oscillations, or extinction. These results underscore the intricate balance of ecological forces and emphasize the significance of resource management and conservation in preserving ecosystem stability.
A Semi-Analytical Study of a Non-Linear Initial Value Problem for the Lassa Fever Model
Mathematical modelling plays a crucial role in comprehending and exercising control over Lassa fever outbreaks. The Homotopy Analysis Method (HAM) is employed to address the system of nonlinear differential equations and generate a semi-analytical solution for the Lassa disease Susceptible-Exposed-Infected-Recovered (SEIR) model. Comparing with numerical simulations, the HAM technique yielded extremely precise approximate solutions, proving its accuracy and efficiency. The numerical simulation is carried out using MATLAB programming. There is a good agreement between the numerical simulation and the approximate analytical results. We also examine the effects of changing model parameters on the different compartments, obtaining insightful knowledge on how the model behaves in various scenarios. This analysis is crucial for understanding how variations in transmission rates, recovery rates, and other key factors affect the overall dynamics of Lassa fever, thereby guiding more effective public health strategies and disease management. Moreover, it underscores the potential of the Homotopy Analysis Method (HAM) as a powerful tool for exploring epidemic models and formulating control strategies.
Fractional Order Solution to Analyze Tumor-Immune Cells by Using Hybrid Natural Transform Method and Residual Power Series Method
The purpose of this study is to examine the impact of fractional-order chemotherapy drug diffusion on tumor-immune cell growth. To address the proposed system of fractional-order tumor-immune cell dynamics, two distinct and robust methods have been employed. Initially, the Hybrid Natural Decomposition method, an analytical approach that merges two effective techniques – Natural Transform and the Adomian Decomposition Method has been applied. Subsequently, the Residual Power Series method, a numerical technique, has been employed to derive the solution in series form. The graphical comparison of the results obtained from both methods with those from classical-order solutions has been provided. The findings clearly demonstrate that the proposed approaches offer a reliable and accurate means of understanding the intricate interactions between tumor growth and immune cell activity.
Properties Around the First Eigenvalue of a Partial Discrete Dirichlet Boundary Value Problem
In this article, we study some results around the first eigenvalue $\lambda_{1,m}$ for the partial discrete Dirichlet boundary value problem, like the constant sign of the eigenfunction associated with $\lambda_{1,m}$, the simplicity of $\lambda_{1,m}$ and the sign change of any eigenfunction associated with $\lambda > \lambda_{1,m}$.
Exact Soliton of Fifth Order (1+1) Dimensional Triple Non-Linear Partial Differential Equations on Modified Truncated Expansion Methods
In this manuscript, modified truncated expansion method-I taking the traveling wave variable $u(x, t)=y(r)=y(kx+wt)$ and modified truncated expansion method-II introducing the traveling wave variable $u(x, t) = y(r)=y(kx − wt)$ are built up to obtain analytical solution in the form of traveling wave solutions with different frequencies and velocities that can be constructed for triple (1+1) dimensional nonlinear partial differential equations (NLPDEs) such as Sawada-Kotera equation (SKE), generalized Korteweg-de Vries equation (GKdVE) and Kaup-Kuperschmidt equation (KKE), which have been widely used in mathematical physics. The present topic minimizes the complex nature and non-integrable characteristics to obtain solutions of NLPDEs. To demonstrate the influence of the parameters, 3D plots are generated for triple NLPDEs. This content is employed in physics such as magneto sound in plasma and nonlinear optics.
triple nonlinear partial differential equations
Dynamical Properties of a Stochastic Tumor-Immune Model with Pulsed Chemotherapeutic Dose Response
Comprehensive pulsed chemotherapy and immunotherapy are widely employed in clinical tumor treatment. Given the periodically pulsed nature of this approach, we propose a stochastic tumor-immune dynamical model with a pulsed chemotherapeutic dose response. The model accounts for the combined effects of pulsed chemotherapy and pulsed immunotherapy, as well as the influence of environmental random disturbances. We prove the existence and uniqueness of a global positive solution to the proposed model. By using comparison theorems for impulsive differential equations, we show the boundedness of the solution’s expectation. Furthermore, we derive sufficient conditions for the extinction and non-persistence in the mean of tumor cells, hunting T-cells, and helper T-cells, as well as for the weak persistence in the mean of tumor cells and helper T-cells, and the stochastic persistence of tumor cells. The results of our study, supported by numerical simulations, demonstrate that random disturbances can effectively inhibit tumor cell growth.
impulsive stochastic differential equation
Three Solutions for a Perturbed Integral Equation with Homogeneous Dirichlet Condition
We consider the integral equation $-\mathcal{L}_K^p u = \lambda f(x,u) + \mu g(x,u),$ with homogeneous Dirichlet condition on a bounded Lipschitz domain of $\mathbb{R}^N$ where $\lambda, \mu \in \mathbb{R},$ $p \geq 2,$ $s \in ]0,1[,$ $N > ps,$ $f,g : \mathbb{R}^N \rightarrow \mathbb{R}$ are Carathéodory functions with subcritical growth and $-\mathcal{L}^p$ denotes a class of operators that includes $(-\Delta)_p^s,$ the fractional $p$-Laplacian. Here $\mu g$ represents a small perturbation of $\lambda f.$ Applying an abstract critical point theorem due to Ricceri, a variational setting developed by Xiang et al. and a Minti-Browder's theorem, we prove the existence of three weak solutions.
On Dynamics of Certain Models via Generalized Conformable Fractional Derivative
This paper deals with the generalized conformable fractional derivative and certain interesting properties which are not compatible with Riemann-Liouville and Caputo fractional derivatives. The newly defined derivative is more efficient than other conformable fractional derivatives and the nonlocal fractional derivatives from a time perspective. To justify the claim, we provide some direct applications, such as population growth, Newton’s body cooling, heat equation and susceptible-infected-removed models. Solutions obtained from models and comparison with respective previous data are demonstrated with the help of graphs or stems.
Minimizing the Eigenvalue Ratio for the $p$-Laplacian Operator
We focus on the minimization problem of the eigenvalue ratio for the $p$-Laplacian operator with Robin boundary conditions on an interval $[0, \hat{π}],$ where $\hat{\pi} = \frac{2\pi}{p \sin(\pi/p)}.$ Using variational techniques and Prüfer-type transformations, we show that the constant weight is not minimizing for the class of concave weights.
A New Study on Common Fixed Points for ($α$-$β$-$F$)-Contraction Mappings
This paper explores the existence of common fixed points for a newly introduced class of mappings called ($α$-$β$-$F$)-contraction mappings in metric spaces. We establish several common fixed point theorems under specific conditions, incorporating ($α$-$β$-$F$)-weak contractions to extend and generalize existing results in fixed point theory. Unlike many classical approaches, our framework retains flexibility while accommodating key structural properties such as compactness and continuity. To support the theoretical findings, illustrative examples are provided. These results enhance the understanding and applicability of fixed point theory.
Bifurcation Analysis of a Discrete Predator-Prey Model with Gompertz Growth and Increased Functional Response
This study examines a discrete predator-prey model that employs a Gompertz growth function for the prey and a Holling type I functional response. Initially, the research explores the existence and local stability of fixed points within the system, employing a fundamental lemma. Subsequently, the conditions necessary for the emergence of transcritical and Neimark-Sacker bifurcations of the system are established through the application of the center manifold theorem and bifurcation theory. Finally, numerical simulations are performed to confirm the existence of the Neimark-Sacker bifurcation.
Spatiotemporal Dynamics of a Memory-Diffusion Predator-Prey System with Two Delays and Nonlocal Competition
Predator gestation delay and memory period, along with nonlocal competition, play key roles in controlling population density and maintaining the stability of ecosystems. To control the population density of the Dendrolimus superans, which causes significant damage to forests, we propose a pest control system incorporating both two delays and nonlocal competition considering Holling II type functional response. We analyze the conditions for the existence of Hopf bifurcation and derive the normal form of Hopf bifurcation using multiple time scales method. Considering the biological significance, we employ appropriate parameters for numerical simulations. Furthermore, we find that varying habitat complexity parameters leads to different moduli of bifurcation periodic solutions. Especially, both two delays contribute to maintaining the stability of steady states.
A New Approach to Hermite-Hadamard-Type Inequality with Proportional Caputo-Hybrid Operator
Fractional calculus plays a crucial role in mathematics and applied sciences as it extends classical analysis, overcoming many of its constraints. Moreover, using the innovative hybrid fractional operator, which merges the proportional and Caputo operators, is beneficial in numerous domains of computer science and mathematics. In this study, we focus on the proportional Caputo-hybrid operator due to its wide range of applications. Firstly, we present a new extension of Hermite-Hadamard-type inequalities for the proportional Caputo-hybrid operator and derive an identity. Then, utilizing this novel generalized identity, we establish significant integral inequalities associated with the right-hand side of Hermite-Hadamard-type inequalities for the proportional Caputo-hybrid operator. Furthermore, we provide illustrative examples accompanied by the graphs to demonstrate the newly established inequalities.
Some Results on Fractional Corrected Euler-Maclaurin-Type Inequalities Related to Various Function Classes
In this paper, some corrected Euler-Maclaurin-type inequalities are established by using various function classes involving Riemann-Liouville fractional integrals. We then present our findings using examples and special cases of the theorems that we have discovered. Moreover, we provide several fractional corrected Euler-Maclaurin-type for bounded functions. Additionally, for Lipschitzian functions, we create a few fractional corrected Euler-Maclaurin-type inequalities. Lastly, we provide some fractional corrected Euler-Maclaurin-type inequalities for functions of bounded variation.
A Global Post Effects of COVID-19: A Mathematical Modelling Study
This study presents the global post-effects of COVID-19 through a mathematical modeling approach. A compartmental model, dividing the total population into six epidemiological compartments, is developed to simulate the dynamics of the disease. These compartments include susceptible humans (S), exposed humans (E), infected humans (I), deceased humans (D), individuals in treatment class (T), and recovered humans (R). The study extensively discusses the post-effects of COVID-19 in Africa and the sensitivity analysis reveals that the contact rate of infection exhibits a positive sensitivity index, indicating that interventions aimed at reducing contact rate $ϕ$ would significantly diminish the spread of the virus within the population. Conversely, the treatment rate $ω$ shows a negative sensitivity index, suggesting that promoting higher treatment rates would lead to an increased recovery rate, thereby effectively controlling the spread of the disease. Numerical simulations carried out using MATLAB further confirms that a high treatment rate, avoidance of contact rate with any infected person or infectious surface, coupled with adherence to COVID-19 control measures, could help prevent the spread and outbreaks of COVID-19 in the future. The implications of these findings extend to healthcare workers, policymakers, and the general public, offering valuable insights into disease transmission dynamics and informing preparedness for future pandemics. Overall, this study considers the importance of proactive measures and effective healthcare interventions in mitigating the impact of COVID-19 and preventing the resurgence of infectious diseases in Africa.
Mathematical Analysis of COVID-19 Fractional Model Incorporating Vaccination, Quarantine and Isolation Measures
This study investigates the role of vaccination, quarantine, and isolation in controlling the spread of COVID-19 using a fractional-order mathematical model. The model consists of six non-linear fractional-order differential equations in the Caputo sense. Stability analysis is conducted using the Ulam-Hyers and modified Ulam-Hyers criteria, and the existence and uniqueness of solutions are explored with the Schauder and Banach fixed-point theorems. The model’s dynamical behavior is analyzed using the fractional Euler method. Dimensional consistency is maintained during the fractionalization process, distinguishing this study from many contemporary investigations. The results show that increasing vaccination rates, improving quarantine protocols, and enhancing isolation facilities are effective strategies for reducing COVID-19 transmission.
Numerical Approximation of the System of Fractional Differential Equations Using the Fibonacci Wavelet Collocation Method
This study uses the Fibonacci wavelet collocation method (FWCM) to solve the system of fractional differential equations. Here, we introduce the innovative Fibonacci wavelet method with the help of an operational matrix of integration generated by the Fibonacci polynomials to compute the approximate solution of the linear and nonlinear fractional differential equations. The Fibonacci wavelet collocation method, initially developed for a system of differential equations of integer order, can approximate the solutions to systems of fractional differential equations of fractional order. This method converts the system of fractional differential equations into a system of algebraic equations. These algebraic equations are then solved using the Newton-Raphson method, and the estimated values of the coefficients are then substituted in the approximation. Numerical outcomes are obtained to illustrate the simplicity and effectiveness of the proposed scheme. The numerical results demonstrate that the method is simple to use and precise. The effectiveness and consistency of the developed strategy’s performance are shown in graphs and tables. The method introduces a potential technique for resolving several linear and nonlinear fractional differential equations. Mathematical software called Mathematica has been used to perform all calculations.
Fractional differential equations (FDEs)
A Multiscale Coupled Reaction-Diffusion Model of Amyloid-Beta and Tau Pathology in Alzheimer’s Disease
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.
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An Analytical Technique on Numerical Solutions for EFKs of Fourth Order and Higher
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.
Three-Dimensional Polynomial Differential Systems with an Isolated Compact Invariant Algebraic Surface
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.
Controllability of Nonlinear Fractional Systems with Multiple State and Control Delays
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.
On Local and Nonlocal Robin Boundary Value Problem with Critical Nonlinearity
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.
Existence, Uniqueness and Fourth-Order Numerical Method for Solving Fully Third-Order Nonlinear ODE with Integral Boundary Conditions
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.
On a Class of Nonlinear Elliptic Problems Involving the $α(z)$-Biharmonic Operator with an $l(z)$-Hardy Term
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.
Fractional Order Dynamical Behavior of Dengue Hemorrhagic Fever with Saturation Factor
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.
Optimal Control Strategies for Reducing Toxic Formation in Acetaminophen Metabolism
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.
On a Class of Eigenvalue Elliptic Systems in Fractional Sobolev Spaces with Variable Exponents
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.
Modeling Divorce Dynamics Due to Extramarital Affairs with Piecewise Classical-Fractional Operator
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.