Our paper introduces a mathematical governance model for digital regulation in Africa by extending the previous G&G model to capture digital propagation effects on social behavior. The model derives from a coupled pseudohyperbolic fractional-type equations with memory and stochastic noise, reduced via a latent potential Φ to link temporal connectivity and spatial dynamics. Analytical and numerical methods, including Hamilton–Jacobi–Bellman theory, show that multiplicative noise and entropy significantly shape system behavior and optimal strategies. The results provide an operational framework for adaptive, resilient digital governance by African Multistakeholder Groups.
This paper develops a noncommensurate time-fractional SEIQR reaction–diffusion model for infectious disease dynamics with artificial intelligence (AI)–assisted inverse identification and forecasting. The susceptible, exposed, infected, quarantined, and recovered classes are governed by distinct Caputo fractional orders so that latency, infection progression, quarantine response, and recovery may carry different memory intensities, whereas spatial mobility is described by classical Laplacian diffusion. A positivity-preserving fractional nonstandard finite difference scheme is constructed by combining the L1 approximation of the Caputo derivative with an implicit Laplacian and nonlocal discretizations of nonlinear incidence terms. The scheme preserves nonnegative population densities and gives the forward reference solution with consistency order Ok2−ϑmax+h2. The qualitative structure of the model is discussed through positivity, finite-horizon admissibility, the disease-free equilibrium, the basic reproduction number, and noncommensurate fractional stability. The AI component is used as a data-assimilation and surrogate layer rather than as a replacement for the structure-preserving solver: it identifies epidemiological parameters, diffusion coefficients, and fractional orders and provides fast repeated forecasts once trained on NSFD-generated reference solutions. Numerical experiments demonstrate compartment-dependent memory effects, positivity preservation against a standard explicit scheme, recovery of model parameters and fractional orders, manufactured-solution verification, and neural-operator speedup for repeated forecasting.
The Caputo fractional derivative is employed to investigate the role of memory effects in modeling the transmission dynamics of recurrent malaria in interacting human–vector populations. The model generalizes an existing classical version accounting for the three major forms of recurrent malaria, including re-infection, recrudescence, and relapse. The formulated noninteger-order nine-dimensional system is shown to be biologically and mathematically well-posed through the generalized mean value theorem, while the existence and uniqueness of the model solutions are established with the help of Banach fixed point theory. The Ulam–Hyers stability approach is adopted to investigate the stability of the unique solution of the fractional-order model. In particular, the study investigates the fractional-order optimal control by incorporating four time-varying intervention strategies such as the use of insecticide-treated bednets (ITNs), antirelapse intervention, effective treatment effort, and mosquito reduction measures using indoor residual spraying. To examine the significance of different control combinations in stemming the problem of malaria recurrence, the time-varying control interventions are further re-classified into four intervention strategies combining any three of the control measures. The influence of implementing each of the intervention strategies with and without memory effects is examined on the subpopulations of the fractional-order recurrent malaria model. The findings show that optimizing intervention measures with memory significantly minimizes recurrent malaria transmission in the population.
This paper develops a Laplace–Residual Power Series Method (L-RPSM) for solving nonlinear time-fractional wave systems in the Caputo sense. The method combines the Laplace transform with a residual power series expansion to construct approximate analytical solutions, where the coefficients are determined recursively through residual conditions. The approach is applied to two benchmark models: the modified Boussinesq system and the approximate long-wave system. The influence of the fractional parameter is examined for different values of α, and convergence behavior is discussed to justify the validity of the method. Numerical results show that the proposed method produces accurate approximations with small absolute errors and stable convergence. Comparisons with established techniques such as ADM, VIM, OHAM, and NIM demonstrate the competitiveness and reliability of the proposed framework for nonlinear fractional wave equations.
Potato plant diseases are a significant concern because they reduce crop yields and quality, putting the global food supply at risk. Our goal in developing this deterministic, fractional-order epidemic model is to explain the transmission of these diseases by simulating the interactions between susceptible, exposed, infected, treated, and recovered potato plants. The deterministic model captures the mainstream dynamics of epidemics. At the same time, the fractional formulation based on the Atangana–Baleanu–Caputo (ABC) derivative represents the memory-dependent and nonlocal effects typical of interactions between plants and pathogens. We find that the fundamental reproduction number can be derived analytically, that there is an endemic and disease-free equilibrium, that solutions are positive and bounded, and that stability can be evaluated locally and globally using Jacobian eigenvalues and Lyapunov functions. Bifurcation theory reveals a forward bifurcation, and game theory is applied to identify a Nash equilibrium strategy for disease control. Transmission rates and disease progression are the two most important variables impacting epidemic outcomes, as shown by a sensitivity analysis. Bifurcation diagrams and Lyapunov exponents show that oscillatory or chaotic behavior can undergo transitions in numerical simulations of fractional dynamics. The fractional-order dynamics model outperforms the deterministic model in terms of persistence, decay rate, and richness of transient behavior. The results show that memory-driven fractional models can help develop agricultural intervention strategies by more accurately depicting the spread of potato diseases.
We propose a nonlinear size-structured fishery model for externally recruited stocks under size-selective harvesting. The population density satisfies a McKendrick–von Foerster transport equation in which growth and natural mortality depend on a nonlocal crowding index, while harvesting acts as a bounded size-dependent mortality control. Unlike standard self-recruiting formulations, recruitment is prescribed as a lower-boundary inflow, making the model suitable for enhancement fisheries or analyses conditional on juvenile input. For the no-harvest baseline, we derive the stationary size profile and reduce the nonlinear equilibrium problem to a scalar closure equation, proving existence and uniqueness under a net monotonicity condition. We introduce an intrinsic replacement index and show why, in this externally forced setting, it is a viability diagnostic rather than a persistence threshold. A formal state–adjoint system yields a bang-bang switching rule; under weak coupling and single crossing, the optimal policy has a threshold structure. Numerical experiments validate the approximation and sensitivity trends.
This work investigates the existence of infinitely many nontrivial weak solutions to a boundary value problem involving the right-sided and left-sided ψ-Hilfer fractional derivative and ψ-Riemann–Liouville fractional integral operators. Using variational methods and critical point theory, we obtain new existence criteria for solutions of the given problem. Our approach is applicable to a variety of nonlinearities and boundary conditions. The results obtained in this article are novel in the given configuration and enrich the literature on the topic.
This study examines precise analytical traveling wave solutions of two significant nonlinear space-time fractional partial differential equations, the fractional generalized reaction Duffing model and the fractional Fisher equation. The fractional derivatives are defined using Jumarie’s modified Riemann–Liouville operator, which facilitates the inclusion of memory and nonlocal effects in the governing equations. Utilizing an appropriate fractional wave transformation and the generalized Bernoulli equation method, several categories of closed-form solutions are obtained in terms of hyperbolic, exponential, and rational functions. The impact of the fractional-order parameter on wave dynamics is analyzed by graphical and parametric methods, in addition to the development of precise solutions. The findings indicate that alterations in the fractional order substantially influence wave shape, resulting in transitions between kink, breather, soliton, peakon, and multi-peakon structures. Specifically, peakon-type solutions arise from fractional nonlocality, demonstrating the fractional framework’s capacity to represent sharp and localized wave fronts that are lacking in classical integer-order models. To further validate the analytical findings, a numerical investigation is carried out using a FDM. The numerical solutions are compared with the corresponding exact solutions at selected grid points, and the absolute error is evaluated. The results demonstrate that the numerical approximations are in excellent agreement with the analytical solutions, with very small error magnitudes. The results indicate that fractional-order nonlinear models offer a more comprehensive characterization of wave propagation in complex media, with possible applications in nonlinear dynamical systems and biological processes. The suggested method provides a systematic and efficient framework for investigating analytical solutions and wave phenomena of nonlinear fractional evolution equations.
We consider a class of partial differential equations that arise from one-dimensional generalized square root problems, featuring drift and diffusion. In particular, we investigate models whereby a diffusion coefficient is described by a power law. Our central aim is to discover coordinate transformations that convert these equations to the heat equation. This is realized by obtaining a necessary and sufficient condition for the underlying drift function. As a direct consequence, we prove how to recover transition probability densities that satisfy a Cauchy problem involving the Dirac measure.
We investigate the existence and multiplicity of bifurcating solutions in a Hopfield-Cohen-Grossberg network with n identical components and time delays. Applying the equivariant degree method, we derive sufficient conditions guaranteeing periodic solutions and their multiplicities. Two examples are provided to exemplify the theoretical results.
Boger fluids have various applications in the scientific and engineering fields, including biomedical engineering, materials sciences, chemical process engineering, and other disciplines. The present study aims to examine the 2-dimensional, steady, incompressible circulation of Boger nanofluids across a curved Riga stretching sheet with the influence of radiation, viscous dissipation, thermophoretic particle deposition, and porous medium. The nonlinear partial differential equations are transformed into ordinary differential equations employing appropriate similarity variables. The Runge-Kutta-Fehlberg fourth-fifth order technique and the shooting scheme are employed to solve these equations. Graphical representations will be utilized to describe the effects of various nondimensional constraints on their respective profiles. The significant engineering coefficients are also analyzed. The results indicate that enhancing the solvent fraction parameter upsurges the velocity profile but significantly drops the velocity profile with increasing the value of relaxation time ratio. The radiation constraints will improve the temperature profile. The thermophoretic constraint decreases the concentration profile. Skin friction reduces with higher values of solid volume fraction and porosity parameter. The Sherwood number decreases with an increment in the values of the solid volume fraction and thermophoretic constraint.
In this work, using a spline-based discretization, we develop a computational approach for singularly perturbed Fredholm integro-differential equations. The scheme addresses the challenges of the singular perturbation parameter & varepsilon; through a tension and compression spline technique, coupled with Simpson's rule for quadrature approximations. We analyze the stability and convergence properties of the proposed algorithm. Through the computation of maximum absolute errors on varying mesh sizes, we demonstrate the method's effectiveness. Numerical results indicate that the scheme yields accurate solutions and exhibits a consistent rate of convergence for arbitrarily small values of & varepsilon;.
We consider the dynamics of a tumor therapy model using oncolytic viruses with time delay. The model is a two-dimensional system of ordinary differential equations with a delay term that represents the latent period required for viral replication following the infection of tumor cells. Our study starts with the derivation of the positive equilibrium point and analyzes its local stability in both the absence and presence of delay terms. Subsequently, by using Pontryagin’s criterion, we establish the necessary and sufficient conditions for the asymptotic stability of the equilibrium point under the delay term. The numerical bifurcation analysis identifies a transcritical bifurcation in the nondelayed case, whereas the analytical analysis reveals a Hopf bifurcation in the delayed case, leading to sustained oscillations in the tumor cell population. These results suggest that the success of the virotherapy is sensitive to the delay effects. It implies that we need to consider time-dependent dynamics to determine effective treatment strategies.
Microstructured solids exhibit complex wave propagation dynamics due to the interplay between nonlinearity, dispersion, dissipation, and higher-order spatiotemporal effects induced by their internal architecture. In this work, we investigate how these properties influence the propagation of hybrid solitary waves governed by a generalized strain-wave equation. The main objective is to identify physically admissible hybrid waveforms and to relate each solution family to the characteristic coefficients of the medium. To achieve this, we employ the implicit Bogning (iB) function method, which provides a unified analytical framework for constructing both hyperbolic and trigonometric wave solutions using a hybrid ansatz that combines pulse-like and kink-like components. The analysis reveals that the nonlinear coefficient plays a dominant role in controlling wave amplitude and localization, while the remaining coefficients modulate the waveform structure, including steepness, periodicity, and spatial extent. Several classes of solutions are obtained, including kink-type, antikink, compacton-like, traveling, periodic, and pulse-soliton profiles. For each admissible solution branch, a reduced waveguide equation is derived, allowing a direct physical interpretation of the governing balance between nonlinear and dispersive effects. The novelty of this work lies in the unified analytical treatment of hybrid solitary waves and in the explicit connection established between solution families and coefficient-resolved reduced models. These results provide new insight into wave propagation in engineered microstructured media and offer practical guidelines for designing waveguides with tunable nonlinear responses.
Transient chaos is a phenomenon in which chaotic dynamics persists for a finite time before transitioning to periodic or steady-state behavior. TS has profound implications across disciplines, from neuroscience to quantum physics and machine learning. Recent studies have highlighted its role in crisis-induced transitions, early-time entanglement growth in quantum system, and pathological neuronal activity. In this paper, we present a novel jerk model characterized by a single nonlinearity in the form of a Lambert function. This system exhibits a short-time chaotic transient followed by convergence to a regular regime. For the proposed system, the conditions under which it exhibits a zero-Hopf equilibrium at the origin are identified. Furthermore, it is shown that applying the first-order averaging theory leads to the emergence of a unique periodic solution bifurcating from this zero-Hopf equilibrium point. Finally, the study extends to fractional order cases both commensurate and incommensurate revealing that one equilibrium point exhibits nonchaotic behavior while another presents chaos. The Gru..nwald-Letnikov method is employed to compute Lyapunov exponents and visualize phase trajectories, confirming the complex fractional dynamics of the system.
This paper investigates two nonlinear reaction-diffusion systems: (i) a spatially extended competitive species model and (ii) the FitzHugh-Nagumo system, which serves as a canonical model of excitable media. For the first system, we derive exact closed-form solutions using the exponential function method; these solutions exhibit spatially periodic structures and reflect fundamental features of competitive and diffusion-driven pattern formation. To handle both systems numerically, particularly where analytical solutions are intractable, we propose an enhanced variant of the modified Adomian decomposition method (E-MADM), incorporating adaptive decomposition and convergence-accelerating modifications. The accuracy and robustness of E-MADM are rigorously assessed by benchmarking its results against (a) the exact analytical solution (where available) and (b) high-resolution numerical approximations obtained via a second-order finite difference method (FDM). This comparative study confirms that E-MADM not only preserves structural properties of the solution but also achieves significant gains in computational efficiency and stability over classical approaches. A genetic algorithm is employed to determine the optimal diffusion coefficient (mu = 0.2281) for the competitive model. The results demonstrate that MADM achieves higher accuracy than FDM while maintaining comparable computational efficiency, confirming its effectiveness for solving nonlinear reaction-diffusion systems, as supported by the comparative analysis with established methods presented in Table 1.
This study presents a modified Laplace transform homotopy perturbation method (MLT-HPM) for obtaining approximate solutions for fractional-order Bratu-type ordinary differential equations involving Caputo fractional derivatives. The proposed modification introduces a specific rule for selecting the initial solution, replacing the conventional random choice. This modification significantly improves the convergence rate, resulting in more accurate and computationally efficient approximate solutions. Several numerical examples of fractional Bratu-type differential equations are provided to demonstrate the accuracy, reliability, and effectiveness of the proposed approach. Comparative analyses reveal that MLT-HPM achieves high-precision results with substantially fewer iterations than the standard LT-HPM, underscoring its superior computational performance and practical applicability.
This work presents conditions under which the Volterra integral equation of the second kind admits a unique solution in the class of locally bounded second kappa-variation functions on [0, +infinity). Our approach relies on successive Picard iterations to obtain such a solution on a compact interval, and then to prolong it to [0, +infinity). We also identify criteria that guarantee the existence and uniqueness of solutions to an integro-differential equation with infinite delay, considered within the same class of functions.
In this paper, utilizing the concept of neutrosophic metric spaces proposed by Kirisci and Simsek, we establish a fixed point theorem of Edelstein type within this framework. A numerical example is provided to illustrate the validity of the main result. Furthermore, we present an application to a general SIR-type epidemic model. By employing the proposed fixed point theorem, we investigate the existence of solutions for this model, demonstrating the theorem’s utility in the mathematical analysis of dynamical systems.
In the present article, a new amplitude expansion-based homotopy perturbation method (AE-HPM) is used to study the nonlinear behavior of a damped oscillator. The traditional homotopy perturbation method is extended, considering a simple amplitude expansion to determine the solution and amplitude frequency relationship for the damped nonlinear system, which could not be solved by the traditional approach. The simplicity, efficiency, and validity of the present AE-HPM are verified by applying it to the cubic-quintic oscillator and pendulum equation with linear damping. The analytical results obtained by the present method show that the oscillation amplitude decays exponentially with the damping parameter, while the frequency response is very much influenced by the system's nonlinearity. The comparison of the results obtained by AE-HPM with numerical results and He's frequency formulation (HFF) solution shows that the present method is accurate, converges faster, and can be used in a wider range of problems, while the traditional HPM fails. Therefore, the proposed method serves as a simple and dependable analytical approach for analyzing nonlinear damped oscillatory systems.
