In this research paper, we utilize an analytical technique to investigate the behavior of the Drinfeld-Sokolov-Wilson equation of arbitrary order. The implemented technique is an adequate composition of the Kharrat-Toma transform and the q-homotopy analysis approach. Here, a regularized form of the Hilfer-Prabhakar derivative of arbitrary order is used to formulate the problem. The Drinfeld-Sokolov-Wilson equation of arbitrary order is utilized to model the dispersive water waves and plays a very significant role in fluid dynamics. The results of the discussed model are presented graphically to show the efficiency and reliability of the obtained results.

Although an Oleinik inequality is not expected for general heterogeneous scalar balance laws, there is a favorable case, when the source term g(x) is smooth and nondecreasing. In this setting, we are able to obtain an Oleinik-type inequality and, consequently, a smoothing effect.

This paper presents a new numerical method for simulating the dynamic behavior of the fractional-in-time Gierer-Meinhardt reaction-diffusion model with periodic boundary conditions. A recursive algorithm for binomial coefficients is introduced, avoiding numerical instabilities associated with Gamma functions. High-precision polynomial expansions and the short-memory principle are employed to enhance efficiency and accuracy. Numerical simulations reveal diverse pattern formation.

In this work, we offer the novel class of (p,q)-Hermite-Appell polynomials. Some attributes of this class are constructed, along with the generating function, series definition, (p,q)-derivative properties, (p,q)-integral representation, summation formulas, and determinate representation. Additionally, we consider a few components for the (p,q)-Hermite-Appell polynomials and infer certain elements of their traits. The generating function and series expansions of some classes of two-dimensional (p,q)-Hermite-Appell polynomials are provided. Moreover, we acquire a (p,q)-differential operator formula for (p,q)-Hermite-Appell polynomials. Finally, the Wolfram Mathematica software is used to plot the graphical diagrams of select components of (p,q)-Hermite- Appell, along with two-dimensional (p,q)-Hermite-Appell polynomials.

The Kuramoto-Velarde equation describes the spatio-temporal evolution of step morphology on crystal surfaces, as well as the dynamics of spinodal decomposition in phase-separating systems subjected to an external field. In this paper, we prove the well-posedness of the solutions for the initial-boundary value problem for this equation, under several possible boundary conditions.

This paper focused on an hp-version mixed discontinuous Galerkin method without penalty terms for the biharmonic equation with Navier boundary conditions. By introducing the auxiliary variable v = u_xx, we reduced the fourth-order problem to a second-order system and derived its penalty-free variational formulation. The analysis replaces the standard coercivity condition with a polynomial-degree-dependent inf-sup condition for the bilinear form B(·,·). While h-convergence rates for both u_h and v_h were optimal, the p-convergence exhibited contrasting behavior: suboptimal in L²- norm but optimal in the energy norm, regardless of the p² scaling in the inf-sup condition. Numerical results revealed that p-convergence order-doubling for boundary-aligned singularities significantly enhanced the efficacy for singular solutions. Furthermore, the method was shown to extend to nonlinear biharmonic equations, while the treatment of Dirichlet boundary conditions necessitated the introduction of penalty terms.

We perform a numerical study of autoencoder deep neural networks (DNNs) when the input and the output vectors have the same dimension. Our focus is on fixed points (FPs) arising in these DNNs. We show that the existence and the number of these FPs depend on the distribution of randomly initialized DNNs’ weight matrices. We first consider initialization with the identically and independently distributed (i.i.d.) light-tailed distributions of weights (e.g., Gaussian) and show existence of a single stable FP for a wide class of DNN architectures. In contrast, for heavy-tailed distributions (e.g., Cauchy), which typically appear after the training of DNNs, a number of stable FPs emerge. We observe an intriguing non-monotone dependence of the number of FPs on the DNN’s depth. Finally, we link our result for untrained DNNs to the trained ones by showing that a number of FPs emerge after training of DNNs with light-tailed initialization.

This paper investigates the dynamic behavior of a fractional-order reaction-diffusion system for vegetation pattern formation, incorporating interactions between plant biomass, soil water, and salt concentration. The model utilizes Grünwald–Letnikov fractional derivatives to capture memory effects and anomalous diffusion. A novel high-order numerical scheme is developed, featuring a high-accuracy, short-memory time discretization with a nine-point finite difference method in space to enhance stability. Bifurcation analysis is performed to determine equilibrium stability and identify Turing instability thresholds. Numerical simulations illustrate the emergence of diverse vegetation patterns, highlighting how different fractional orders influence spatiotemporal dynamics. Overall, the proposed framework provides an effective computational tool for analyzing complex fractional ecological systems.

This article is concerned with the reachable set estimation (RSE) for delayed memristive neural networks (MNNs). By exploiting the differential inclusion theory and inequality techniques, the RSE problem of MNNs was investigated. A memoryless adaptive controller was designed to realize that states of MNNs converge to a bounded region. Based on this result, an updated memoryless adaptive controller was designed, which further removed the restriction that the delay derivative must be less than 1, leading to a more general result. The new results were presented in the form of algebraic criteria, which were straightforward to verify. Ultimately, the effectiveness of the proposed criteria was demonstrated through two numerical simulations.

In this paper, we first study the complex center and complex isochronous center problems for a complex quartic polynomial differential system. More precisely, by calculating and decomposing the variety of the ideal generated by the singular point quantities (resp. complex period quantities), we obtain the necessary conditions for the resonant elementary equilibrium (resp. complex center) of the system to be a complex center (resp. a complex isochronous center). Using time-reversibility and the Darboux integrable theory, we rigorously prove that these conditions are also sufficient. Furthermore, when the coefficients and the variables of the system are complex conjugates, we not only derive the necessary and sufficient conditions for the center-type equilibrium to be both a center and an isochronous center, but we also give the parametric conditions under which the center-type equilibrium of the system becomes a weak focus of order 10. In this case, although the independence of the focal quantities is not satisfied, we still prove that there exist parametric conditions under which 10 limit cycles can bifurcate from this weak focus.

This work focused on the analysis of a nonlinear wave equation of Hartree-type that includes a distributed delay term, where the delay effects are governed with fractional conditions. Such a formulation allows the model to incorporate long-range memory effects and anomalous dissipation phenomena, which are characteristic of complex media. The model captures complex memory and nonlocal interaction effects that arise in various physical systems, such as quantum mechanics and nonlinear optics. In particular, the fractional delay mechanism provides a more accurate description of hereditary effects than classical integer-order delay models. We worked under a framework that allows for initial data with negative energy and imposed suitable assumptions on the kernel functions and nonlinear terms. Using energy methods and a concavity argument, we rigorously proved that the solution to the system cannot exist globally in time and must blow up in finite time. Compared with the classical Hartree wave equation without delay or fractional effects, our results show that the combined presence of distributed delay and fractional damping significantly enhances the instability mechanism.

This work proposes a high-order discontinuous Galerkin (DG) formulation employing generalized alternating numerical fluxes for approximating solutions to the distributed-order diffusion equation. Such models often arise in ultraslow diffusion, where solutions decay only logarithmically as t →∞. Utilizing the Grünwald–Letnikov scheme and the DG method, we construct a fully discrete numerical algorithm. Using a rigorous induction argument, we prove unconditional stability and convergence of the proposed scheme. A comprehensive set of computational experiments is presented to validate the efficacy and performance of the method.

In this study, we investigate two mathematical models, formulated using delay differential equations, to capture HIV-1 transmission dynamics. Both models incorporate CD4⁺T cells and macrophages as target cells and saturated non-linear infection transmission terms. The first model introduces a discrete time delay, while the second employs a distributed delay to reflect more realistic biological insights. Our theoretical analysis explores key properties such as the basic reproduction number, non-negativity, boundedness, and the existence of equilibria. The global stability analysis of the disease-free and endemic equilibrium are analysed using the LaSalle invariance principle by proposing Lyapunov functions. We conducted sensitivity analysis to find the significant parameters related to the infection dynamics. Numerical simulations are performed to validate our theoretical results and visualize the behavior of the system under different parametric conditions. This study shows the critical role of time delays in shaping HIV-1 infection dynamics. Incorporation of such delays into mathematical models are essential for accurately capturing the progression of the disease and proposing effective intervention strategies for HIV-1.

In this paper, we propose a linearly implicit scheme that preserves the local energy dissipation property for the generalized nonlinear wave equation. By employing the energy quadratization approach, we reformulated the original equation into an equivalent form. A semi-discrete structure-preserving system was constructed via finite difference discretization in space. Subsequently, we derived a fully discretized scheme using the Crank-Nicolson method combined with the extrapolation technique, ensuring the preservation of local energy dissipation law. Under appropriate boundary conditions, such as homogeneous Dirichlet or periodic boundary conditions, the proposed method also maintained the global energy dissipation law. Furthermore, the unique solvability, fast implementation, and convergence theorem of the discrete scheme were analyzed rigorously. Numerical experiments are presented to validate our theoretical results, demonstrating that the proposed scheme outperforms traditional methods in terms of stability and efficiency.