A predictive model for debris dispersion from suborbital vehicle breakups is developed in this study through the novel integration of the Spacecraft Collision Breakup Model (SCBM) with minimum-volume enclosing ellipsoid (MVEE) theory. The key innovation lies in the development of a computationally efficient algorithmic framework that captures nonlinear fragment dynamics (0.03 to $$-$$0.99m) under complex aerodynamic and atmospheric interactions, incorporating size-dependent drag and stochastic perturbations. Distinct behavioral regimes are identified: smaller fragments (0.03 to $$-$$0.1m) exhibit prolonged atmospheric residence, while larger fragments (0.1 to $$-$$0.99m) display rapid spatial divergence due to initial separation energies. Compared to conventional Monte Carlo or pure probabilistic approaches, the proposed hybrid model reduces collision probability errors by 32–41%, and establishes quantitative relationships between fragment size, dispersion patterns, and dynamically evolving risk fields. These advances provide critical insights into high-altitude breakup phenomena, significantly improving risk assessment capabilities for suborbital operations. The methodology is also applicable to re-entry analysis and space debris modeling, contributing to the understanding of nonlinear astrodynamical systems with explicit mathematical-computational linkages.

We consider in this article the numerical analysis of the Boussinesq equation. The numerical results are based on a finite element discretization in space and a finite difference scheme in time. Furthermore, we introduce a Physics-Informed Neural Network (PINN) and explore its use in solving the Boussinesq equation, comparing its performance with the traditional numerical approach.

The non-relativistic quantum theory has been interpreted causally by de Broglie, David Bohm, and others, where a quantum entity is viewed as a particle with a definite position and momentum. This interpretation contrasts with the Copenhagen orthodox interpretation, which conveys uncertainty through the commutator relationship between position and momentum operators. This is further exacerbated by the de Broglie-Bohm interpretation, which uses no operators corresponding to these observables. We reconcile these opposing viewpoints by introducing mathematical nonlinear transforms in Bohmian mechanics for the momentum observable. While these nonlinear operator-like transforms cannot be easily represented as matrices in the Hilbert space, we show that they can be utilized in the de Broglie-Bohm interpretations of quantum theory.

Compared to single-scroll chaotic systems, multi-scroll chaotic systems (MSCSs) exhibit enhanced pseudorandomness. This study introduces a novel five-dimensional (5D) Hamiltonian conservative chaotic system (HCCS), which achieves multi-directional multi-scroll conservative chaotic systems (MSCCSs) distributed along single, dual, and triple axes through the incorporation of piecewise functions. The system allows for precise control over the number and size of scrolls by adjusting the parameters of the piecewise functions. Additionally, scrolls with varying energy levels can be generated by altering the system’s initial conditions (ICs). It was observed that the system’s Maximum Lyapunov exponent (MLE) increases with the number of scrolls and decreases with the width of the scrolls. After the National Institute of Standards and Technology (NIST) validation, the pseudo-random sequences based on this system demonstrated superior performance, successfully passing all NIST test criteria. Furthermore, this research implemented the system’s hardware circuit using Field Programmable Gate Array (FPGA) technology and experimentally verified the system’s physical feasibility.

This study introduces a mathematical model for disease dynamics, incorporating a non-linear incidence rate and time delay to analyze infectious disease transmission and recovery processes. Local stability is examined using the Routh-Hurwitz criterion, while global stability is established using Lyapunov functionals and the LaSalle invariance principle. The phase portraits and domain of attraction plots illustrate system dynamics, including equilibrium states, stability, intervention effects, and long-term predictions. The sensitivity analysis of $$\textbf{R}_0$$ with respect to the natural death rate (d) and disease-induced death rate ( $$d_1$$ ) provides a significant contribution to their relative impact on the transmission potential of the disease. Numerical simulations support the theoretical results, providing a comprehensive and robust mathematical framework for the analysis of epidemic dynamics, with valuable insights into the stability and control of infectious diseases.

We consider the dynamical behavior of fermionic instanton solutions of the Thirring Model with two forcing terms. In particular, the study focuses on the effect of the frequency and amplitude of the forcing terms on the behavior of fermionic instanton solutions. Numerical analysis based on the Smaller Alignment Index (SALI) method, Bifurcation Diagram and Permutation Entropy (PE) are used to show how and which dynamical behaviors occur in the system. Color maps and diagrams of the SALI time (St) (the time St required for the SALI index to below a threshold value of 10−12) and PE of the system with respect to varying of frequency and amplitude values of the forcing terms are plotted comparatively to determine different dynamical behavior of the system. The study shows that fermionic instanton solutions exhibit a wide variety of dynamical behavior due to two forcing terms. Furthermore, it is emphasized that SALI time (St) can be easily compared with bifurcation diagram and complexity method as a fast, efficient and precise method to investigate different degrees of chaos. In general, the instanton solutions with two forcing terms have been observed to exhibit different type of dynamical behavior. In this study, identical and symmetric coexisting attractors are demonstrated for these different types of behavior of the forced Thirring instanton.

This article is concerned with the numerical approximation of the Schnakenberg model under the influence of time white noise. This model consists of nonlinear stochastic differential equations of Itô type and due to randomness, these equations are rarely solved, exactly. The Schnakenberg models are used in various fields of life such as pattern formation in skin analysis, embryogenesis, and morphogen. Schauder’s fixed point theorem is used to ensure that a solution exists. The proposed stochastic weighted average scheme and nonstandard finite difference scheme are developed to deal with the numerically underlying model. The stability of the given schemes is shown by the von Neumann criteria. The proposed stochastic Weighted Average finite difference scheme is stable for $$\lambda \ge \frac{1}{2}$$ and von Neumann stable for $$\lambda < \frac{1}{2}$$ . The proposed stochastic finite difference scheme is von Neumann stable. In the mean square sense, the consistency of the proposed schemes is proved. The 2D and 3D plots for the different values of the parameters depict the efficacy of time-efficient numerical schemes.

The formats of references [3], [6], and [12] need to be modified, that is "The unique solution of a class of the new generalized absolute value equation"--> "The unique solution of a class of the new generalized absolute value equation"; "A class of maximum-based iteration methods for the generalized absolute value equation" --> "A class of maximum-based iteration methods for the generalized absolute value equation"; "Bounds for the solutions of absolute value equations" --> "Bounds for the solutions of absolute value equations".2. The mathematical formula in reference [14] is garbled and needs to be modified as: "|Ax| -|B||x|=b".3. In the lower part of page 5, that is line 27, "It easy to see....." should be modified as "It is easy to see....."4. On the penultimate line of page 11, the quotation marks on both sides of "diag" need to be modified.5. In the middle and lower parts of page 5, that is, line 18 and line 27, does the beginning of the paragraph need to be moved backward?

An optimization framework is presented for minimizing the energy functional developed around a generalized equation governing physical systems such as fluid dynamics, particle transport, phase transition, and other related systems. The convexity of the energy functional is investigated to derive the necessary conditions for a smooth and global optimum solution. Furthermore, the Total Variation (TV) regularization term is introduced to gain insights into the solution space and convergence analysis of convection-dominated problems. We demonstrate the practical application of our method by applying it to some selected examples such as the Boltzmann, Navier-Stokes, and Maxwell equations.

This paper demonstrates a numerical stratagem for the solution of two dimensional single and coupled partial differential equations, using the new version of the Haar wavelets namely: the scale-3 Haar wavelets (S3HW), combined with the finite difference formulation. The proposed method consists of two phases. The first phase deals with the numerical estimation of the temporal derivative via finite difference which converts the problem to time discrete form. The second phase describes, the approximation of the spatial derivatives along with solution, adopting S3HW. Then, the collocation technique is implemented to transform the resultant system to the set of linear algebraic equations. Solution of the linear system gives the unknown wavelet coefficients which utilized to determine the numerical solutions. Afterwards, the error, convergence, and stability analysis are conducted and deduced a new error estimate. Besides, the numerical simulations are done to verify the scheme and the obtained theoretical findings (convergence and stability). To validate, the performance of the present scheme different error measures , and relative error are determined numerically. The scheme is also compared in terms of error with the scale-2 Haar wavelets and radial basis functions based algorithms. Overall judgement shows, that the numerical results of the developed scheme are in good agreement with the exact solution and the aforementioned methods in the literature.