We develop a delay-memory model for non-stationary solid-propellant combustion. An exponential Volterra kernel is represented by an auxiliary state, yielding a finite-dimensional retarded system. Local well-posedness, global continuation under dissipativity, linear stability, and a Hopf crossing criterion are established. The characteristic equation is derived explicitly, the Lyapunov constants are specified, and the transversality derivative is evaluated numerically. Improved simulations distinguish transient oscillations from a closed post-transient phase portrait.
This paper presents a mathematical framework for quantifying individual impact in collaborative learning networks. Classroom interactions are modeled as a bidirectional weighted graph and then formulated as a cooperative game. The main methodological contribution is the construction of a graph-based characteristic function that represents learning interactions, together with the systematic use of the Shapley value to compute a unique quantitative impact index for each student based on marginal contributions. Although the Shapley value is a classical concept in cooperative game theory, the novelty of this work lies in its structured integration with interaction networks in educational settings and in the formal translation of collaborative learning mechanisms into a cooperative game framework. Numerical simulations using hypothetical classroom data illustrate the behavior of the proposed index and show consistency between Shapley values and students’ structural roles in the interaction network. The reliance on synthetic data and modeling assumptions is acknowledged as a limitation, and directions for empirical validation, sensitivity analysis, and methodological extensions are outlined. The proposed approach provides a mathematically consistent tool for the fair assessment of individual contributions in collaborative learning environments.
A basic issue in theoretical physics and applied mathematics is the linearization of third-order nonlinear ordinary differential equations (ODEs). For second-order ODEs, classical Lie symmetry techniques work well; however, more complex transformations are needed to extend them to third-order equations. The Generalized Sundman Transformation (GST) is examined in this article as a methodical approach to linearizing third-order nonlinear ODEs. Several worked examples showing fully explicit solutions are presented, together with explicit determining equations for the transformation functions and necessary and sufficient conditions for linearizability. To aid in comprehension, a graphical representation of the transformation process is provided.
Euler’s method and its variants are widely used for solving first-order ordinary differential equations, although their accuracy is often limited by truncation errors. In this paper, a geometric interpretation of the classical Heun predictor-corrector method (HM) is presented, using forward prediction and backward correction based on tangent-line approximations. The method is applied to a benchmark initial value problem and compared with the classical Euler and fourth-order Runge-Kutta (RK4) methods. Numerical and graphical results demonstrate that the Heun method provides improved accuracy and reduced error compared with the classical Euler method. A local truncation error analysis confirms that the method is second-order accurate. All computations and graphical illustrations are carried out using MATLAB.
This article presents a stochastic modeling approach for the lateral dynamics of a vehicle in cornering maneuvers, based on the deterministic half-vehicle model developed by Rossa et al. [1]. Uncertainties associated with the steering angle and lateral forces are modeled using Brownian processes, thereby transforming the ordinary differential equations into stochastic differential equations (SDEs). The latter are solved numerically in the MATLAB environment using the Euler-Maruyama method and Monte Carlo simulations. The results reveal that 75% of the simulated trajectories exhibit unstable behavior (oversteer or understeer), whereas the initial deterministic model classified them as stable. This work underscores the importance of stochastic approaches in mathematical modeling for road safety, particularly in cornering decision-making processes.
The variational data assimilation problem provides a standard approach to compute the unknown initial value for prediction of natural phénomena. We use an ill-posed optimal control problem for the determination of the initial state at the time of the first available measurements. In this paper, an efficient algorithmic schema to approximate the initial condition for linear evolution equations of reaction-diffusion and convection type is presented. In this paper, we propose a decision-support tool for identifying the initial condition of an inverse problem governed by a linear parabolic partial differential equation (PDE) modeling pollution concentration in a bounded domain . More precisely, the objective is to reconstruct , where denotes the final observation time, throughout the entire domain , using only the available data from a subdomain . This reconstruction is based on a non-standard approach to null-controllability.
This paper is devoted to study the fractional -Laplacian equation with singular nonlinearities. We use the direct method of moving planes to derive the symmetry and monotonicity result of positive solutions to the fractional -Laplacian equations with singular nonlinearities. Compared to the work proposed in Hu [13], we extend the results of fractional Laplacian to -Laplacian in a bounded domain. In addition, we also consider a singular nonlinear elliptic equation with fractional -Laplacian term in .
This paper presents a matched asymptotic analysis of dendritic crystal growth in an undercooled melt affected by buoyancy-driven convection. Focusing on small Grashof numbers ( ), we derive asymptotic solutions for the flow and temperature fields by dividing the domain into near-field and far-field regions and matching solutions in between. The analysis explicitly incorporates buoyancy effects, revealing how gravity alters flow patterns, enhances heat transfer, and can cause interface instability. This extended model broadens classical dendritic growth theories by including gravitational influences, with implications for materials processing and geological phenomena.
Analyzing steady-state concentration profiles in diffusion-reaction systems is crucial for designing and optimizing many chemical engineering processes, including catalytic reactors and separation units. These phenomena are frequently described by Helmholtz or modified Helmholtz equations. This study investigates the practical application of a meshless numerical technique, the radial basis function-method of approximate particular solutions (RBF-MAPS), for simulating such systems. The inherent meshless nature of RBF-MAPS offers significant advantages for handling the complex geometries often encountered in chemical engineering equipment. We apply the method to model two representative scenarios: reactant diffusion and first-order reaction within a square catalytic plate reactor incorporating an internal source term, and within a circular catalyst pellet with surface boundary conditions. The RBF-MAPS approach effectively discretizes the governing Helmholtz-type equations, and the resulting algebraic systems are robustly solved using singular value decomposition. To establish the reliability of the simulations for engineering analysis, the method’s implementation is rigorously verified using the method of manufactured solutions, demonstrating high-order convergence. Furthermore, convergence studies on specific application problems confirm the stability of the computed concentration profiles. This work validates RBF-MAPS as a flexible and accurate computational tool for chemical engineers needing to analyze diffusion-reaction behavior governed by Helmholtz equations in diverse reactor geometries.
The Sudoku is a very popular puzzle all over the world and it is also an interesting subject in discrete mathematics. Recently, an excellent algorithm called the Inoue algorithm has been developed for solving Sudoku puzzles based on boolean Groebner bases, and successfully applied to the mathematical evaluation of the difficulty level of Sudoku puzzles.
We investigate the Carson-Cambi equation, a second-order linear differential equation arising in the modeling of frequency modulation circuits with a constant inductance and a periodically varying capacitance. We provide a comprehensive classification of nontrivial even and odd solutions with period or semi-period . Using SturmLiouville theory and the framework of Hill and Ince equations, we analyze the spectral properties of the equation and describe the structure and distribution of its eigenvalues. We examine the coexistence of linearly independent periodic solutions, providing necessary and sufficient conditions via an associated polynomial criterion. In contrast to earlier studies limited by perturbative methods or numerical breakdown near , we employ a nonperturbative framework based on self-adjoint operator theory and Prüfer angle analysis. This approach yields a Hochstadt-type estimate for instability intervals and supports accurate numerical computations even for modulation parameters approaching , thus significantly extending the effective range of prior analyses.
In this article, by bringing in a nonlinear transformation containing two trial functions, we put forward a new trial function technique named as a dual trial function technique to search for the explicit and accurate travelling wave solutions of NPDEs. In order to illustrate the feasibility of this technique, we apply it to solve the KdV-Burgers equation, KdV-Burgers-Kuramoto equation and Kuramoto-Sivashinsky equation. As a result, a lot of more general explicit and accurate travelling wave solutions of these three equations, including the single wave solutions and the singular travelling wave solutions, etc, are successfully constructed in a systematic and simple way. The obtained solutions are the same as those given in the existing references. In addition, compared with the proposed approaches in the existing references, the technique described herein appears to be less calculative. Our technique may provide a novel way of thinking for solving NPDEs. Furthermore, it is worth noting that our dual trial function technique is different from the existing trial function method containing one trial function. Compared with the existing trial function method, our technique is of certain advantages that this technique is more flexible and convenient for solving NPDEs and that it can be applied to search for the explicit and accurate travelling wave solutions of more NPDEs. It is our firm conviction that the procedure used herein can be utilized to search for the explicit and accurate travelling wave solutions of other NPDEs as well. We try to generalize this technique to search for the explicit and accurate travelling wave solutions of other NPDEs.
This article develops a method for obtaining an approximate solution for a Ginzburg-Landau type partial differential equation through an application of the generalized method of lines. More specifically, we address the issue of setting a boundary condition on a non-circular domain boundary, in the context of polar and Cartesian coordinates.
Lombok Island is one of the superpriority areas of tourism destinations which is of competitive advantage and a leading sector of the economy of West Nusa Tenggara (NTB) Province, Indonesia. This research aims to determine the optimal tourism transportation route on Lombok Island as a part of the development of its tourism system on Lombok Island. The Lombok Island Tourism Network (Jaringan Pariwisata Pulau Lombok, JP2L) in this study was built by 55 tourist destinations. Furthermore, JP2L is represented by a digraph with the set V representing the set of tourist destinations and the set E representing the road section connecting each tourist destination pair. The application of Floyd’s algorithm to JP2L provides an optimal tourist route on the island of Lombok along with all possible destinations to be passed on the selected travel route. Finally, based on whether or not a tourist location is visited from and to a tourist location (destination among favorites), this study produces recommendations for destinations between favorites, both from the destination tourist session and the original tourist destination.
The Burgers equation, KdV equation and Burgers-KdV equation are real physical modes concerning many branches in physics. In this paper, rich types of explicit and exact travelling wave solutions for these three equations, including the solitary wave solutions, the singular travelling wave solutions, the triangle function periodic wave solutions, etc., are presented by a direct trial function approach. Among them, some are new travelling wave solutions.
In this work, using the “Extreme Measurement Theory”, a previously unknown simple differential equation of field theory was obtained, from which a single universal value of the electromagnetic field is determined for any massive bodies of arbitrary physical and chemical composition, guaranteeing weightlessness for all bodies located in the zone on the Earth’s surface in which this field is generated. It is most likely that the builders of the Egyptian pyramids moved huge stone blocks using this electromagnetic field. The equations found can also be used to develop electromagnetic accelerators that will replace the first stage of large multistage rockets.
Many kinds of travelling wave solutions of the KdV-Burgers-Kuramoto equation including the solitary wave solutions are presented by applying the trial function approach. The results obtained are in agreement with those given in existing reference.
The use of Big Data and the engineering of business processes are becoming increasingly important for managing the business of companies and large international corporate groups. For more successful outcomes, even for complex interventions such as those of the PNRR (National Recovery and Resilience Plan), the use of mathematical models to assist management decisions represents a significant lever to optimize time, costs, and planned overall performance. This analytical approach strengthens traditional qualitative techniques like SWOT analysis and finds useful application in the optimal definition of any form of PPP (Public-Private Partnership), such as the recent “Partnership for Innovation” (PPI) included in the new procurement code. The scientific work developed from the systematic analysis of processes proposes a particular multi-objective mathematical model that uses a system of vectors in an m-dimensional reference space.
