Mathematical Modelling and Analysis

A joint discrete limit theorem for Epstein and Hurwitz zeta-functions

Hany Gerges — 2025-04-18

In the paper, we obtain a joint limit theorem on weak convergence for probability measure defined by discrete shifts of the Epstein and Hurwitz zeta-functions. The limit measure is explicitly given. For the proof, some linear independence restriction is required. The proved theorem extends and continues Bohr–Jessen’s classical results on probabilistic characterization of value distribution for the Riemann zeta-function.

Solutions of the attraction-repulsion-chemotaxis system with nonlinear diffusion

Yadhavan Karuppusamy — 2025-04-18

In this study, we consider the well-posedness of the attraction-repulsion chemotaxis system. This paper explores the dynamics of species movement in reaction to two chemically opposing substances, incorporating nonlinear diffusion. Our primary objective is to establish the existence of a global-in-time weak solution for the proposed model in an unbounded three-dimensional spatial domain. Our study has confirmed the existence of a global-in-time weak solution for the proposed system in three dimensions. Furthermore, we demonstrate that global-in-time weak solutions are also attainable for the proposed system in a bounded domain with a smooth boundary.

Calderón-Zygmund estimates for Schrödinger equations revisited

Le Xuan Truong — 2025-04-18

We establish a global Calderón-Zygmund estimate for a quasilinear elliptic equation with a potential. If the potential has a reverse Hölder property, then the estimate was known in [6]. In this note, we observe that the estimate remains valid when the potential is merely Lebesgue integrable. Our proof is short and elementary.

A class of nonlinear systems with new boundary conditions: existence of solutions, stability and travelling waves

Abdelkader Lamamri — 2025-04-18

In this work, we begin by introducing a new notion of coupled closed fractional boundary conditions to study a class of nonlinear sequential systems of Caputo fractional differential equations. The existence and uniqueness of solutions for the class of systems is proved by applying Banach contraction principle. The existence of at least one solution is then accomplished by applying Schauder fixed point theorem. The Ulam Hyers stability, with a limiting-case example, is also discussed. In a second part of our work, we use the tanh method to obtain a new travelling wave solution for the coupled system of Burgers using time and space Khalil derivatives. By bridging these two aspects, we aim to present an understanding of the system’s behaviour.

Development and analysis of an efficient Jacobian-free method for systems of nonlinear equations

Janak Raj Sharma — 2025-04-24

A multi-step derivative-free iterative technique is developed by extending the well-known Traub-Steffensen iteration for solving the systems of nonlinear equations. Keeping in mind the computational aspects, the general idea to construct the scheme is to utilize the single inverse operator per iteration. In fact, these type of techniques are hardly found in literature. Under the standard assumption, the proposed technique is found to possess the fifth order of convergence. In order to demonstrate the computational complexity, the efficiency index is computed and further compared with the efficiency of existing methods of similar nature. The complexity analysis suggests that the developed method is computationally more efficient than their existing counterparts. Furthermore, the performance of method is examined numerically through locating the solutions to a variety of systems of nonlinear equations. Numerical results regarding accuracy, convergence behavior and elapsed CPU time confirm the efficient behavior of the proposed technique.

A numerical scheme to simulate the distributed-order time 2D Benjamin Bona Mahony Burgers equation with fractional-order space

Hais Azin — 2025-04-24

In this study, a new class of the Benjamin Bona Mahony Burgers equation is introduced, which considers the distributedorder in the time variable and fractional-order space in the Caputo form in the 2D case. The 2D-modified orthonormal normalized shifted Ultraspherical polynomials are derived from 1Dmodified orthonormal normalized shifted Ultraspherical polynomials and 2D-modified orthonormal normalized shifted Ultraspherical polynomials and the orthonormal normalized shifted Ultraspherical polynomials are applied to approximate of the space and time variables, respectively. Moreover, the convergence analysis of these basis functions is investigated. Due to the time variable being in the distributed-order mode and the space variable being in the fractional-order case, to apply the desired numerical algorithm for this type of equation, operational matrices of ordinary, fractional and distributed-order derivatives are computed. In the proposed method, once the unknown function is approximated using the mentioned polynomial, the matrix form of the residual function is derived and then a system of algebraic equations is adopted by applying the collocation approach. An approximate solution is extracted for the original problem by solving constructed equation system. Several examples are examined to demonstrate the accuracy and capability of the method.

Boundary feedback stabilization of quasilinear hyperbolic systems with zero characteristic speed

Zhiqiang Wang — 2025-04-24

In this paper, we investigate the boundary feedback stabilization of a quasilinear hyperbolic system with zero characteristic speed and a partially dissipative structure. This structure enables us to construct a Lyapunov function that guarantees exponential stability for the H2 solution. We also introduce another set of stability conditions by restricting terms corresponding to zero eigenvalues to the dissipative part, which still ensures exponential stability. As an application, we achieve feedback stabilization for the modified model of neurofilament transport in axons.

Optimizing chaotic systems by orbit counting and Fourier spectrum: FPGA implementation and image encryption application

Ricardo Rojas-Galván — 2025-04-24

The optimization of chaotic systems has been performed by considering dynamical characteristics of the mathematical models. The proposed work shows the application of genetic algorithms (GAs) to optimize the chaotic behavior of three well-known systems, namely: Lorenz, Chen and Lü. The parameters of the chaotic systems are varied in a specific range of values considered as the search space, and the evaluation of the mathematical model is performed by applying the Forward Euler method. The contribution presented herein is that the chaotic behavior is evaluated by counting the orbits in an attractor and the sparsity of them. In addition, the chaotic behavior is guaranteed by evaluating the Fourier spectrum of the time series. The solutions provided by the GA, are then implemented on a field-programmable gate array (FPGA) to verify the experimental generation of chaotic attractors. Finally, two optimized chaotic systems are synchronized and used to encrypt an image, thus confirming the appropriateness of optimizing the chaotic behavior by orbit counting and Fourier spectrum analysis.

Analyzing Helmholtz phenomena for mixed boundary values via graphically controlled contractions

Mudasir Younis — 2025-04-24

Helmholtz’s problem helps us to completely understand how sound behaves in a cylinder that is closed from one of its ends and opened at another. This paper aims to employ some novel convergence results to the Helmholtz problem with mixed boundary conditions and demonstrate the existence and uniqueness of the solution by applying graph-controlled contractions. For this purpose, we enunciate graphically Reich type and graphically Ćirić type contractions in the realm of graphical-controlled metric type spaces. In our study, we showcase the existence and uniqueness of fixed point results by employing these graphical contractions. This is demonstrated through extensive examples that a graphicalcontrolled metric-type space is distinct from traditional controlled metric-type spaces. We also exhibit an example of a graphically Reich contraction that is not a classical Reich contraction. Similarly, a decent example of graphical Ćirić contraction is presented, which is distinct from the classical Ćirić contraction. Concrete illustrative examples solidify our theoretical framework.

Approximation of a system of nonlinear Carrier wave equations by approximating the Carrier terms with their integral sums

Le Thi Phuong Ngoc — 2025-04-24

This paper is concerned with the approximation of a system of nonlinear Carrier wave equations (CEs) by approximating the Carrier terms with their integral sums. At first, under suitable conditions, the linear approximate method, the Galerkin method, and compactness arguments provide the unique existence of a weak solution (uⁿ,vⁿ) of the problem (P_n), for each n ∈ N, for a system of nonlinear wave equations related to Maxwell fluid between two infinite coaxial circular cylinders. Next, we prove that {(uⁿ,vⁿ)}_nconverges to the weak solution (u,v)of the problem for a system of CEs in a suitable function space. This proof is done by using the compactness lemma of Aubin-Lions and the method of continuity with a priori estimates. We end the paper with a remark related to open problems.

Analysis study of hybrid Caputo-Atangana-Baleanu fractional pantograph system under integral boundary conditions

Sabri T. M. Thabet — 2025-04-24

This manuscript investigates the qualitative analysis of a new hybrid fractional pantograph system involving AtanganaBaleanu-Caputo derivatives, complemented by hybrid integral boundary conditions. Dhage’s fixed point theorem is employed to investigate the existence theorem of the solutions, while uniqueness is proven by using Perov’s approach and Lipschitz’s matrix. The Hyers-Ulam (HU) stability is also demonstrated using the Lipschitz’s matrix and techniques from nonlinear analysis. Finally, illustrative example is enhanced to examine the effectiveness of the obtained results.