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Approximation of a system of nonlinear Carrier wave equations by approximating the Carrier terms with their integral sums
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 (Pn), 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)}n converges 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.
Copyright (c) 2025 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
Spectral algorithm for fractional BVPs via novel modified Chebyshev polynomials fractional derivatives
This paper introduces a spectral algorithm tailored for solving fractional boundary value problems (BVPs) using the fractional derivatives of modified Chebyshev polynomials. Specifically, it addresses linear and non-linear BVPs and Bratu equations in one dimension via spectral methods. The approach employs basis functions derived from first-kind shifted polynomials that satisfy the homogeneous boundary conditions. The fractional derivatives are formulated to facilitate the solution process. The convergence analysis is studied for the suggested basis expansion; some numerical results are exhibited to verify the applicability and accuracy of the method.
Periodic travelling waves of a forced Cahn-Hilliard equation
We prove analytically the existence of a uniparametric family of periodic travelling waves for a Cahn-Hilliard equation with an external forcing term modelling the phase separation of a binary mixture of fluids with thermal diffusion. Some quantitative estimates on the solutions are derived.
Global existence and uniqueness of mild solution for a fractional Keller-Segel system in Besov-Morrey spaces
This paper investigates the fractional Keller-Segel system involving both temporal and spatial variables. We examine fractional dissipation mechanisms for the physical variables, including chemotactic diffusion with fractional dissipation, and incorporate a time-fractional variation in the Caputo sense. Our analysis centers on the fractional heat semigroup, deriving time-decay and integral estimates for Mittag-Leffler operators in Besov-Morrey spaces. Furthermore, we establish a bilinear estimate arising from the nonlinearity of the Keller-Segel system, avoiding reliance on auxiliary norms. These results are then employed to demonstrate the existence and uniqueness of mild solution in Besov-Morrey spaces.
existence and uniqueness of mild solution
Robust computing technique for reaction diffusion 2D parabolic problems with shift
Higher dimensional singularly perturbed problems frequently appears in many mathematical modelling. Solving such higher dimensional problem is not as much easy as possible. So order reducing technique namely alternating direction method is one such good choice for solving them. Further singular perturbation problem has its own complexities like boundary and/or interior layers, hence it requires a fitted method on special mesh. Reaction diffusion type singular perturbation problem with space shift is considered in this article. The presence of space shift leads strong interior layer in the solution. To take care of interior and boundary layers, a special mesh is constructed. Hence the problem considered in this article is solved by alternating direction method and fitted difference method with bilinear interpolation. Further the convergence analysis also carried out with rate one in both time and space. Computational validation is also done.
Convergence analysis of a class of iterative methods: a unified approach
In this paper, we study the convergence of a class of iterative methods for solving the system of nonlinear Banach space valued equations. We provide a unified local and semi-local convergence analysis for these methods. The convergence order of these methods are obtained using the conditions on the derivatives of the involved operator up to order 2 only. Further, we provide the number of iterations required to obtain the given accuracy of the solution. Various numerical examples including integral equations and Caputo fractional differential equations are considered to show the performance of our methods.
Dynamical analysis and adaptive synchronization of a new 6D hyperchaotic system with the cosine function
This paper presents a novel 6D dynamic system derived from modified second-type 3D Lorenz equations using state feed- back control. While these original 3D equations are structurally simpler than the classical Lorenz equations, they generate more topologically complex attractors with a distinctive two-winged butterfly structure. The proposed system is the most compact of its kind in the literature, containing only 11 terms: two cross-product nonlinearities, two piecewise linear functions, one cosine function, five linear terms, and one constant. The newly developed 6D hyper- chaotic system exhibits rich dynamic properties, including hidden attractors and dissipative behavior. A detailed dynamic analysis has identified two unstable hyperbolic equilibrium points, indicating the potential for self-exciting attractors. Additionally, bifurcation diagrams were constructed, Lyapunov exponents were computed, and the maximum Kaplan-Yorke dimension DKY = 3.23 was obtained at parameter value a = 0.5, revealing the high complexity of the hyperchaotic dynamics. Furthermore, multistability and offset boosting control were examined to gain deeper insights into the system’s behavior. Finally, synchronization between two identical 6D hyperchaotic systems was successfully achieved using an adaptive control method.
A predation model considering a generalist predator and the Rosenzweig functional response
This work deals with the dynamics of an ordinary differential equation system describing a Leslie-Gower predator-prey model with a generalist predator and a non-differentiable functional response proposed by M. L. Rosenzweig, given by h(x) = qxα with 0 < α < 1. Two aspects have a significant impact on the model: (1) the predator’s carrying capacity depends on both the favorite prey population and an alternative food source, and (2) consumers have access to an alternative food source. Among the main results, a separatrix curve Σ arises dividing the phase plane into regions with different dynamic behaviors. Trajectories above the separatrix curve Σ reach the vertical axis in finite time, while those below Σ may converge to positive equilibrium points, limit cycles, or homoclinic connections. Furthermore, the system is non-Lipschitz, implying non-uniqueness of solutions at points of the vertical axis. Several bifurcations, including saddle-node, homoclinic, Hopf, generalized Hopf, and Bogdanov-Takens bifurcations, are identified through the use of computational techniques. The dynamics of the system are visualized by presenting a bifurcation diagram in a convenient parameter space.
One-signed rotationally symmetric solutions of singular Dirichlet problems with the prescribed higher mean curvature operator in Minkowski spacetime
We investigate the existence, uniqueness and multiplicity of one-signed rotationally symmetric solutions of singular Dirichlet problems with the prescribed higher mean curvature operator in Minkowski spacetime. The main tools are the Schauder fixed point theorem along with cut-off technique and the Leggett-Williams fixed point theorem. In addition, we give some practical models to illustrate the effectiveness of our results.
one-signed rotationally symmetric solutions
Variational analysis of a frictional contact problem with wear and damage
We study a quasistatic problem describing the contact with friction and wear between a piezoelectric body and a moving foundation. The material is modeled by an electro-viscoelastic constitutive law with long memory and damage. The wear of the contact surface due to friction is taken into account and is described by the differential Archard condition. The contact is modeled with the normal compliance condition and the associated law of dry friction. We present a variational formulation of the problem and establish, under a smallness assumption on the data, the existence and uniqueness of the weak solution. The proof is based on arguments of parabolic evolutionary inequations, elliptic variational inequalities and Banach fixed point.
Copyright (c) 2021 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
A history-dependent frictional contact problem with wear for thermoviscoelastic materials
In this manuscript we study a contact problem between a deformable viscoelastic body and a rigid foundation. Thermal effects, wear and friction between surfaces are taken into account. A variational formulation of the problem is supplied and an existence and uniqueness result is proved. The idea of the proof rested on a recent result on history-dependent quasivariational inequalities. Finally, a perturbation of the data is initiated and a convergence result is demonstrated when the perturbation parameter converges to zero.
history-dependent quasivariational inequality
Copyright (c) 2019 The Author(s). Published by Vilnius Gediminas Technical University. https://creativecommons.org/licenses/by/4.0
Stability analysis and numerical simulations of IVGTT glucose-insulin interaction models with two time delays
This paper presents a systematic study of a mathematical model of glucose and insulin interaction with two time delays, with a focus on analytical studies, bifurcation analysis, and very well numerical simulations. This model based on the Intra-Venous Glucose Tolerance Test (IVGTT) and is presented with two time delays. One delay is the insulin response time to an increase in glucose concentration, and the hepatic glucose production time delay is the other. Then, we establish results on positivity, boundedness, and persistence. We also provide suﬃcient stability analysis conditions for both local and global asymptotic stability of the proposed models. For the latter, two diﬀerent strategies are used: stability bifurcation analysis and Lyapunov-Krasovskii functionals. We investigate diﬀerent regions of parameter space using two approaches, that yield diﬀerent sets of suﬃcient conditions for global stability. The bifurcation graphs generated from our extensive and carefully designed simulations complement and conﬁrm these analytical results. The insulin concentration level peaks after the glucose concentration level, according to the numerical simulations.
Copyright (c) 2022 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
A reproducing kernel method for solving singularly perturbed delay parabolic partial differential equations
In this article, we put forward an efficient method on the foundation of a few reproducing kernel spaces(RK-spaces) and the collocation method to seek the solution of delay parabolic partial differential equations(PDEs) with singular perturbation. The approximated solution to the equations is formulated and proved the exact solution is uniformly convergent by the solution. Furthermore, the partial differentiation of the approximated solution is also proved the partial derivatives of the exact solution is uniformly convergent by the solution. Meanwhile, we show that the accuracy of our method is in the order of T/n where T is the final time and n is the number of spatial (and time) discretization in the domain of interests. Three numerical examples are put forward to demonstrate the effectiveness of our presented scheme.
Copyright (c) 2023 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
Fractional multiwavelet methods for solving spatiotemporal fractional diffusion equations with non-smooth solutions
This introduces a new method that effectively solves spatiotemporal fractional diffusion equation(FDE) using fractional Lagrange interpolation and fractional multiwavelets. The method effectively addresses situations with non-smooth solutions. The approach begins by discretizing the time variable t using the fractional piecewise parabolic Lagrange interpolation method. For the spatial variables, we construct fractional multiwavelets. Through the least residue method, we obtain approximate solutions, while also conducting convergence analysis. Numerical demonstrations validate the high accuracy achieved by the proposed method, notably showcasing the better approximation capability of fractional polynomials compared to their integer counterparts.
A boundary value problem with an integral condition for a certain fractional differential equation
The aim of this work is to prove the existence and the uniqueness of the solution of one dimensional initial boundary value problem for a parabolic equation with a Caputo time fractional differential operator supplemented by periodic nonlocal boundary condition and integral condition. First, an a priori estimate is established for the associated problem. Secondly, the density of the operator range generated by the considered problem is proved by using the functional analysis method.
Analysis study of hybrid Caputo-Atangana-Baleanu fractional pantograph system under integral boundary conditions
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.
coupled hybrid fractional differential system
Analyzing Helmholtz phenomena for mixed boundary values via graphically controlled contractions
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.
mixed boundary conditions graphic-contractions
Optimizing chaotic systems by orbit counting and Fourier spectrum: FPGA implementation and image encryption application
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.
Boundary feedback stabilization of quasilinear hyperbolic systems with zero characteristic speed
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.
A numerical scheme to simulate the distributed-order time 2D Benjamin Bona Mahony Burgers equation with fractional-order space
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.
2D Benjamin Bona Mahony Burgers equation
Development and analysis of an efficient Jacobian-free method for systems of nonlinear equations
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 class of nonlinear systems with new boundary conditions: existence of solutions, stability and travelling waves
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.
Calderón-Zygmund estimates for Schrödinger equations revisited
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.
Solutions of the attraction-repulsion-chemotaxis system with nonlinear diffusion
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.
A joint discrete limit theorem for Epstein and Hurwitz zeta-functions
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.
On the stability and efficiency of high-order parallel algorithms for 3D wave problems
In this work, we investigate the stability conditions for four new high-order ADI type schemes proposed to solve 3D wave equations with a non-constant sound speed coefficient. This analysis is mainly based on the spectral method, therefore a basic benchmark problem is formulated with a constant sound speed coefficient. For a case of general non-constant coefficient the stability analysis is done by using the energy method. Our main conclusion states that the selected ADI type schemes use different factorization operators (mainly due to the need to approximate the artificial boundary conditions on the split time levels), but the general structure of the stability factors are similar for all schemes and thus the obtained CFL conditions are also very similar. The second goal is to compare the accuracy and efficiency of the selected ADI solvers. This analysis also includes parallel versions of these schemes. Two schemes are selected as the most effective and accurate.
Equivalents of the Riemann hypothesis involving the Gram points
The Riemann hypothesis (RH) on zeros of the zeta-function $\zeta(s)$, $s=\sigma +it$, states that all zeros of $\zeta(s)$ in the strip $0< \sigma < 1$ lie on the line $\sigma =1/2$. Several equivalents of RH are known. In the paper, we obtain equivalents of RH in terms of self-approximation of $\zeta(s)$ by shifts $\zeta(s+iht_k)$, $k\in \mathbb{N}$, where $\{t_k, k\in \mathbb{N}\}$ is the sequence of the Gram points.
weak convergence of probability measures
Fixed point approximation of contractive-like mappings using a stable iterative family and its dynamics via quadratic polynomials
This study aims at presenting a novel bi-parametric family of iterative methods for computing the fixed points of a contractive-like mapping. We thoroughly analyze the strong and stable convergence of the proposed technique and explore its applicability across various problem domains. Regarding convergence, it is proven that for several operators, the Mann iteration is analogous to the proposed multi-step class, and vice-versa. Moreover, numerical tests demonstrate the superior performance of the new procedures compared to existing three-step schemes. We further examine the dynamic behavior of several fixed-point iterative techniques when applied to quadratic polynomials. Based on the outcomes of these experiments, it can be concluded that the proposed family demonstrates both validity and effectiveness.
Constitutive memory equations for auxetic materials
In this note we suggest a set of constitutive equations for anelastic materials whose internal structure can present anomalous variations resulting from external effects of tension/compression, as in auxetic media. For these problems we require of a new function related with the variation of internal structure, which allows us to define a threshold separating the internal structure media from the classic ones. The equations are formulated in the case of static problems but also when the material has memory or plastic properties in addition to auxetic ones. In order to limit the complexity of the formulae, the discussion is limited to the case where the perturbation is one dimensional, which however does not limit the significance of the results.
A partially derivative-free cyclic block coordinate descent method for nonseparable composite optimization
In this paper, we address a composite optimization problem in which the objective function consists of two terms: the first presents a function with a gradient that satisfies a Lipschitz–Hölder composition, while the second one is a convex function. Under general settings, we propose and analyze a new coordinate descent method that can operate without the use of derivatives. The algorithm is an adaptation of the coordinate proximal gradient method, specifically designed to consider the composite form of the objective function. We perform a complete worst-case complexity analysis, assuming that the coordinates (or blocks of coordinates) are selected in a cyclic manner. In addition, we present academic numerical examples that illustrate the efficiency of our algorithm in practical problems.