Mathematical Modelling and Analysis publishes original research on all areas of mathematical modelling and analysis. More information ...
Authors who publish with this journal agree to the following terms that this article contains no violation of any existing copyright or other third party right or any material of a libelous, confidential, or otherwise unlawful nature, and that I will indemnify and keep indemnified the Editor and THE PUBLISHER against all claims and expenses (including legal costs and expenses) arising from any breach of this warranty and the other warranties on my behalf in this agreement; that I have obtained permission for and acknowledged the source of any illustrations, diagrams or other material included in the article of which I am not the copyright owner. on behalf of any co-authors, I agree to this work being published in the above named journal, Open Access, and licenced under a Creative Commons Licence, 4.0 https://creativecommons.org/licenses/by/4.0/legalcode. This licence allows for the fullest distribution and re-use of the work for the benefit of scholarly information. For authors that are not copyright owners in the work (for example government employees), please contact VILNIUS TECHto make alternative agreements.
Generalized dynamic inequalities similar to Hardy’s inequality involving a convex function on time scales
In this paper, we establish some new generalizations of dynamic inequalities similar to Hardy's inequality on a time scale $\mathbb{T}$, by applying Jensen's inequality, integration by parts and chain rule on time scales. In particular, when $\mathbb{T}=\mathbb{R}$, we get the classical inequalities known from the literature, while in the discrete case $\mathbb{T}=\mathbb{N}$, the obtained inequalities are essentially new. In addition, we show that our results are more accurate than some recent dynamic inequalities known from the literature. Finally, we establish the corresponding relations in quantum calculus, when $\mathbb{T}=q^{\mathbb{N}_{0}}$, $q>1$.
Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
Optimal control and stability analysis of a glucose-insulin-FFA dynamic model with GLP-1 incretin effects
Obesity and its associated metabolic dysregulations, particularly Type 2 Diabetes Mellitus (T2DM), constitute a global health crisis. Understanding the intricate interplay of key metabolic components is crucial for effective management strategies. This study presents a novel mathematical model capturing the dynamic interactions among plasma glucose, insulin, and free fatty acids (FFAs), critically integrating the regulatory influence of Glucagon-Like Peptide-1 (GLP-1). Through qualitative analysis, we established the model’s physiological relevance and demonstrated the existence of a stable equilibrium point, confirmed by numerical simulations across various initial conditions. Sensitivity analysis revealed that FFA-related parameters (e.g., lipolysis rates and FFA-induced insulin impairment) and insulin secretion/clearance rates profoundly affect glucose homeostasis, underscoring the detrimental role of elevated FFAs in hyperglycemia. Furthermore, we applied optimal control theory, using Pontryagin’s Maximum Principle, to design GLP-1 receptor agonist intervention strategies. We evaluated two scenarios that balance the cost of intervention with the effectiveness of glucose regulation. Results show that GLP-1 agonism effectively lowers glucose and FFA levels, with greater glucose reduction achieved when control cost and glucose deviation are equally weighted. This research provides a comprehensive mathematical framework for analyzing complex glucose-insulin-FFA-GLP-1 dynamics. Our findings highlight the interconnectedness of insulin sensitivity, lipid metabolism, and incretin action in metabolic health and offer valuable insights for optimizing therapeutic interventions. The developed optimal control strategies suggest potential to improve glycemic control and to inform future clinical approaches to prevent and manage metabolic disorders.
Efficient spectral-Galerkin method for the two-dimensional Gray-Scott model
The Gray-Scott (GS) model is a nonlinear reaction diffusion system widely used in applied sciences. This paper delves into the investigation of a spectral method for the GS model with homogeneous Neumann boundary conditions. The proposed numerical scheme integrates a spectral approach based on generalized Jacobi polynomials for spatial discretization with the two-step backward differentiation formula (BDF2) for temporal discretization. We prove the boundedness, generalized stability, and convergence of this new method. Extensive numerical results demonstrate the efficiency of the new proposed scheme and provide numerical validation of the theoretical analysis. The key advantages of our new approach are twofold: (i) it utilizes generalized Jacobi polynomials with indices α = β = −2, which naturally satisfy the boundary conditions and thereby not only simplify the theoretical analysis but also yield a well-conditioned discrete system that enhances computational efficiency; and (ii) the numerical errors exhibit exponential decay in space.
Extension of a Bohr-Jessen type theorem for the Epstein zeta-function in short intervals
Let $Q$ be a positive definite $n \times n$ matrix, $n \in 2\mathbb{N}$, $n \geqslant4$. The Epstein zeta-function $\zeta(s; Q)$, defined for $\mathrm{Re}\,s > \tfrac{n}{2}$, is given by $\zeta(s; Q) = \sum_{\underline{x} \in \mathbb{Z}^n \setminus \{\underline{0}\}} (\underline{x}^T Q{\underline{x}})^{-s},$ and has a meromorphic continuation to the whole complex plane. Let $T^{{27}/{82}} \leqslant H \leqslant T^{{1}/{2}}$. In this paper, we prove a limit theorem on weak convergence for $\frac{1}{H} \mathrm{meas}\left\{t \in [T, T+H]: \zeta(\sigma + it; Q) \in A \right\},\; A \in \mathcal{B}(\mathbb{C}),$ as $T\to\infty$, where $\mathcal{B}(\mathbb{C})$ is the Borel $\sigma$-algebra on $\mathbb{C}$. The limit measure is explicitly given. The result extends a known theorem obtained for the interval $[0, T]$.
A successive midpoint-based method for the numerical analysis of chaotic systems with local and nonlocal operators
In this study, we examine the uniqueness conditions for solutions of fractal differential equations using the Krasnoselskii-Krein uniqueness theorem. The analysis establishes sufficient criteria that guarantee the existence of unique solutions. Additionally, we employ the successive midpoint method to numerically solve chaotic systems governed by both fractal and global derivatives. To evaluate the effectiveness of the proposed approach, graphical simulations are presented for various derivative orders. These results illustrate the method’s accuracy, stability, and reliability in capturing the intricate dynamics of the considered systems.
Well-posedness and exponential stability for the logarithmic Lamé system with a time varying delay
The focus of this paper revolves around the initial–boundary value problem associated with a logarithmic Lamé system within a bounded domain, and incorporating a time-varying delay. We demonstrate the system’s well-posedness through the application of semigroup theory. Subsequently, we establish the existence of global solutions by employing the well-depth method. Furthermore, we establish exponential decay of solutions under adequate constraints concerning the weight of the time-varying delay and the frictional damping.
Numerical algorithms to solve inverse problems for parabolic equations
This paper presents and analyzes robust numerical algorithms for solving inverse problems for parabolic equations, specifically focusing on the determination of an unknown time-dependent source function from an integral flux condition. The study is motivated by mathematical models based on Navier-Stokes equations, particularly those exhibiting Poiseuille-type solutions. We employ a variational approach, formulating the inverse problem as the minimization of a Tikhonov regularization cost functional. Discrete approximation schemes are rigorously derived using finite volume methods in space and both backward Euler and Crank-Nicolson schemes in time. A key contribution of this work is the strict justification of the gradient formula for the cost functional by deriving the adjoint problem directly from the fully discrete scheme, rather than discretizing the continuous adjoint problem. This methodology is extended to problems involving fractional powers of elliptic operators and two-dimensional domains. Numerical experiments are conducted to compare the efficiency of Gradient Descent and Conjugate Gradient methods. The results demonstrate that the Conjugate Gradient method significantly outperforms standard gradient descent, maintaining high accuracy and convergence rates even with the inclusion of regularization terms and complex diffusion operators.
parabolic partial differential equations
A new closed-form solution for optimal portfolio selection with liquidity risk
In the literature on optimal portfolio selection problems, it is rare that closed-form solutions are found. It is even more so when liquidity risk needs to be taken into consideration. In this paper, we present a closed-form solution for the optimal weights of a portfolio that consists of a risky and riskless asset under a new key assumption that the liquidity risk is directly proportional to the wealth of the portfolio invested in the risky asset. The solution found is for the Constant Relative Risk Aversion (CRRA) utility function, after successfully solving the associated HJB (HamiltonJacobi-Bellman) equation exactly. Due to the presence of liquidity risk, the research findings reveal that the optimal weights are consistently lower than those found by [19]. Finally, the quantitative impact of the proposed solution is discussed.
Re-iterated approximation methods for nonlinear Volterra integral equations
In this article, the Newton-iteration scheme based upon iterated Galerkin operator is applied for solving non-linear Volterra Urysohn integral equations of the second kind for smooth and weakly singular kernels. A one step of improvement by iteration to the Galerkin method, named as iterated Galerkin method is a well discussed method and it gives improved convergence rates than Galerkin method. But if we iterate them one more time, then there is no guarantee that we get any improved convergence rates. The proposed Newton-iteration scheme based upon iterated Galerkin operator ensures improved convergence rates at every step of iteration. Specifically, we establish that the convergence rate in iterated Galerkin method increases by O(hr) for smooth kernel, and O(h1−α) for weakly singular kernel, in each step of reiteration, where h is the norm of the partition. Numerical examples are provided to justify the reliability and efficiency of the proposed technique.
Analysis of the Darcy-Brinkman flow with viscous dissipation and non-homogeneous thermal boundary condition
This study investigates the steady-state Darcy-Brinkman flow within a thin, saturated porous domain, focusing on the effects of viscous dissipation and non-homogeneous boundary condition for the temperature. Employing asymptotic techniques with respect to the domain’s thickness, we rigorously derive the simplified coupled model describing the fluid flow. The mathematical analysis is based on deriving the sharp a priori estimates and proving the compactness results of the rescaled functions. The resulting limit model incorporates contributions of viscous dissipation and thermal boundary conditions and thus could prove useful in the engineering applications involving porous media.
Generalized practical stability of Hopfield-type neural networks differential equations
This paper investigates the boundedness and practical stability properties of solutions for a class of neural differential equations inspired by Hopfield-type neural networks. Specifically, we develop a novel analytical framework that extends beyond traditional Lyapunov stability theory, Barbalat-type arguments, and fixed-point methods by relaxing common structural assumptions such as smoothness and global Lipschitz continuity. Our approach broadens the class of admissible systems to include nonlinearities with weaker growth conditions and time-varying perturbations that are not easily handled by classical techniques. Sufficient conditions are established to ensure the existence of a globally exponentially stable neighborhood of the origin, even in the presence of varying perturbation conditions. Furthermore, numerical examples are provided to demonstrate and validate the main result.
Analytical formulas for polynomial coefficients in radial basis function interpolation
Radial basis functions (RBF) are used in many areas, including interpolation and approximation, solution of partial differential equations, neural networks, and machine learning. RBFs are based on the sum of weighted kernel functions. Additional orthogonal polynomials are added for robustness, numerical stability, and computational efficiency improvement. This contribution gives a new analytical formula specifying values of the polynomial coefficients used in RBF interpolation. The zerodegree polynomial coefficient is related to the sigmoid function used in RBF-neural networks (RBF-NN). Unlike prior works where polynomial augmentation is only used to guarantee solvability, this paper provides explicit closed-form formulae for polynomial coefficients (with special focus on the zerodegree case). This new analytical treatment clarifies their role as global bias terms in both interpolation and RBF neural networks. Expected applicability is in data interpolation and approximation, RBF-neural networks, scientific computing and PDE solutions, geostatistics & spatial interpolation, machine learning, and data fitting and signal processing.
Weak solutions to degenerate p(t)-Laplacian elliptic equations involving (q, r(t)) double phase Hardy terms
This paper is devoted to establishing novel existence criteria for weak solutions to a class of weighted quasilinear degenerate elliptic equations featuring double phase Hardy-type singular coefficients. These types of problems are rarely discussed in variable exponent Sobolev spaces in previous work. We prove the existence of at least one and at least two weak solutions via variational methods and critical point theory, under appropriate assumptions on the weight function and the nonlinearity.
On improved P1-interpolation error estimates in W1,p(0, 1): application to the finite element method
Based on a new Taylor-like formula, we derived an improved interpolation error estimate in W1,p. We compare it with the classical error estimates based on the standard Taylor formula, and also with the corresponding interpolation error estimate, derived from the mean value theorem. We then assess the improvement in accuracy we can get from this formula, leading to a significant reduction in finite element computation costs.
Variable s-step CGNR method for solving the matrix equation AXB = C arising in image processing
The matrix equation $ {AXB} = {C}$ is widely utilized in signal and image processing. In this paper, we present a variable s-step algorithm based on the CGNR method for solving this matrix equation by employing normalization techniques. This algorithm is subsequently enhanced through the application of s-step and regularization methods. By varying the number of basic matrices involved (denoted as s), both the accuracy and speed of the algorithm are improved. The proposed algorithm effectively computes solutions to the matrix equation, demonstrating superior performance when the problem matrices are symmetric. Finally, we investigate the performance and efficacy of these techniques through several numerical examples.
On shifts of periodic zeta-function in short intervals
The periodic zeta-function $\zeta(s; a)$, $s = \sigma + it$, $a = \{a_m \in \mathbb{C} : m \in \mathbb{N}\}$, in the half-plane $\sigma > 1$ is defined by Dirichlet series with periodic coefficients $a_m$, and has the meromorphic continuation to the whole complex plane. The function $\zeta(s; a)$ is a generalization of the Riemann zeta-function and Dirichlet $L$-functions. In the paper, using only the periodicity of the sequence $a$, we obtain that the shifts $\zeta(s + i\tau; a)$, $\tau \in \mathbb{R}$, approximate a certain class of analytic functions, defined in the strip $\{s \in \mathbb{C} : 1/2 < \sigma < 1\}$. For $T^{23/70} \leqslant H \leqslant T^{1/2}$, the set of such shifts has a positive lower density in the interval $[T, T + H]$, $T \to \infty$. The case of positive density is also discussed. For the proof, the mean square estimate in short intervals for the Hurwitz zeta-function, and probabilistic limit theorems are applied.
Study of a class of nonlinear heterogeneous diffusion with mixed phases under L∞ − data
In this paper we investigate a class of nonlinear degenerate parabolic equations involving heterogeneous (p,q)-Laplacian operators and subject to Dirichlet boundary conditions. These equations model complex diffusion phenomena with mixed-phase behavior in heterogeneous media. Our aim is to establish existence and uniqueness results for weak solutions under minimal regularity assumptions on the source term f, without requiring any control at infinity. The main difficulties stem from the degeneracy of the operator, the non-standard (p,q)-growth conditions, and the discontinuity of material phases. To overcome these challenges, we develop a variational framework based on Orlicz–Sobolev space theory and employ a generalized version of the Minty–Browder theorem to ensure the surjectivity of the nonlinear operator. Our approach yields new energy estimates, compactness results in non-reflexive settings, and stability under L∞-perturbations of the data. This work provides a rigorous mathematical foundation for analyzing nonlinear diffusion problems in complex and irregular environments.
A uniformly accurate hybrid difference approximation of a system of singularly perturbed reaction-diffusion equations with delay using grid equidistribution
This paper presents a uniformly accurate difference approximation for a system of singularly perturbed reaction-diffusion equations with delay. The proposed method utilizes an appropriate combination of exponential and cubic spline difference schemes. It employs grid equidistribution to address the challenges posed by the multiscale nature of these systems, which often feature sharp gradients and boundary layers. The grid is generated based on the equidistribution of a positive monitor function, a linear combination of a constant floor and a power of the second derivative of the solution. By using adaptive mesh generation and a spline difference method, the approach enhances the accuracy of the numerical solutions while maintaining computational efficiency. Numerical experiments validate the uniform convergence and theoretical findings, demonstrating the method’s robustness irrespective of the perturbation parameter size.
A second order method for a drug release process defined by a differential Maxwell-Wichert stress-strain relation
Polymeric drug delivery platforms offer promising capabilities for controlled drug release thanks to their ability to be custom-designed with specific properties. In this paper we present a model to simulate the complex interplay between solvent absorption, polymer swelling, drug release and stress development within these platforms. A system of nonlinear partial differential equations coupled with an ordinary differential equation is introduced to avoid drawbacks from other models found in the literature. These incorporated a memory effect but from a numerical standpoint, required storing all previous time steps, making them computationally expensive. This paper proposes a new numerical method to simulate such devices based on nonuniform grids and an implicit midpoint time discretization. Our main results are the second order convergence of the method for nonsmooth solutions and the scheme’s stability under the assumption of quasiuniform grids and a small enough timestep.
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