Mathematical Modelling and Analysis
type: channel
description: Mathematical Modelling and Analysis publishes original research on all areas of mathematical modelling and analysis. More information ...
publisher: Vilnius Gediminas Technical University
language: en
publication name: Mathematical Modelling and Analysis
issn: 1392-6292
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items: Two regularization methods for identifying the initial value of Caputo–Hadamard time-fractional diffusion equation
items: On the put-call duality of the game options
items: On an inverse coefficient problem for a drug war reaction-diffusion system via an optimization approach
items: Mapped Legendre collocation methods for linear Volterra–Fredholm integral equations with delays on a semi-infinite interval
items: New existence and exponential stability results for periodic solutions in recurrent neural networks with generalized piecewise constant delay via coincidence degree theory
items: Existence results for nonlinear Fourier problems with variable exponent growth
items: A measure theoretic approach to Lipschitz regularity and its Haar type wavelet analysis
items: Solvability of two-dimensional nonlinear singular Volterra integral equations with fractional order in Banach space and approximation of the solution of it
items: Uniform numerical method for singularly perturbed parabolic time delay reaction-diffusion problems arising in control theory
items: Second derivative two-step peer methods
items: Representation of solutions of systems of linear differential equations with multiple delays and nonpermutable variable coefficients
items: Fractional integro-differential equations with nonlocal conditions and ψ–Hilfer fractional derivative
items: On fractional Volterra integrodifferential equations with fractional integrable impulses
items: Generalized dynamic inequalities similar to Hardy’s inequality involving a convex function on time scales
items: Optimal control and stability analysis of a glucose-insulin-FFA dynamic model with GLP-1 incretin effects
items: Efficient spectral-Galerkin method for the two-dimensional Gray-Scott model
items: Extension of a Bohr-Jessen type theorem for the Epstein zeta-function in short intervals
items: A successive midpoint-based method for the numerical analysis of chaotic systems with local and nonlocal operators
items: Well-posedness and exponential stability for the logarithmic Lamé system with a time varying delay
items: Numerical algorithms to solve inverse problems for parabolic equations
items: A new closed-form solution for optimal portfolio selection with liquidity risk
items: Re-iterated approximation methods for nonlinear Volterra integral equations
items: Analysis of the Darcy-Brinkman flow with viscous dissipation and non-homogeneous thermal boundary condition
items: Generalized practical stability of Hopfield-type neural networks differential equations
items: Analytical formulas for polynomial coefficients in radial basis function interpolation
items: Weak solutions to degenerate p(t)-Laplacian elliptic equations involving (q, r(t)) double phase Hardy terms
items: On improved P1-interpolation error estimates in W1,p(0, 1): application to the finite element method
items: Variable s-step CGNR method for solving the matrix equation AXB = C arising in image processing
items: On shifts of periodic zeta-function in short intervals
items: Study of a class of nonlinear heterogeneous diffusion with mixed phases under L∞ − data

Two regularization methods for identifying the initial value of Caputo–Hadamard time-fractional diffusion equation
type: item
description: In this paper, the inverse problem of identifying the unknown initial value for time fractional diffusion equation with Caputo-Hadamard derivative is considered. This problem is illposed and two regularization methods are used to solve it. Firstly, we prove that this problem is ill-posed. Secondly, the conditional stability result and the optimal error bound are given. Then, the error estimates of the Quasi-boundary regularization method and the fractional Landweber iterative regularization method under a priori and a posteriori regularization parameter selection rules are given respectively. Finally, numerical examples are given to illustrate the effectiveness of two regularization methods.
subject: Articles
  topic: section
subject: Caputo-Hadamard derivative
  topic: keywords
subject: initial value identification
  topic: keywords
subject: quasi-boundary regularization method
  topic: keywords
subject: fractional Landweber iterative regularization method
  topic: keywords
creator: Ruo-Hong Li
creator: Ying Cao
creator: Fan Yang
creator: Xiao-Xiao Li
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-06-19
publication date: 2026-06-19
starting page: 561
ending page: 582
doi: 10.3846/mma.2026.23999

On the put-call duality of the game options
type: item
description: The purpose of this paper is to obtain a duality between the game put and call options assuming three component penalties – proportion of the usual option payoff, shares of the underlying asset, and a fixed amount. We examine separately the cases of finite and infinite maturities. For the perpetual options, we need to derive a polynomial-style equations for the optimal boundaries. We prove the existence and uniqueness of their solutions as well as provide a method for their deriving. This result is important in itself since the current literature in the field is based on inverting of several functions or on solving of non-linear systems which may lead to some computational difficulties and significant errors for extreme parameter values. Furthermore, the duality is established under a finite time horizon too. It is important to note that this duality does not hold under the classical assumption of a fixed penalty.
subject: Articles
  topic: section
subject: put-call duality
  topic: keywords
subject: game options
  topic: keywords
subject: optimal boundaries
  topic: keywords
subject: polynomial-style formula
  topic: keywords
subject: pricing
  topic: keywords
creator: Tsvetelin Zaevski
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-06-19
publication date: 2026-06-19
starting page: 542
ending page: 560
doi: 10.3846/mma.2026.24619

On an inverse coefficient problem for a drug war reaction-diffusion system via an optimization approach
type: item
description: In this paper, we study a coefficients inversion problem of a coupled system controlled by three reaction-diffusion equations describing a simple dynamic model of a drug epidemic in an idealized community from the final measurement data. Firstly, the optimization theory is used to transform the given problem into an optimal control problem, and the existence of minimizer is established. Then the stability estimates of the Lipschitz type for the three spatially varying coefficients are proved, where the upper bounds are given by some Lebesgue norms of the final measure.
subject: Articles
  topic: section
subject: coefficient inverse problem
  topic: keywords
subject: coupled reaction-diffusion system
  topic: keywords
subject: optimal control problem
  topic: keywords
subject: stability
  topic: keywords
creator: Zhaoqi Zhang
creator: Liangliang Sun
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-06-19
publication date: 2026-06-19
starting page: 521
ending page: 541
doi: 10.3846/mma.2026.24516

Mapped Legendre collocation methods for linear Volterra–Fredholm integral equations with delays on a semi-infinite interval
type: item
description: In this paper, we present a collocation method for linear Volterra–Fredholm integral equations with delay on a semi-infinite interval. The method employs orthogonal mapped Legendre basis functions together with a mapped Gauss quadrature rule adapted to the Volterra operator, leading to a stable and well-conditioned linear system. The convergence properties of both the collocation and iterated collocation solutions are investigated in the L2- and L∞-norms, and algebraic convergence rates are derived under mild regularity assumptions. Several numerical examples are presented and discussed to show the accuracy and efficiency of the methods.
subject: Articles
  topic: section
subject: Volterra–Fredholm integral equations
  topic: keywords
subject: delay term
  topic: keywords
subject: mapped Legendre functions
  topic: keywords
subject: semi-infinite interval
  topic: keywords
subject: spectral collocation
  topic: keywords
subject: convergence analysis
  topic: keywords
creator: Sofiane Boutarcha
creator: Azedine Rahmoune
creator: Ahmed Guechi
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-06-19
publication date: 2026-06-19
starting page: 499
ending page: 520
doi: 10.3846/mma.2026.24076

New existence and exponential stability results for periodic solutions in recurrent neural networks with generalized piecewise constant delay via coincidence degree theory
type: item
description: The present work investigates recurrent neural systems incorporating generalized piecewise constant delay, with particular emphasis on establishing periodic behaviors and verifying their exponential convergence on a global scale. The existence of periodic solutions is established via Mawhin’s coincidence degree in combination with sharp a priori estimates, while uniqueness and exponential attractivity are derived through a Lyapunov functional approach supported by differential inequalities adapted to the delay structure. The obtained criteria are concise, verifiable, and applicable in practice. Representative computational experiments are provided to substantiate the analytical findings.
subject: Articles
  topic: section
subject: recurrent neural networks
  topic: keywords
subject: generalized piecewise constant delay
  topic: keywords
subject: periodic solutions
  topic: keywords
subject: coincidence degree theory
  topic: keywords
subject: global exponential stability
  topic: keywords
subject: method of Lyapunov functions
  topic: keywords
creator: Kuo-Shou Chiu
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-06-18
publication date: 2026-06-18
starting page: 476
ending page: 498
doi: 10.3846/mma.2026.25296

Existence results for nonlinear Fourier problems with variable exponent growth
type: item
description: We study nonlinear problems which use the generalized $\alpha(x)-$Laplacian operator under Fourier boundary conditions with $L^{1}$ data. Our research establishes both weak and entropy solutions through the framework of variable exponent Sobolev spaces. Our solution method uses monotone operator theory and appropriate approximation methods to develop a unified approach for dealing with nonlinearities that exhibit variable growth. These findings help expand knowledge about nonlinear Fourier-type problems while demonstrating how entropy formulations enable well-posedness for problems with minimal data requirements.
subject: Articles
  topic: section
subject: Lebesgue and Sobolev spaces with variable exponent
  topic: keywords
subject: Fourier boundary conditions
  topic: keywords
subject: weak solutions
  topic: keywords
subject: entropy solution
  topic: keywords
subject: nonlinear elliptic problems
  topic: keywords
creator: Mohamed Badr Benboubker
creator: Omar Benslimane
creator: Maria Alessandra Ragusa
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-06-18
publication date: 2026-06-18
starting page: 452
ending page: 475
doi: 10.3846/mma.2026.25157

A measure theoretic approach to Lipschitz regularity and its Haar type wavelet analysis
type: item
description: The $\alpha$−Lipschitz character of a time series or an image summarizes, in the single parameter α, some persistence properties of the original function modeling the given signal. Such is the case of the Hurst exponent widely used in hydrology and other sciences. It is known that the continuous wavelet transform is useful to characterize the $\alpha$−Lipschitz condition of a function. Recent results show that, in one dimension, the behavior of the Haar coefficients characterize the $\alpha$−Lipschitz regularity of functions with respect to the induced dyadic metric. In this note, we extend the characterization of dyadic Lipschitz regularity of functions to non-atomic probability spaces, using generalized Haar systems. We also provide some two dimensional examples that can be designed and used to reflect specific textures in images.
subject: Articles
  topic: section
subject: Lipschitz regularity
  topic: keywords
subject: Haarlet analysis
  topic: keywords
subject: wavelets
  topic: keywords
creator: Hugo Aimar
creator: Juliana Boasso
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-06-17
publication date: 2026-06-17
starting page: 431
ending page: 451
doi: 10.3846/mma.2026.24576

Solvability of two-dimensional nonlinear singular Volterra integral equations with fractional order in Banach space and approximation of the solution of it
type: item
description: In this article, existence and uniqueness of the solution for two-dimensional non-linear singular Volterra integral equations with fractional orders in a Banach space is discussed by utilizing the concept of the measure of non-compactness and fixed-point theorem. In fact, this kind of equations is a generalization of two-dimensional Riemann-Liouville fractional non-linear integral equations. To approximate the solution of the above problem, we use modified homotopy perturbation with the help of Adomian polynomials. To validity of the derived results, we introduce an example in the field of singular non-linear integral equations. Hence, a semianalytic solution for given example is obtained ensuring satisfactory accuracy. Also, to ensure the effectiveness of the proposed method the results are compared with some other works.
subject: Articles
  topic: section
subject: measure of non-compactness
  topic: keywords
subject: two dimensional non-linear singular integral equation
  topic: keywords
subject: fractional order
  topic: keywords
subject: iterative algorithm
  topic: keywords
creator: Samaneh Darvishi Khezri
creator: Mohsen Rabbani
creator: Reza Arab
creator: Vahid Dadashi
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-06-17
publication date: 2026-06-17
starting page: 407
ending page: 430
doi: 10.3846/mma.2026.24019

Uniform numerical method for singularly perturbed parabolic time delay reaction-diffusion problems arising in control theory
type: item
description: This paper introduces a uniform numerical scheme to find approximate solutions for singularly perturbed parabolic time delay reaction-diffusion problems. The scheme utilizes the Crank-Nicolson method for approximating time derivatives, combined with a novel finite difference method for spatial discretization. The stability and uniform convergence of the proposed scheme are investigated. The primary objective of this work is to demonstrate that the proposed scheme achieves a parameter-free error bound of order O(k2 + N−2). To validate the theoretical results, various numerical experiments have been conducted, showing that the proposed scheme yields superior results compared to some existing methods in the literature.
subject: Articles
  topic: section
subject: singular perturbation problem
  topic: keywords
subject: reaction-diffusion equation
  topic: keywords
subject: Crank-Nicolson method
  topic: keywords
subject: time delay
  topic: keywords
creator: Mousa J. Huntul
creator: Imiru Takele Daba
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-06-17
publication date: 2026-06-17
starting page: 392
ending page: 406
doi: 10.3846/mma.2026.24165

Second derivative two-step peer methods
type: item
description: This paper is devoted to extending two-step peer methods, for the numerical solution of ordinary differential equations, for the case where the second derivative of the solution is incorporated into the formula of the methods. The main features including consistency, zero-stability, and convergence of the proposed methods together with their order conditions and stability analysis are examined. Construction of explicit methods within the proposed class of the methods, possessing the Runge–Kutta stability property, is investigated, and examples of such methods up to order five are provided. The efficiency and accuracy of the constructed methods are validated through various numerical experiments conducted in both fixed and variable stepsize environments.
subject: Articles
  topic: section
subject: two-step peer methods
  topic: keywords
subject: second derivative methods
  topic: keywords
subject: non-stiff or mildly stiff ODEs
  topic: keywords
subject: Runge–Kutta stability
  topic: keywords
creator: Mohammad Sharifi
creator: Ali Abdi
creator: Michał Braś
creator: Gholamreza Hojjati
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-06-17
publication date: 2026-06-17
starting page: 369
ending page: 391
doi: 10.3846/mma.2026.24590

Representation of solutions of systems of linear differential equations with multiple delays and nonpermutable variable coefficients
type: item
description: Solutions of nonhomogeneous systems of linear differential equations with multiple constant delays are explicitly stated without a commutativity assumption on the matrix coefficients. In comparison to recent results, the new formulas are not inductively built, but depend on a sum of noncommutative products in the case of constant coefficients, or on a sum of iterated integrals in the case of time-dependent coefficients. This approach shall be more suitable for applications. Representation of a solution of a Cauchy problem for a system of higher order delay differential equations is also given.
subject: Articles
  topic: section
subject: Laplace transform
  topic: keywords
subject: multiple delays
  topic: keywords
subject: matrix polynomial
  topic: keywords
subject: nonhomogeneous equation
  topic: keywords
subject: noncommutative product
  topic: keywords
creator: Michal Pospíšil
rights: Copyright (c) 2020 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2020-03-18
publication date: 2020-03-18
starting page: 303
ending page: 322
doi: 10.3846/mma.2020.11194

Fractional integro-differential equations with nonlocal conditions and ψ–Hilfer fractional derivative
type: item
description: Considering a fractional integro-differential equation with nonlocal conditions involving a general form of Hilfer fractional derivative with respect to another function. We show that weighted Cauchy-type problem is equivalent to a Volterra integral equation, we also prove the existence, uniqueness of solutions and Ulam-Hyers stability of this problem by employing a variety of tools of fractional calculus including Banach fixed point theorem and Krasnoselskii's fixed point theorem. An example is provided to illustrate our main results.
subject: Articles
  topic: section
subject: fractional integro-diﬀerential equations
  topic: keywords
subject: ψ -Hilfer fractional derivative
  topic: keywords
subject: ψ-fractional integral
  topic: keywords
subject: existence and and Ulam-Hyers stability
  topic: keywords
subject: ﬁxed point theorem
  topic: keywords
subject: Mittag-Leﬄer function
  topic: keywords
subject: least squares method
  topic: keywords
creator: Mohammed S. Abdo
creator: Satish K. Panchal
creator: Hussien Shafei Hussien
rights: Copyright (c) 2019 The Author(s). Published by Vilnius Gediminas Technical University. https://creativecommons.org/licenses/by/4.0
date: 2019-10-25
publication date: 2019-10-25
starting page: 564
ending page: 584
doi: 10.3846/mma.2019.034

On fractional Volterra integrodifferential equations with fractional integrable impulses
type: item
description: We consider a class of nonlinear fractional Volterra integrodifferential equation with fractional integrable impulses and investigate the existence and uniqueness results in the Bielecki’s normed Banach spaces. Further, Bielecki-Ulam type stabilities have been demonstrated on a compact interval. A concrete example is provided to illustrate the outcomes we acquired.
subject: Articles
  topic: section
subject: fractional Volterra integrodifferential equation
  topic: keywords
subject: integrable impulses
  topic: keywords
subject: Banach contraction principle
  topic: keywords
subject: existence of solutions
  topic: keywords
subject: Bielecki norm
  topic: keywords
subject: Bielecki-Ulam type stability
  topic: keywords
creator: Sagar T. Sutar
creator: Kishor D. Kucche
rights: Copyright (c) 2019 The Author(s). Published by Vilnius Gediminas Technical University. https://creativecommons.org/licenses/by/4.0
date: 2019-06-06
publication date: 2019-06-06
starting page: 457
ending page: 477
doi: 10.3846/mma.2019.028

Generalized dynamic inequalities similar to Hardy’s inequality involving a convex function on time scales
type: item
description: 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 $\mathbb{T}=q^{\mathbb{N}_{0}}$, $q>1$.
subject: Articles
  topic: section
subject: Hardy-type inequalities
  topic: keywords
subject: time scales
  topic: keywords
subject: Jensen's inequality
  topic: keywords
subject: chain rule
  topic: keywords
creator: Ahmed I. Saied
creator: Mario Krnić
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-04-27
publication date: 2026-04-27
starting page: 348
ending page: 368
doi: 10.3846/mma.2026.24795

Optimal control and stability analysis of a glucose-insulin-FFA dynamic model with GLP-1 incretin effects
type: item
description: 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.
subject: Articles
  topic: section
subject: mathematical model
  topic: keywords
subject: glucose
  topic: keywords
subject: insulin
  topic: keywords
subject: free fatty acids
  topic: keywords
subject: GLP-1
  topic: keywords
subject: sensitivity analysis
  topic: keywords
subject: optimal control
  topic: keywords
creator: Salma Al-Tuwairqi
creator: Khlood Al Amer
creator: Hunida Malaikah
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-04-27
publication date: 2026-04-27
starting page: 326
ending page: 347
doi: 10.3846/mma.2026.24231

Efficient spectral-Galerkin method for the two-dimensional Gray-Scott model
type: item
description: 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.
subject: Articles
  topic: section
subject: spectral method
  topic: keywords
subject: Gray-Scott model
  topic: keywords
subject: generalized Jacobi polynomials
  topic: keywords
subject: nonlinear system
  topic: keywords
subject: reaction-diffusion system
  topic: keywords
creator: Yujian Jiao
creator: Mengshu Jia
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-04-27
publication date: 2026-04-27
starting page: 303
ending page: 325
doi: 10.3846/mma.2026.24728

Extension of a Bohr-Jessen type theorem for the Epstein zeta-function in short intervals
type: item
description: 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 $[0, T]$.
subject: Articles
  topic: section
subject: Epstein zeta-function
  topic: keywords
subject: limit theorem
  topic: keywords
subject: Haar probability measure
  topic: keywords
subject: weak convergence
  topic: keywords
creator: Aidas Balčiūnas
creator: Virginija Garbaliauskienė
creator: Renata Macaitienė
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-04-27
publication date: 2026-04-27
starting page: 288
ending page: 302
doi: 10.3846/mma.2026.24285

A successive midpoint-based method for the numerical analysis of chaotic systems with local and nonlocal operators
type: item
description: 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.
subject: Articles
  topic: section
subject: fractal differential equations
  topic: keywords
subject: Krasnoselskii-Krein uniqueness theorem
  topic: keywords
subject: successive midpoint method
  topic: keywords
creator: Seda İğret Araz
creator: Mehmet Akif Çetin
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-04-24
publication date: 2026-04-24
starting page: 267
ending page: 287
doi: 10.3846/mma.2026.24857

Well-posedness and exponential stability for the logarithmic Lamé system with a time varying delay
type: item
description: 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.
subject: Articles
  topic: section
subject: logarithmic Lamé system
  topic: keywords
subject: global existence
  topic: keywords
subject: exponential stability
  topic: keywords
subject: nonlinear equations
  topic: keywords
subject: time varying delay
  topic: keywords
creator: Fares Yazid
creator: Salah Mahmoud Boulaaras
creator: Mohammad Shahrouzi
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-02-10
publication date: 2026-02-10
starting page: 194
ending page: 213
doi: 10.3846/mma.2026.24819

Numerical algorithms to solve inverse problems for parabolic equations
type: item
description: 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.
subject: Articles
  topic: section
subject: inverse problems
  topic: keywords
subject: parabolic partial differential equations
  topic: keywords
subject: Tikhonov regularization
  topic: keywords
subject: adjoint problem
  topic: keywords
subject: gradient-based optimization
  topic: keywords
creator: Raimondas Čiegis
creator: Olga Suboč
creator: Vadimas Starikovičius
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-02-12
publication date: 2026-02-12
starting page: 246
ending page: 266
doi: 10.3846/mma.2026.25585

A new closed-form solution for optimal portfolio selection with liquidity risk
type: item
description: 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.
subject: Articles
  topic: section
subject: portfolio choice problem
  topic: keywords
subject: log-return assumption
  topic: keywords
subject: liquidity cost
  topic: keywords
subject: closed-form solution
  topic: keywords
creator: Dhruv Goel
creator: Song-Ping Zhu
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-02-11
publication date: 2026-02-11
starting page: 228
ending page: 245
doi: 10.3846/mma.2026.24351

Re-iterated approximation methods for nonlinear Volterra integral equations
type: item
description: 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.
subject: Articles
  topic: section
subject: nonlinear Volterra integral equations
  topic: keywords
subject: Newton method
  topic: keywords
subject: smooth kernel
  topic: keywords
subject: weakly singular kernel
  topic: keywords
subject: iterated Galerkin method
  topic: keywords
creator: Samiran Chakraborty
creator: Kapil Kant
creator: Ritu Nigam
creator: Gnaneshwar Nelakanti
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-02-10
publication date: 2026-02-10
starting page: 214
ending page: 227
doi: 10.3846/mma.2026.22594

Analysis of the Darcy-Brinkman flow with viscous dissipation and non-homogeneous thermal boundary condition
type: item
description: 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.
subject: Articles
  topic: section
subject: viscous dissipation
  topic: keywords
subject: porous medium
  topic: keywords
subject: non-homogeneous boundary conditions
  topic: keywords
subject: asymptotic modeling
  topic: keywords
creator: Igor Pažanin
creator: Francisco J. Suárez-Grau
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-01-21
publication date: 2026-01-21
starting page: 172
ending page: 193
doi: 10.3846/mma.2026.24073

Generalized practical stability of Hopfield-type neural networks differential equations
type: item
description: 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.
subject: Articles
  topic: section
subject: equilibrium
  topic: keywords
subject: Hopfield neural networks
  topic: keywords
subject: generalized practical stability
  topic: keywords
subject: boundedness
  topic: keywords
creator: Hanen Damak
creator: Mohamed Ali Hammami
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-01-21
publication date: 2026-01-21
starting page: 149
ending page: 171
doi: 10.3846/mma.2026.23045

Analytical formulas for polynomial coefficients in radial basis function interpolation
type: item
description: 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.
subject: Articles
  topic: section
subject: Radial basis functions
  topic: keywords
subject: RBF
  topic: keywords
subject: meshless methods
  topic: keywords
subject: RBF interpolation
  topic: keywords
subject: polynomial
  topic: keywords
subject: RBF neural networks
  topic: keywords
subject: RBF-NN
  topic: keywords
subject: invariance
  topic: keywords
subject: scattered spatio-temporal data
  topic: keywords
creator: Vaclav Skala
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-01-21
publication date: 2026-01-21
starting page: 130
ending page: 148
doi: 10.3846/mma.2026.22788

Weak solutions to degenerate p(t)-Laplacian elliptic equations involving (q, r(t)) double phase Hardy terms
type: item
description: 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.
subject: Articles
  topic: section
subject: weighted Sobolev space
  topic: keywords
subject: degenerate p(t)-Laplacian operators
  topic: keywords
subject: r(t)-Hardy terms
  topic: keywords
subject: variational methods
  topic: keywords
creator: Lili Qi
creator: Jian Liu
creator: Khaled Kefi
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-01-21
publication date: 2026-01-21
starting page: 116
ending page: 129
doi: 10.3846/mma.2026.24048

On improved P1-interpolation error estimates in W1,p(0, 1): application to the finite element method
type: item
description: 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.
subject: Articles
  topic: section
subject: Taylor-like formula
  topic: keywords
subject: error estimates
  topic: keywords
subject: interpolation error
  topic: keywords
subject: approximation error
  topic: keywords
subject: finite elements
  topic: keywords
creator: Joel Chaskalovic
creator: Franck Assous
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-01-21
publication date: 2026-01-21
starting page: 96
ending page: 115
doi: 10.3846/mma.2026.22775

Variable s-step CGNR method for solving the matrix equation AXB = C arising in image processing
type: item
description: 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.
subject: Articles
  topic: section
subject: matrix equation
  topic: keywords
subject: CGNR method
  topic: keywords
subject: variable s-step algorithm
  topic: keywords
subject: regularization method
  topic: keywords
creator: Hojjatollah Shokri Kaveh
creator: Masoud Hajarian
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-01-21
publication date: 2026-01-21
starting page: 79
ending page: 95
doi: 10.3846/mma.2026.23695

On shifts of periodic zeta-function in short intervals
type: item
description: 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.
subject: Articles
  topic: section
subject: approximation of analytic functions
  topic: keywords
subject: Dirichlet series
  topic: keywords
subject: Hurwitz zeta-function
  topic: keywords
subject: periodic zeta-function
  topic: keywords
subject: weak convergence
  topic: keywords
creator: Marius Grigaliūnas
creator: Antanas Laurinčikas
creator: Darius Šiaučiūnas
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-01-21
publication date: 2026-01-21
starting page: 63
ending page: 78
doi: 10.3846/mma.2026.24070

Study of a class of nonlinear heterogeneous diffusion with mixed phases under L∞ − data
type: item
description: 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.
subject: Articles
  topic: section
subject: double phase operator
  topic: keywords
subject: weak solution
  topic: keywords
subject: existence
  topic: keywords
subject: semi-discretization
  topic: keywords
subject: Rothe’s method
  topic: keywords
subject: weighted Sobolev space
  topic: keywords
creator: El Mehdi Hassoune
creator: Ahmed Jamea
creator: Abdelghali Ammar
creator: Adnane Kaddiri
rights: Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University. http://creativecommons.org/licenses/by/4.0
date: 2026-01-20
publication date: 2026-01-20
starting page: 47
ending page: 62
doi: 10.3846/mma.2026.23889