Matrix Differential Equation Research Articles (Page 1)

Preconditioning Low Rank Generalized Minimal Residual Method (GMRES) for Implicit Discretizations of Matrix Differential Equations

This paper introduces three novel numerical approaches for solving linear matrix differential equations. Leveraging exponential functions and collocation points, these methods transform the original problem into a system of algebraic equations through matrix operations. The first method employs negative exponential functions, the second adopts positive exponential functions, and the third combines both into extended exponential functions. Error analysis is provided, and an error problem formulated via the residual function is solved using the proposed techniques to estimate errors. Numerical examples demonstrate the effectiveness of the methods and the accuracy of the error estimation.

In this article, we consider eigenvector centrality for the nodes of a graph and study the robustness (and stability) of this popular centrality measure. For a given weighted graph G \mathcal {G} (both directed and undirected), we consider the associated weighted adjacency matrix A A , which by definition is a non-negative matrix. The eigenvector centralities of the nodes of G \mathcal {G} are the entries of the Perron eigenvector of A A , which is the (positive) eigenvector associated with the eigenvalue with largest modulus. They provide a ranking of the nodes according to the corresponding centralities. An indicator of the robustness of eigenvector centrality consists in looking for a nearby perturbed graph G ~ \widetilde {\mathcal {G}} , with the same structure as G \mathcal {G} (i.e., with the same vertices and edges), but with a weighted adjacency matrix A ~ \widetilde A such that the highest m m entries ( m ≥ 2 m \ge 2 ) of the Perron eigenvector of A ~ \widetilde A coalesce, making the ranking at the highest level ambiguous. To compute a solution to this matrix nearness problem, a nested iterative algorithm is proposed that makes use of a constrained gradient system of matrix differential equations in the inner iteration and a one-dimensional optimization of the perturbation size in the outer iteration. The proposed algorithm produces the optimal perturbation (i.e., the one with smallest Frobenius norm) of A A which causes the looked-for coalescence, which is a measure of the sensitivity of the graph. As a by-product of our methodology we derive an explicit formula for the gradient of a suitable functional corresponding to the minimization problem, leading to a characterization of the stationary points of the associated gradient system. This in turn gives us important properties of the solutions, revealing an underlying low rank structure. Our numerical experiments indicate that the proposed strategy outperforms more standard approaches based on algorithms for constrained optimization. The methodology is formulated in terms of graphs but applies to any non-negative matrix, with potential applications in fields like population models, consensus dynamics, economics, etc.

Stability to Quaternion Matrix Differential Equations with Singular Coefficients

In this study, we aim to construct a finite set of orthogonal matrix polynomials for the first time, along with their finite orthogonality, matrix differential equation, Rodrigues’ formula, several recurrence relations including three-term relation, forward and backward shift operators, generating functions, integral representation and their relation with Jacobi matrix polynomials. Thus, the concept of “finite”, which is used to impose parametric constraints for orthogonal polynomials, is transferred to the theory of matrix polynomials for the first time in the literature. Moreover, this family reduces to the finite orthogonal M polynomials in the scalar case when the degree is 1, thereby providing a matrix generalization of finite orthogonal M polynomials in one variable.

Abstract Due to its reduced memory and computational demands, dynamical low-rank approximation (DLRA) has sparked significant interest in multiple research communities. A central challenge in DLRA is the development of time integrators that are robust to the curvature of the manifold of low-rank matrices. Recently, a parallel robust time integrator that permits dynamic rank adaptation and enables a fully parallel update of all low-rank factors was introduced. Despite its favorable computational efficiency, the construction as a first-order approximation to the augmented basis-update & Galerkin integrator restricts the parallel integrator’s accuracy to order one. In this work, an extension to higher order is proposed by a careful basis augmentation before solving the matrix differential equations of the factorized solution. A robust error bound with an improved dependence on normal components of the vector field together with a norm preservation property up to small terms is derived. These analytic results are complemented and demonstrated through a series of numerical experiments.

In this article, an important type of fuzzy parabolic differential equations will be discussed, which is the one-dimensional fuzzy reaction-diffusion equation with fuzzy boundary conditions. This equation is one of the most widespread chemical fuzzy reaction-diffusion equations, as well as, studying the possibility of controlling and reducing the chemical pollution occurred in the chemical reactions. In order to reduce chemical contamination in the reaction medium, we observed that investigating this equation's stability is essential. In order to achieve stability, the fuzzy backstepping approach is proposed, which transforms the unstable system into a stable system after controlling the boundary conditions. Therefore, two different cases of Hukuhara derivatives must be considered, which are important in the study of fuzzy differential equations. Two cases are considered depending on the comparison between the lower and upper variable solution time derivative. Also, the proposed backstepping approach is applied based on the interval analysis of α-level sets. For this purpose, and in order to avoid the difficulty of separating the upper and lower solutions, the resulting non-fuzzy or crisp differential equations are converted into matrix differential equations, and then Consequently, we are able to remove the residual terms that are responsible for the instability of the open-loop. Moreover, this backstepping transformation is continuously invertible. Thus, the inverse transformation is used to obtain stabilizing state feedback for the original partial differential equation.

Damped wave equations have been used in many real-world simulations. In this paper, we study a low-rank solution method of the strongly damped wave equation with the damping term, visco-elastic damping term and mass term. Firstly, a second-order finite difference method is employed for spatial discretization. Then, we receive a system of second-order matrix differential equations. Next, we transform it into an equivalent system of first-order matrix differential equations, and split the transformed system into three subproblems. Applying a Strang splitting to such subproblems and combining a dynamical low-rank approach, we obtain a low-rank algorithm. Numerical experiments are reported to demonstrate that the proposed low-rank algorithm is robust and accurate, and has the second-order convergence rate in time.

This paper introduces two semi-Markov-modulated short rate processes and discusses the pricing of a zero-coupon bond. Specifically, a semi-Markov-modulated Hull-White (HW) model and a semi-Markov-modulated Cox-Ingersoll-Ross (CIR) model for short-term interest rates are proposed. Under the semi-Markov-modulated HW model, the mean-reverting level and the volatility of short rate process are modulated by a continuous-time, finite-state, semi-Markov chain. Under the semi-Markov-modulated CIR model, only the mean-reverting level is modulated by the semi-Markov chain. Using forward measures, stochastic flows of diffeomorphisms and fundamental matrix solutions of linear matrix differential equations, semi-analytical approximate formulas for the bond price are obtained under the two short rate processes. The formulas are of semi-Markov-modulated exponential affine forms. Numerical studies, comparisons and sensitivity analyses are provided and discussed.

Regarding this investigation, the Moore-Gibson-Thompson (MGT) model was developed with the impact of acoustic pressure. This research’s light is spotted on semiconductor material undergoing thermo-acoustic and optical deformation in the context of a theory of photo-thermoelasticity (PTE). The governing equations are formulated using a modified photo-excitation model, where (MGT) equation represents the heat conduction during processes of optical transport. This model represents the coupling between plasma, thermal, mechanical-elastic, and acoustic wave propagation. Analytical solutions for the main physical quantities are obtained utilizing the Laplace transform method combined with the vector–matrix differential equation method. Boundary conditions for the acoustic, plasma, and thermo-mechanical effects are applied at the outer surface of the medium. Numerical inversion of Laplace transforms is performed to obtain complete space–time solutions for primary fields. Silicon is utilized as a representative semiconductor material for numerical computations, with the results presented graphically and discussed with various influencing parameters. This study is significant because it provides a novel way to analyze the behavior of semiconducting materials under photo-acoustic excitation, applying the eigenvalue approach to a system previously modeled using simple methods. It fills existing gaps in the literature related to the application of the MGT model in semiconducting photo-acoustics and provides more detailed and reliable predictions for real-world applications.

Real physical events, such as earthquakes, sea waves, and wind, are often random in nature and can be defined as a realization of a stochastic process. The simplest way to model them is to use stationary processes. However, in some cases, it’s necessary to consider their evolutionary nature over time to properly account for their non-stationary nature, as in the case of seismic records. In such circumstances, it’s common to assume the process as a non-stationary separable process modulated by a deterministic function that can represent the time variation of the physical event. Solving this problem is a complex task, and there are a few numerical approaches proposed with this aim. In this paper, the case of the dynamic structural response of a linear multi-degree-of-freedom system subject to non-stationary random Gaussian dynamic actions is analyzed. In the case of non-stationary inputs, the evaluation of second-order spectral moments requires the solution of a Lyapunov matrix differential equation. In this work, numerical schemes for its resolution are proposed. The numerical computational effort is minimized by taking into account the symmetry characteristic of the state space covariance matrix. As an application of the proposed method, a multi-storey building is analyzed to determine the reliability of ensuring that the maximum inter-storey drift does not exceed a specified acceptable limit.

The paper considers the problem of analytical design of optimal regulators (ADOR) in the formulation of A. A. Krasovsky for the studied stable multidimensional objects described by a matrix differential equation with polynomial nonlinearities in phase coordinates. This class of control objects, called polynomial, is quite wide for applications: these models are used to describe the movement of systems of a very different nature, for example, electromechanical devices, chemical reactors, industrial facilities with recycling, biological and environmental systems, etc. To solve this problem of ADOR in the early work of the author, a method is proposed for synthesizing quasi-optimal controllers, which largely weakens the disadvantages of the power series method (a large volume of operations with polynomials that are not suitable for programming) through the use of a procedure for multidimensional linearization of the description of polynomial objects, carried out by expanding the space state of the object with new coordinates, which are products of the original phase coordinates, and the application of the apparatus of matrix theory with the Kronecker (direct) product. In this paper, a modification of this method of system synthesis is carried out by using a reduced rather than a full Kronecker degree of the object state vector and taking into account the block-diagonal structure of the parameter matrix of the applied linearized model of the object, which ensures a further multiple reduction in the volume of calculations in the synthesis of control systems. The proposed modified synthesis method makes it possible to find, in the form of a polynomial function, an approximate solution to the ADOR problem with high accuracy, and its implementation is extremely simple due to the use of mainly well-known software for solving linear-quadratic optimal control problems (procedures for solving Lyapunov and Sylvester matrix equations).The features of the method are illustrated using a specific example of the synthesis of a quasi-optimal control system.

This article explores the fixed point theorem for a novel class of interpolative relation theoretical convex mappings in TVS-valued cone metric spaces, integrating relational theory, convexity, and interpolation properties to offer fresh perspectives and possible uses in theoretical and applied mathematics. An application of the results to differential equations and matrix equations in the context of orthogonal TVS-valued cone metric spaces is presented, along with a constructive example to support the findings.

Dynamical low-rank approximation allows for solving large-scale matrix differential equations (MDEs) with significantly fewer degrees of freedom and has been applied to a growing number of applications. However, most existing techniques rely on explicit time integration schemes. In this work, we introduce a cost-effective Newton’s method for the implicit time integration of stiff, nonlinear MDEs on low-rank matrix manifolds. Our methodology is focused on MDEs resulting from the discretization of random partial differential equations (PDEs), where the columns of the MDE can be solved independently. Cost-effectiveness is achieved by solving the MDE at the minimum number of entries required for a rank- r approximation. We present a novel cross low-rank approximation that requires solving the parametric PDE at r strategically selected parameters and O ( r ) grid points using Newton’s method. The selected random samples and grid points adaptively vary over time and are chosen using the discrete empirical interpolation method or similar techniques. The proposed methodology is developed for high-order implicit multi-step and Runge–Kutta schemes and incorporates rank adaptivity, allowing for dynamic rank adjustment over time to control error. Several analytical and PDE examples, including the stochastic Burgers’ and Gray–Scott equations, demonstrate the accuracy and efficiency of the presented methodology.

In recent work, we demonstrated that a spectral variety for the Berry connection of a 2d \mathcal{N}=(2,2)𝒩=(2,2) GLSM with Kähler vacuum moduli space XX and Abelian flavour symmetry is the support of a sheaf induced by a certain action on the equivariant quantum cohomology of XX. This action could be quantised to first-order matrix difference equations obeyed by brane amplitudes, and by taking the conformal limit, vortex partition functions. In this article, we elucidate how some of these results may be recovered from a 3d perspective, by placing the 2d theory at a boundary and gauging the flavour symmetry via a bulk A-twisted 3d \mathcal{N}=4𝒩=4 gauge theory (a sandwich construction). We interpret the above action as that of the bulk Coulomb branch algebra on boundary twisted chiral operators. This relates our work to recent constructions of actions of Coulomb branch algebras on quantum equivariant cohomology, providing a novel correspondence between these actions and spectral data of generalised periodic monopoles. The effective IR description of the 2d theory in terms of a twisted superpotential allows for explicit computations of these actions, which we demonstrate for Abelian GLSMs.

In this paper, practically computable low-order approximations of potentially high-dimensional differential equations driven by geometric rough paths are proposed and investigated. In particular, equations are studied that cover the linear setting, but we allow for a certain type of dissipative nonlinearity in the drift as well. In a first step, a linear subspace is found that contains the solution space of the underlying rough differential equation (RDE). This subspace is associated to covariances of linear Ito-stochastic differential equations which is shown exploiting a Gronwall lemma for matrix differential equations. Orthogonal projections onto the identified subspace lead to a first exact reduced order system. Secondly, a linear map of the RDE solution (quantity of interest) is analyzed in terms of redundant information meaning that state variables are found that do not contribute to the quantity of interest. Once more, a link to Ito-stochastic differential equations is used. Removing such unnecessary information from the RDE provides a further dimension reduction without causing an error. The resulting reduced order rough equation can be solved numerically much faster than the original system. Therefore, our approach provides enormous savings in computing time and is hence beneficial from the practical point of view. Finally, we discretize a linear parabolic rough partial differential equation and a rough wave equation in space. The resulting large-order RDEs are subsequently tackled with the exact reduction techniques studied in this paper. We illustrate the enormous complexity reduction potential in the corresponding numerical experiments.

We analyze, using the Lyapunov-Krasovskii method, the conditions for the stability, boundedness and periodicity of solutions to a class of nonlinear matrix differential equation of third order with variable delay. Criteria under which the solutions to the equation considered possess solutions that are stable and bounded on the real line as well as existence of at least one periodic solution are given. Our results generalize and extend many existing results in the literature on scalar, vector and matrix differential equations with or without delay. The integrity of our results is demonstrated by two numerical examples included.

A generalized differential scheme for the effective conductivity of percolating microinhomogeneous materials with the Hall effect

في هذا البحث تناولنا طريقة فعالة وجديدة وهي الدمج بين طريقتي الادوميان والهوموتوبي مع إستخدام طريقة الخطوات لتسهيل المسألة والتي تخص المعادلات التفاضلية الاعتيادية التباطئية لحل معادلة المصفوفات التباطئية التربيعية الغير خطية . كلتا الطريقتين على درجة عالية من التأثير والفعالية. جزء التكلمل الكلي لطريقة الهوموتوبي سيستخدم بدلا من جزء التكامل الخاص ب الادوميان . الميزة الرئيسية لهذه التقنية هي الحصول على نتائج اكثر دقة و لفترة و منطقة اوسع واطول ولمعرفة دقة هذه النتائج تحت تأثير التأخير. الجزء الخاص بالتأخير يختفي بعد استخدام طريقة الخطوات. تم حساب الخطأ المتبقي . لتقليل الوقت والعمليات الحسابية المعقدة تم أستخدام تحويلات لابلاس. أخيرا النتائج التي تم الحصول عليها بينت ان التقنية فعالة وسريعة التقارب للحل المظبوط ولفترة ومنطقة اوسع . يمكن استخدام هذه التقنية لحل مسائل غير خطية مختلفة. طريقة الادوميان هي تقنية شبه تحليلية لحل معادلات تفاضلية مختلفة أعتيادية ؛ جزئية ؛ كسرية و تباظؤية . هذه الطريقة تطورت بواسطة جورج ادوميان. هي سريعة التقارب للحل المظبوط وتستخدم للخطية و غير الخطية و المتجانسة و غيلر المتجانسة. متعددة ادوميان تسمح للحل التقارب للحل المظبوط للمسألة قيد الدراسة دون الحاجة الى أي تحوير. طريقة الهوموتوبي هي تقنية شبه تحليلية لحل معادلات تفاضلية مختلفة أعتيادية؛ جزئية ؛ كسرية و تباطؤية وانواع مختلفة. هذه الطريقة تطورت عن طريق العالم ليو . هي سريعة التقارب للحل المظبوط وتستخدم للخطية وغير الخطية و المتجانسة وغير المتجانسة. جاء مفهوم الهوموتوبي من مفهوم التبلوجي في توليد متسلسلة متقاربة للحل المظبوط. هذه الطريقة تم ايجادها من قبل ليو خلال اطروحته. تحتوي الطريقة على متقير او معلمة خلاله تمكننا التقارب.

In this study, a biomechanical model mimicking the human hand-arm system under heavy disk excitation is developed to define the stability threshold between the preload vibration and the stress of the human hand-arm system. The fully electromechanical system, consisting of a DC motor, a transmission system, and three discrete masses representing the upper arm, forearm, and hand lifting a heavy disk, is connected by shock absorbers and various springs. The challenge of mimicking the angular activity of the elbow joint in torsion and flexion was also included to assess its contribution to system stability and to study the absorption of mechanical energy in the human hand-arm system. Finally, the movements of the model were described by a differential matrix equation, and its analysis facilitates the understanding of the threshold of the chaotic behaviors observed in an oscillating arms system. Specifically, it explores how the motion of a DC motor can be regulated using a single controller parameter within the context of a multi-degree-of-freedom human hand-arm system. The study uses numerical integrations to analyze system behaviors, with an emphasis on the use of frequency spectra, Poincaré maps, and bifurcation diagrams for visualization and interpretation. The results indicate that the system exhibits chaotic behavior when subjected to certain conditions, particularly when the mass load exceeds 35 kg. Imposing motion on the mass block further intensifies the chaos, leading to manifestations such as strong oscillations and frequency-synchronization effects. These characteristics, exhibited by the idealized model, which imitates the human body through a mechanical model and computation of the motions of the body, play a vital role in various fields of ergonomics and sports biomechanics. Understanding the dynamic behaviors of such systems, especially in response to varying conditions, has significant implications for engineering applications.