Matrix Ordinary Differential Equations Research Articles

Performance analysis of multilayered transversely isotropic saturated media under temperature and horizontally circular loads

AbstractThis study constructs a multilayered transversely isotropic saturated model under thermal and horizontally circular loads, and further investigates the model's thermo‐hydro‐mechanical coupling response. Firstly, the ordinary differential matrix equations of thermoelastic saturated media in the integral transformed domain are derived. Secondly, the solution for multilayered thermoelastic saturated media is developed using the extended precise integration method (EPIM), along with the boundary conditions at both ends of the foundation and the continuity conditions between adjacent layers. After that, the solution in the physical domain is further attained with the use of Laplace‐Hankel integral transform inversion. Finally, the accuracy of the proposed theory is confirmed through numerical examples, and the influences of anisotropic parameters, the soil's stratification and porosity on the thermo‐hydro‐mechanical coupling response of the media are studied.

Phase change in a one-dimensional functionally graded material

Functionally graded materials (FGMs) are a class of materials with engineered microstructure to achieve spatial variation in properties so as to serve a desired function. While FGMs have been well-researched for their structural properties, there is a lack of investigation of FGMs in solid-liquid phase change heat transfer processes. A well-designed functionally graded phase change material (FG-PCM) may overcome some of the well-known limitations of the phase change process. In order to investigate this possibility, this work presents a theoretical model to predict phase change propagation in a one-dimensional FG-PCM containing an arbitrary spatial variation in thermal conductivity, thermal diffusivity and latent heat. The problem is solved using the method of eigenfunction expansion to determine the temperature distribution, wherein, spatial variation in thermophysical properties listed above is accounted for by deriving a first-order matrix ordinary differential equation in the coefficients of the series solution. Results are found to correctly reduce to the well-known Stefan solution under special conditions. The impact of variation in thermophysical properties is investigated in detail. It is found that while spatial variation in thermal conductivity and latent heat has a significant effect on the rate of phase change, the impact of the thermal diffusivity function is negligible for realistic values of the Stefan number. The effect of the Stefan number governing this problem on the rate of melting is investigated. The nature of phase change under linear (1+aξ), polynomial (1+aξn) and exponential (eaξ) functional forms of thermophysical properties is investigated. It is shown that an appropriate choice of these spatial functions may help overcome the well-known self-limiting nature of phase change and produce a nearly-linear phase change propagation curve. For example, results show 33% increase in phase change propagation for a linear (1+5ξ) variation in thermal conductivity compared to baseline. The theoretical tools developed here may help design and optimize functionally graded PCMs towards improved phase change heat transfer in applications of practical interest.

Stability analysis of Cohen-Grossberg neural networks with time-varying delay by flexible terminal interpolation method

<abstract><p>In the paper, the existence and uniqueness of the equilibrium point in the Cohen-Grossberg neural network (CGNN) are first studied. Additionally, a switched Cohen-Grossberg neural network (SCGNN) model with time-varying delay is established by introducing a switched system to the CGNN. Based on reducing the conservativeness of the system, a flexible terminal interpolation method is proposed. Using an adjustable parameter to divide the invariant time-delay interval into multiple adjustable terminal interpolation intervals $ (2^{\imath +1}-3) $, more moments when signals are transmitted slowly can be captured. To this end, a new Lyapunov-Krasovskii functional (LKF) is constructed, and the stability of SCGNN can be estimated. Using the LKF method, a quadratic convex inequality, linear matrix inequalities (LMIs) and ordinary differential equation theory, a new form of stability criterion is obtained and specific instances are given to prove the applicability of the new stability criterion.</p></abstract>

Generic Solving of Physiologically-based Kinetic Models in Support of Next Generation Risk Assessment Due to Chemicals

Increasing confidence in using in vitro and in silico model-based data to aid the chemical risk assessment process is one of the most significant challenges faced by regulatory authorities. A crucial concern is taking full advantage of scientifically valid physiologically-based kinetic (PBK) models. Nevertheless, risk assessors remain unwilling to employ PBK models in their daily work. Indeed, PBK models are not often included in current official guidance documents. In addition, most users have limited experience with modeling in general. So, the complexity of PBK models and a lack of evaluation methods for their performance certainly contribute to their limited use in practical risk assessment. This paper proposes an innovative and unified modeling framework for writing PBK equations as matrix ordinary differential equations (ODE) and their solutions, expressed with matrix products. This generic PBK solution considers as many state variables as needed to quantify chemical absorption, distribution, metabolism, and excretion processes within living organisms when exposed to chemical substances. This generic PBK model makes possible any compartmentalization to be considered, as well as all appropriate interconnections between compartments and with the external medium. We first introduce our PBK modeling framework, with all the intermediate steps from the matrix ODE to the exact solution. Then we apply this framework to bioaccumulation testing before illustrating its concrete use through complementary case studies regarding species, compounds, and model complexity.

Nonzero Solutions for Nonlinear Systems of Fourth-Order Boundary Value Problems

This study is devoted to the investigation of nonlinear systems of fourth-order boundary value problems. Namely, using some techniques from matrix analysis and ordinary differential equations, a Lyapunov-type inequality providing a necessary condition for the existence of nonzero solutions is obtained. Next, an estimate involving generalized eigenvalues is derived as an application of our main result.

Free vibration analysis of functionally graded magneto-electro-elastic plates with in-plane material heterogeneity

A three dimensional elasticity analysis for the transverse free vibration characteristics of functionally graded magneto-electro-elastic plates based on the scaled boundary finite element method (SBFEM) incorporated with the precise integration algorithm (PIA) is presented. The material properties of magneto-electro-elastic plates are changing along the in-plane direction with arbitrary mathematical functions. In the proposed methodology, the strategies of only discretizing the in-plane surface and utilizing the two-dimensional spectral elements to construct diagonal coefficient matrices are adopted, which contributes to decreasing the calculation effort. The derivation process begins with the three dimensional governing equations of magneto-electro-elastic materials. Neither the plate mechanical kinematics nor invoking assumptions on the spatial distributions of electric and magnetic quantities are adopted. Built upon the introduced scaled boundary coordinates, the principle of virtual work and the technique of dual vectors, a first order ordinary differential SBFEM matrix equation for the in-plane functionally graded magneto-electro-elastic plates is obtained. Its general solution is analytically denoted as the matrix exponent. To improve the computation accuracy of the matrix exponent, the PIA is utilized to form the stiffness matrix. By virtue of the kinetic energy technique, it is convenient to construct the mass matrix of the in-plane functionally graded magneto-electro-elastic plates based on the SBFEM for the first time. Finally, comparisons of flexural frequency parameters with those from exact solutions and other numerical methods are provided. The accuracy, effectiveness, and versatility of the employed technique are validated. Moreover, additional numerical exercises are conducted to exhibit the influences of boundary conditions, material gradation functions, and aspect ratios on the free vibration behaviors of in-plane functionally graded magneto-electro-elastic rectangular, circular, and perforated plates.

Extended precise integration solution to layered transversely isotropic unsaturated poroelastic media under harmonically dynamic loads

A fully coupled dynamic model is presented in this paper by considering the effect of matric suction, varying parameters in connection with saturation, stratification and transverse isotropy. This model can describe separate flows of two immiscible fluids in layered transversely isotropic unsaturated poroelastic media subjected to harmonically dynamic loads. The ordinary differential matrix equations of this model are derived based on Hankel integral transform and a series of algebraic manipulations. Then the final solution is obtained by conducting the extended precise integration method, and further verified against the existing solution. Numerical examples are carried out to demonstrate the effects of saturation, transversely isotropic parameters, angular frequency and soil thickness on dynamic behavior of unsaturated porous media.

Modal waves in multiconductor transmission lines by using fundamental matrix response

The differential equations that model voltage and current for a multiconductor transmission line are written in matrix form. Supposing a time exponential solution through of the modal analysis the modal waves are obtained and solution of a ordinary matrix differential equation, thus determining the amplitude for voltage and current. The modal waves are given in terms of the fundamental matrix solution associated to the ordinary matrix differential equation. The decomposition of the modal waves in forward and backward propagators are used for determine the reflection and transmission matrices for junction in transmission lines. Circulant symmetric transmission lines are discussed, case in that the values for the self-impedance are the same as well as the mutual-impedance values and the same considerations to the admittance matrix. In particular, for these transmission lines are characterized the propagation constants and is observed that the number of multiconductors has effects only on a specific propagation constant. Numerical example of one multiconductor transmission line is presented allowing to observe important aspects of the methodology developed.

Backstepping Method for Stabilizing System of 2×2 Riccati Matrix Differential Equations

In this research paper, the backstepping method (BSM) will be proposed for stabilizing and solving system of 2×2 Riccati matrix ordinary differential equations. Such equations have many difficulties in the studying their solutions and stability. The basic idea behind of this approach is to use the BSM as a transformation method for transforming the original system into an equivalent one which is stabilizable and solvable based on constructing the Lyapunov function.

Swarms on the 3-sphere for online clustering of multivariate time series and data streams

We propose a novel algorithm named SwarmStream for real-time clustering of multivariate time series and data streams. The algorithm is inspired by phenomena of self-organization in complex systems. The data are processed by a swarm whose motions are governed by the system of matrix ordinary differential equations (ODEs) on the matrix group SU(2) with the group manifold S3. Such systems of matrix ODEs are referred to as non-Abelian Kuramoto models in Physics. With each object we associate one generalized oscillator and encode its attributes into the corresponding frequency matrix. The underlying idea is that oscillators with similar frequencies will synchronize their motions due to the internal communication within the swarm. The method is illustrated in several short videos, demonstrating evolution of clusters in some sets of evolving multivariate data.

Three dimensional mechanical behaviors of in-plane functionally graded plates

A semi-analytical solution procedure to investigate the distributions of displacement and stress components in the in-plane functionally graded plates based on the scaled boundary finite element method (SBFEM) in association with the precise integration algorithm (PIA) is developed in this paper. The proposed approach is applicable to conduct the flexural analysis on functionally graded plates with various geometric configurations, boundary conditions, aspect ratios and gradient functions. The elastic material parameters of functionally graded plates discussed here are mathematically formulated as power law, exponential and trigonometric functions varied along with the in-plane directions in a continuous pattern. Only a surface of the plate parallel to the middle plane is required to be discretized with two dimensional high order spectral elements, which contributes to reducing the computational expense. By virtue of the scaled boundary coordinates, the virtual work principle and the internal nodal force vector, the basic equations of elasticity are converted into a first order ordinary differential SBFEM matrix equation. The general solution of the governing equation is analytically expressed as a matrix exponential with respect to the transverse coordinate z. According to the PIA, the stiffness matrix from the matrix exponential can be acquired. Considering that the PIA is a highly accurate method, any desired accuracy of the displacement and stress field can be obtained. The entire derivation process is built on the three dimensional elasticity equations without importing any assumptions on the plate kinematics. Comparisons with numerical solutions available from prevenient researchers are made to validate the high accuracy, efficiency and serviceability of the employed technique. Additionally, circular and perforated examples are provided to highlight the performance of the developed methodology and depict the influences of boundary conditions, thickness-to-length ratios and gradient indexes on the deformable behaviors of in-plane functionally graded plates.

Localized solutions of the Dirac equation in free space and electromagnetic space-time crystals

Localized solutions of the Dirac equation for an electron moving in free space and electromagnetic field lattices with periodic dependence on space-time coordinates (electromagnetic space-time crystals) are treated using the expansions in basis wave functions. The techniques for calculating these functions with any prescribed accuracy are presented. It is shown that in the crystals created by two counterpropagating plane electromagnetic waves with the same or the opposite circular polarizations, the Dirac equation describing the basis functions reduces to matrix ordinary differential equations. These functions and the corresponding mean values of velocity, momentum, energy, and spin operators are found for the both types of crystals. Localized solutions describing the families of orthonormal beams in electromagnetic space-time crystals and free space, defined by a given set of orthonormal complex scalar functions on a two-dimensional manifold, are obtained. By way of illustration the orthonormal beams in free space and various localized states with complex vortex structure of probability currents, defined by the spherical harmonics, are presented. The obtained solutions have high probability density only in very small core regions. The evolution of wave packets with one-dimensional localization in the both types of crystals created by two circularly polarized waves is described.

Simulink models of technological systems with perfect mixing and plug-flow hydrodynamics

The dynamic models of elements of technological systems with perfect mixing and plug-flow hydrodynamics are based on the systems of algebraic and differential equations that describe a change in the basic technological parameters. The main difficulty in using such models in MathWorks Simulink™ computer simulation systems is the representation of ordinary differential equations (ODE) and partial differential equations (PDE) that describe the dynamics of a process as a MathWorks Simulink™ block set. The study was aimed at developing an approach to the synthesis of matrix dynamic models of elements of technological systems with perfect mixing and plug-flow hydrodynamics that allows for transition from PDE to an ODE system on the basis of matrix representation of discretization of coordinate derivatives. The process of synthesis of the dynamic matrix mathematical model was considered by the example of a sugar syrup cooler, the quality indicator of the finished product are selected as sucrose crystals and their portion in the total volume of caramel mass. Taking into account the dependence of syrup viscosity on temperature, thermal effects as a result of the process of crystallization of sucrose from syrup, design features of a typical caramel machine made it possible to clarify the dynamics of the process of syrup cooling. The model developed with this approach allows to obtain real-time estimates of temperatures at the outlet of the cooler, which makes it possible to study the dynamics of the technological process and synthesize the control system. The presented approach allows to implement mathematical models of ideal reactors in Simulink system and to move to matrix ordinary differential equations, which makes it possible to convert them into Simulink blocks. The approach is also applicable to other models of ideal reactors, which allows to form libraries of typical ideal reactors of Simulink system for synthesis of heat and mass exchange equipment. The proposed approach significantly simplifies the study and modernization of the current and the development of new technological equipment, as well as the synthesis of algorithms for controlling the processes therein.

Dynamic Analysis of the Biomechanical Model of Head Load Impact Using Differential Transform Method

The dynamic analysis of the biomechanical model of the head load impact using the Differential Transform Method is presented in this paper. In many parts of the world, the problem of traumatic brain injuries (TBI) has led to neurodegenerative dementing disorders and diseases as a result of head load impact from sporting activities, accidents involving the head, etc. have serious effects on humanity. The head load impact and its control have been modeled as a rigid linkage head-neck manipulator. This rigid link manipulator is governed by a system of nonlinear ordinary differential matrix equations with three degrees of freedom which requires special techniques for its solution. The system of equations was solved using Differential Transform Method (DTM) and the results were compared with results obtained in earlier studies and validated with the fourth-order Runge-Kutta numerical method (RK4). Good agreements are reached in all these results. From the model, the effects of head loads, head mass, neck mass, upper and lower linkage lengths, head and neck moments of inertia were investigated. As the head loads increased, there were increases in both axial and angular displacement of the head motion and the neck region. The study provides a theoretical basis for the design and understanding of the effects of head load carriage on vital organs that are susceptible to pains, damages, and even failure.

Toward a Unification of System-Theoretical Principles in Biology and Ecology—The Stochastic Lyapunov Matrix Equation and Its Inverse Application

System theory has its roots in mathematical formalisms developed by mathematicians and physicians, such as Leibniz, Euler and Newton, and applied by congenial ecologists and biologists such as Lotka and Bertalanffy. In these approaches the dynamical system – may it be either single organisms or populations of organisms in their ecosystems – is defined and formally translated into an interaction matrix and first order ordinary differential equations (ODE) which are then solved. This provides the background for the quantitative analysis of any linear to non-linear system. In his inspiring article “ can a biologist fix a radio?” Lazebnik made the differences very clear between a “guilt-by association” hypothesis of a modern biologist versus a Signal-Input-Output (SIO) model of an electrical engineer. The drawback of this “Gedankenexperiment” is that two different approaches are compared – a forward approach in case of the SIO model by an engineer and an inverse or reverse approach by the biologist or ecologist. The point is that biological and ecological systems are much too complex to estimate the deterministic input signals that generate a probability distribution. Thus, a combination of reverse data-driven modelling with stochastic simulation is a key process to understand the control of a biological or ecological system. The challenge of the next decades of systems biology is to link these approaches more systematically. Over the last years we and others have developed such an hybrid modelling approach based on the stochastic Lyapunov matrix equation for the analysis of genome-scale molecular data which links forward and reverse strategies such as the genome-scale based metabolic reconstruction of an organism and the calculation of dynamics around a quasi steady state using statistical features of large-scale PANOMICS data. PANOMICS combines all molecular and phenotypical levels of an organism from the genome, transcriptome, proteome, metabolome to morphology and physiology to unambiguously define the genotype-phenotype-relationship. This system-theoretical formalism establishes the generic analysis of the genotype-environment-phenotype-relationship to finally result in predictability of organismal function.

A study of heat transfer during cryosurgery of lung cancer.

In this study, a mathematical model describing two-dimensional bio-heat transfer during cryosurgery of lung cancer is developed. The lung tissue is cooled by a cryoprobe by imposing its surface at a constant temperature or a constant heat flux or a constant heat transfer coefficient. The freezing starts and the domain is distributed into three stages namely: unfrozen, mushy and frozen regions. In stage I where the only unfrozen region is formed, our problem is an initial-boundary value problem of the hyperbolic partial differential equation. In stage II where mushy and unfrozen regions are formed, our problem is a moving boundary value problem of parabolic partial differential equations and in stage III where frozen, mushy, and unfrozen regions are formed, our problem is a moving boundary value problem of parabolic partial differential equations. The solution consists of the three-step procedure: (i) transformation of problem in non-dimensional form, (ii) by using finite differences, the problem converted into ordinary matrix differential equation and moving boundary problem of ordinary matrix differential equations, (iii) applying Legendre wavelet Galerkin method the problem is transferred into the generalized system of Sylvester equations which are solved by applying Bartels-Stewart algorithm of generalized inverse. The complete analysis is presented in the non-dimensional form. The consequence of the imposition of boundary conditions on moving layer thickness and temperature distribution are studied in detail. The consequence of Stefan number, Kirchoff number and Biot number on moving layer thickness are also studied in specific.

Semi-analytical analysis of static and dynamic responses for laminated magneto-electro-elastic plates

The scaled boundary finite element method (SBFEM) incorporated with the precise integration technique (PIT) is further extended to present the semi-analytical analysis of static bending and free vibration behaviors of laminated magneto-electro-elastic composite plates. It is applicable to thin and thick multilayered magneto-electro-elastic plates. The basic equations are formulated by only using the two dimensional model, which helps to make sure that the computation is efficient. The discretization is carried out in terms of spectral elements. Only three displacement components, electric and magnetic potentials are selected as the basic variables. Characterized by the important features of the SBFEM, the elastic displacements, electric and magnetic potentials along the thickness direction can be solved analytically. Differing from most plate models taking a priori assumptions on distributions of the mechanical, electric and magnetic variables, the derivation of governing SBFEM equations strictly follows the 3D theory of magneto-electro-elastic materials without introducing any assumptions on multiphysics fields. By virtue of the scaled boundary coordinates and the principle of virtual work, the key partial differential equations are simplified into the governing ordinary differential matrix equation. To increase the accuracy of the SBFEM governing equation, the PIT is utilized, which ensures that any desired accuracy of the results can be reached. Static and dynamic numerical examples show that variations of mechanical, electric, magnetic fields and natural frequencies predicted by the proposed approach are in excellent agreement with the elastic solutions of other methods. Therefore, the versatility and perfect accuracy of the present technique is fully validated. Furthermore, the influences of aspect ratios, boundary conditions and stacking sequences on the cross-thickness distributions of displacements, stresses, electric potential, electric displacements, magnetic potential, magnetic induction and vibration frequencies in the multilayered magneto-electro-elastic plate are discussed.

Sign‐Consensus of Linear Multiagent Systems under a State Observer Protocol

The sign‐consensus problem for linear time‐invariant systems under signed digraph is considered. The information of the agents’ states is reconstructed, and then, a state observer‐type sign‐consensus protocol is proposed, whose performance is analyzed using matrix analysis and ordinary differential equation theory. Sufficient conditions for ensuring sign‐consensus are given. It is proven that if the adjacency matrix of the signed digraph has strong Perron–Frobenius property or is eventually positive, sign‐consensus can be achieved under the proposed protocol. In particular, conventional consensus is a special case of sign‐consensus under mild conditions.

A study of cryosurgery of lung cancer using Modified Legendre wavelet Galerkin method

In this paper, we have developed a new mathematical model describing bio-heat transfer during cryosurgery of lung cancer. The lung tissue cooled by a flat probe whose temperature decreases linearly with time. The freezing process occurs in three stages and the whole region is divided into solid, mushy and liquid region. The heat released in the mushy region is considered as discontinuous heat generation. The model is an initial-boundary value problem of the hyperbolic partial differential equation in stage 1 and moving boundary value problem of parabolic partial differential equations in stage 2 and 3. The method of the solution consists of four-step procedure as transformation of problem in dimensionless form, the problem of hyperbolic partial differential equation converted into ordinary matrix differential equation and the moving boundary problem of parabolic partial differential equations converted into moving boundary problem of ordinary matrix differential equations by using finite differences in space, transferring the problem into the generalized system of Sylvester equations by using Legendre wavelet Galerkin method and the solution of the generalized system of Sylvester equation are solved by using Bartels-Stewart algorithm of generalized inverse. The whole analysis is presented in dimensionless form. The effect of cryoprobe rate on temperature distribution and the effect of Stefan number on moving layer thickness is discussed in detail.

Thermo-mechanical performance of layered transversely isotropic media around a cylindrical/tubular heat source

We develop a new numerical model based on a precise integration method to investigate the coupled thermo-mechanical performance of layered transversely isotropic media around a cylindrical/tubular heat source. To obtain the relational matrices of the extended precise integration method, we first convert the governing equations of the problem into ordinary differential matrix equations through the Laplace–Hankel transform. Then, the cylindrical heat source is divided into a series of plane heat sources, and the plane temperature load term is added to the state vector between layer elements. By combining the layer elements, we build a layered transversely isotropic numerical model containing a cylindrical heat source in the transformed domain. Finally, we solve the model in the transformed domain and obtain the solution of the problem in the real domain through the Laplace–Hankel transform inversion. The accuracy of this method is verified by comparing the solutions with the results of the analytical method and the finite element method. Then, we study the influence of the anisotropy of thermal parameters, the embedded depth, the length/radius ratio, the type of heat source and the stratification of the medium on the thermo-mechanical coupled performance.