System Of Linear Ordinary Differential Equations Research Articles

Model order reduction for the input–output behavior of a geothermal energy storage

In this article, we consider a geothermal energy storage system in which the spatio-temporal temperature distribution is modeled by a heat equation with a time-dependent convection term. Such storage systems are often embedded in residential heating systems. The control and management of such systems requires knowledge of aggregated characteristics of the temperature distribution in the storage. These describe the input–output behavior of the storage, the associated energy flows, and their response to charging and discharging processes. Our aim is to derive an efficient, approximate description of these characteristics by using low-dimensional systems of ordinary differential equations (ODEs). This leads to a model order reduction problem for a large-scale linear system of ODEs resulting from the semidiscretization of the heat equation combined with a linear algebraic output equation. In a first step, we approximated the nonautonomous system of ODEs by a linear time-invariant system. Then, we applied Lyapunov balanced truncation model order reduction to approximate the output by a reduced-order system that has only a few state equations but almost the same input–output behavior. The results of our extensive numerical experiments show the efficiency of the applied model order reduction method. We found that only a few suitably chosen ODEs are sufficient to achieve good approximations of the input–output behavior of the storage.

A class of single-step hybrid block methods with equally spaced points for general third-order ordinary differential equations

This study presents a class of single-step, self-starting hybrid block methods for directly solving general third-order ordinary differential equations (ODEs) without reduction to first order equations. The methods are developed through interpolation and collocation at systematically selected evenly spaced nodes with the aim of boosting the accuracy of the methods. The zero stability, consistency and convergence of the algorithms are established. Scalar and systems of linear and nonlinear ODEs are approximated to test the effectiveness of the schemes, and the results obtained are compared against other methods from the literature. Significantly, the study shows that an increase in the number of intra-step points improves the accuracy of the solutions obtained using the proposed methods.

Love stories in a differential equations classroom

We believe that developing cultural competencies can help students learn mathematics and conversely that learning mathematical content can help students learn about themselves and others. Using frameworks introduced by Rendón (2009) and Gutiérrez (2018), we present a five-part bundle of activities for undergraduate differential equations course instructors, including one pre-activity reflection assignment, three modeling activities, and a final project. The Pre-Activity Assignment engages students to draw on their own personal and/or cultural experiences with the concept of love. Three activities focus on developing revision skills in mathematical modeling and practicing methods of solving systems of ordinary differential equations (ODEs). In these activities, students would collaborate to construct a relationship model consisting of a system of first-order linear ODEs and solve different model iterations with the characteristic polynomial, matrix form, or Laplace transform method. The final project connects the reflections in Pre-Activity Assignment with the skills developed in the three activities by inviting students to create a relationship scenario, model, revise, solve it, and present the conclusions. By engaging in this set of assignments, students connect personal and cultural experiences with the concept of love and perceive themselves and their peers in the curriculum, fostering a sense of belonging and relevance.

Energy‐based model order reduction for linear stochastic Galerkin systems of second order

AbstractWe consider a second‐order linear system of ordinary differential equations (ODEs) including random variables. A stochastic Galerkin method yields a larger deterministic linear system of ODEs. We apply a model order reduction (MOR) of this high‐dimensional linear dynamical system, where its internal energy represents a quadratic quantity of interest. We investigate the properties of this MOR with respect to stability, passivity and energy dissipation. Numerical results are shown for a system modelling a mass‐spring‐damper configuration.

A new matrix equation expression for the solution of non‐autonomous linear systems of ODEs

AbstractThe solution of systems of non‐autonomous linear ordinary differential equations is crucial in a variety of applications, such us nuclear magnetic resonance spectroscopy. A new method with spectral accuracy has been recently introduced in the scalar case. The method is based on a product that generalizes the convolution. In this work, we show that it is possible to extend the method to solve systems of non‐autonomous linear ordinary differential equations (ODEs). In this new approach, the ODE solution can be expressed through a linear system that can be equivalently rewritten as a matrix equation. Numerical examples illustrate the method's efficacy and the low‐rank property of the matrix equation solution.

Solving Non-Homogeneous Linear Ordinary Differential Equations Using Memristor-Capacitor Circuit

Inhomogeneous linear ordinary differential equations (ODEs) and systems of ODEs can be solved in a variety of ways. However, hardware circuits that can perform the efficient analog computation to solve them are rarely in the literature. To address such problems, this paper proposes a general method of using a memristor-capacitor (M-C) circuit to solve inhomogeneous linear ODEs and systems of ODEs of any order in initial value problems. The M-C circuit can match the coefficients of the equations sought by adjusting the memristor resistance value according to the coefficient formula proposed in the paper, which has higher programmability. Then, some ODEs and systems of ODEs are given in the paper as examples to evaluate the proposed method. According to the comparison results based on MATLAB software simulation and the simulation based on OrCAD software, the designed M-C circuit has an effective improvement in speed and the accuracy exceeds 99.95% in software simulation. Based on practical verification, this paper gives the actual M-C circuit experiment based on PCB. Moreover, the proposed method can be used to quickly solve the object motion state in the spring mass damping system in actual engineering, and the accuracy can reach 99.98%.

Approximation treatment for Linear Fuzzy HIV Infection Model by Variational Iteration Method

There has recently been considerable focus on finding reliable and more effective approximate methods for solving biological mathematical models in the form of differential equations. One of the well-known approximate or semi-analytical methods for solving linear, nonlinear differential well as partial differential equations within various fields of mathematics is the Variational Iteration Method (VIM). This paper looks at the use of fuzzy differential equations in human immunodeficiency virus (HIV) infection modeling. The main advantage of the method lies in its flexibility and ability to solve nonlinear equations easily. VIM is introduced to provide approximate solutions for linear ordinary differential equation system including the fuzzy HIV infection model. The model explains the amount of undefined immune cells, and the immune system viral load intensity intrinsic that will trigger fuzziness in patients infected by HIV. CD4+T-cells and cytototoxic T-lymphocytes (CTLs) are known for the immune cells concerned. The dynamics of the immune cell level and viral burden are analyzed and compared across three classes of patients with low, moderate and high immune systems. A modification and formulation of the VIM in the fuzzy domain based on the use of the properties of fuzzy set theory are presented. A model was established in this regard, accompanied by plots that demonstrate the reliability and simplicity of the methods. The numerical results of the model indicate that this approach is effective and easily used in fuzzy domain.

A Liouville’s Formula for Systems with Reflection

In this work, we derived an Abel–Jacobi–Liouville identity for the case of two-dimensional linear systems of ODEs (ordinary differential equations) with reflection. We also present a conjecture for the general case and an application to coupled harmonic oscillators.

Optimization of Right-Hand Sides of Nonlocal Boundary Conditions in a Controlled Dynamical System

We study an optimal control problem described by a system of linear ordinary differential equations with boundary conditions containing point and integral values of the state variable. The controls occurring in the differential equations and the values of the right-hand sides of the nonlocal boundary conditions are determined in this problem. Necessary conditions for the existence and uniqueness of a solution of the boundary value problem and for the convexity of the objective functional, as well as necessary conditions for the optimality of the parameters to be optimized in the control problem, are studied. The formulas obtained for the gradient of the objective functional of the problem are used in the numerical solution of an illustrative problem. The results of numerical experiments are provided.

Equivalence between the DPG method and the exponential integrators for linear parabolic problems

The Discontinuous Petrov-Galerkin (DPG) method and the exponential integrators are two well established numerical methods for solving Partial Differential Equations (PDEs) and stiff systems of Ordinary Differential Equations (ODEs), respectively. In this work, we apply the DPG method in the time variable for linear parabolic problems and we calculate the optimal test functions analytically. We show that the DPG method in time is equivalent to exponential integrators for the trace variables, which are decoupled from the interior variables. In addition, the DPG optimal test functions allow us to compute the approximated solutions in the time element interiors. This DPG method in time allows to construct a posteriori error estimations in order to perform adaptivity. We generalize this novel DPG-based time-marching scheme to general first order linear systems of ODEs. We show the performance of the proposed method for 1D and 2D + time linear parabolic PDEs after discretizing in space by the finite element method.

High-order partitioned spectral deferred correction solvers for multiphysics problems

We present an arbitrarily high-order, conditionally stable, partitioned spectral deferred correction (SDC) method for solving multiphysics problems using a sequence of pre-existing single-physics solvers. This method extends the work in [1,2], which used implicit-explicit Runge-Kutta methods (IMEX) to build high-order, partitioned multiphysics solvers. We consider a generic multiphysics problem modeled as a system of coupled ordinary differential equations (ODEs), coupled through coupling terms that can depend on the state of each subsystem; therefore the method applies to both a semi-discretized system of partial differential equations (PDEs) or problems naturally modeled as coupled systems of ODEs. The sufficient conditions to build arbitrarily high-order partitioned SDC schemes are derived. Based on these conditions, various of partitioned SDC schemes are designed. The stability of the first-order partitioned SDC scheme is analyzed in detail on a coupled, linear model problem. We show that the scheme is conditionally stable, and under conditions on the coupling strength, the scheme can be unconditionally stable. We demonstrate the performance of the proposed partitioned solvers on several classes of multiphysics problems with moderate coupling strength. They include a stiff linear system of ODEs, advection-diffusion-reaction systems, and fluid-structure interaction problems with both incompressible and compressible flows, where we verify the design order of the SDC schemes and study various stability properties. We also directly compare the accuracy, stability, and cost of the proposed partitioned SDC solver with the partitioned IMEX method in [1,2] on this suite of test problems. The results suggest that the high-order partitioned SDC solvers are more robust than the partitioned IMEX solvers for the numerical examples considered in this work, while the IMEX methods require fewer implicit solves.

Reduced Ordered Modelling of Time Delay Systems Using Galerkin Approximations and Eigenvalue Decomposition

In this paper, we develop ordinary differential equations (ODEs) based approximations for linear delay differential equations (DDEs). Using the shift of time transformation, the DDE is converted into an equivalent partial differential equation (PDE) with boundary conditions (BCs). The obtained PDE along with BCs is then approximated into a system of linear ODEs using Galerkin approximations. The eigenvalues of these ODEs approximate the characteristic roots of the original DDE. By only retaining the dynamics of these ODEs corresponding to converged eigenvalues of the original DDE, these ODEs are further reduced using eigenvalue decomposition method. The time responses obtained for the Galerkin approximated system models are compared with the same obtained using the direct numerical simulation and the results are found to be satisfactory.

Parameter estimation in tree graph metabolic networks.

We study the glycosylation processes that convert initially toxic substrates to nutritionally valuable metabolites in the flavonoid biosynthesis pathway of tomato (Solanum lycopersicum) seedlings. To estimate the reaction rates we use ordinary differential equations (ODEs) to model the enzyme kinetics. A popular choice is to use a system of linear ODEs with constant kinetic rates or to use Michaelis–Menten kinetics. In reality, the catalytic rates, which are affected among other factors by kinetic constants and enzyme concentrations, are changing in time and with the approaches just mentioned, this phenomenon cannot be described. Another problem is that, in general these kinetic coefficients are not always identifiable. A third problem is that, it is not precisely known which enzymes are catalyzing the observed glycosylation processes. With several hundred potential gene candidates, experimental validation using purified target proteins is expensive and time consuming. We aim at reducing this task via mathematical modeling to allow for the pre-selection of most potential gene candidates. In this article we discuss a fast and relatively simple approach to estimate time varying kinetic rates, with three favorable properties: firstly, it allows for identifiable estimation of time dependent parameters in networks with a tree-like structure. Secondly, it is relatively fast compared to usually applied methods that estimate the model derivatives together with the network parameters. Thirdly, by combining the metabolite concentration data with a corresponding microarray data, it can help in detecting the genes related to the enzymatic processes. By comparing the estimated time dynamics of the catalytic rates with time series gene expression data we may assess potential candidate genes behind enzymatic reactions. As an example, we show how to apply this method to select prominent glycosyltransferase genes in tomato seedlings.

The active disturbance rejection control approach to stabilisation of coupled heat and ODE system subject to boundary control matched disturbance

We consider stabilisation for a linear ordinary differential equation system with input dynamics governed by a heat equation, subject to boundary control matched disturbance. The active disturbance rejection control approach is applied to estimate, in real time, the disturbance with both constant high gain and time-varying high gain. The disturbance is cancelled in the feedback loop. The closed-loop systems with constant high gain and time-varying high gain are shown, respectively, to be practically stable and asymptotically stable.

System of linear ordinary differential and differential-algebraic equations and pseudo-spectral method

In this paper, first we introduce, briefly, pseudo-spectral method to solve linear Ordinary Differential Equations (ODEs), and then extend it to solve a system of linear ODEs and then Differential-Algebraic Equations (DAEs). Furthermore, because of appropriate choice of Chebyshev–Gauss–Raudo points we will show that this method can be used to solve a DAE whenever some coefficient functions in constraint are not analytic by providing some numerical examples.

Mathematical modeling and numerical computation for the vibration of a magnetostrictiveactuator

Mathematical modeling to analyze the vibration problem of a Terfenol-D actuator ispresented, using Hamilton’s variational principle. The total kinetic, strain and magneticenergy stored in the rod is expressed as a function of the displacement. The materialconstitutive relation of the magnetostrictive rod is assumed to be cubic nonlinear. Boththe governing equation and the boundary condition derived are of time variablecoefficient. A numerical approach which combines the finite difference method with thetransfer matrix method is proposed for solving the excited vibration problem of amagnetostrictive rod. By discretizing the displacement in space and making thefinite difference formulation, the nodal displacement is obtained in terms of asystem of linear second-order ordinary differential equations (ODEs), which can besubsequently transformed into a system of linear first-order ODEs in the time domain.The time domain within a period is then discretized with the numerical solutionbeing expressed by the transfer matrix method. As a numerical example, thevibration of a Terfenol-D rod excited by a harmonic current is analyzed. Thenumerical results show that the induced displacement in the rod is periodic, withits frequency being roughly twice that of the exciting current. Such a doublefrequency effect has been observed in experiments. The stress behavior and the peakdisplacement within the rod are also numerically analyzed, with an emphasison the effect of the involved magnetostrictive and magnetoelastic parameters.The good agreement between the numerical results and the experimental dataavailable in the literature verifies the validity of the present modeling and method.

Stiffness and transfer matrices of a non-naturally curved 3D-beam element

This article deals with the 3D-curved beam element with non-naturally curved differential geometry governing equations, i.e. which arc length is a function of a parameter, as for example the classical case of the ellipse. The derivative of the arc length with respect to a parameter chosen, meaning the Frenet frame velocity along the central-line, is not unitary. A twelve linear ordinary differential equation system governing its mechanical behaviour is presented. Exact analytical and numerical approximate procedures are proposed. Both transfer and stiffness matrices are provided to determine the internal forces and displacements in a 3D-curved beam element defined by its parametric equations with varying cross-section area and generalized loads applied. Both matrix methods offer directly the load transmission and equivalent vectors, respectively. The semi-circular arch subjected to a distributed load in and out of its plane, a semi-elliptical arch and an elliptic–helical beam examples are given for verification.

Master equation solution of a plant disease model

We develop the exact solution for a stochastic plant disease model with nonlinear density dependence and time-decaying susceptibility. In our biological application only the transient behaviour, rather than the stationary state, is relevant. To model the population noise we use the Master equation, leading to a linear ordinary differential equation system with finally time-independent coefficients. A numerically stable procedure is given by the Padé approximation of the solution in form of an exponential of a matrix. On the basis of this solution the parameter estimation is performed using experimental data from epidemiological microcosms. In the Master equation formalism, in order to generalize to more complicated models, for example, including a time-independent latent period as a first multivariate model, we finally suggest a numerical procedure to estimate the likelihood comparing data time series with simulated Master equation time series.

A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations

Waveform relaxation is a technique to solve large systems of ordinary differential equations (ODEs) in parallel. The right hand side of the system is split into subsystems which are only loosely coupled. One then solves iteratively all the subsystems in parallel and exchanges information after each step of the iteration. Two classical convergence results state linear convergence on unbounded time intervals for linear systems of ODEs under some dissipation assumption and superlinear convergence on bounded time intervals for nonlinear systems under a Lipschitz condition on the splitting. To apply waveform relaxation to partial differential equations (PDEs), one traditionally discretizes the PDE in space to get a large system of ODEs, to which then the waveform relaxation algorithm is applied using a matrix splitting. There are two problems with this approach: first information about how to split the right hand side is lost during the discretization; second the convergence results derived in this fashion depend in general on the mesh parameter and convergence rates deteriorate when the mesh is refined. To avoid those problems a new waveform relaxation algorithm is formulated directly at the PDE level. The differential operator on the right hand side is split using domain decomposition. It is shown for a scalar reaction diffusion equation with variable diffusion coefficient that the new waveform relaxation algorithm converges superlinearly for bounded time intervals and linearly for unbounded time intervals, extending the two classical convergence results to this type of PDE. Interestingly the superlinear convergence rate is faster than the superlinear convergence rate obtained by the traditional matrix splitting methods. It is shown how the convergence rates depend on the overlap of the domain decomposition and a Lipschitz condition on the reaction function. The splitting of the right hand side is naturally given by the domain decomposition and the convergence rates are robust with respect to mesh refinement when the algorithm is discretized.

Solutions of Finitely Smooth Nonlinear Singular Differential Equations and Problems of Diagonalization and Triangularization

It is known that existence of a formal power series solution $\hat y(x)$ to a system of nonlinear ordinary differential equations (ODEs) with analytic or infinitely smooth coefficients at an irregular singular point implies the existence of an actual solution y(x), which possesses the asymptotic expansion $\hat y(x)$. In the present paper we extend this result for systems with finitely smooth coefficients. In this case one cannot speak about a formal power series solution $\hat y(x)$; it has therefore to be replaced by the requirement of existence of an approximate solution y0 (x). The existence of a corresponding actual solution is a subject of certain conditions that link the smoothness of the system, the accuracy of the approximation y0 (x), and the degeneracy of the system, linearized with respect to y0 (x). As applications, problems of reduction of linear time dependent systems of ODEs into diagonal and triangular forms, as well as some other problems, are considered. In particular, the well-kn...