Linear Ordinary Differential Systems Research Articles

Semiclassical quantification of some two degree of freedom potentials: A differential Galois approach

In this work we explain the relevance of the Differential Galois Theory in the semiclassical (or WKB) quantification of some two degree of freedom potentials. The key point is that the semiclassical path integral quantification around a particular solution depends on the variational equation around that solution: a very well-known object in dynamical systems and variational calculus. Then, as the variational equation is a linear ordinary differential system, it is possible to apply the Differential Galois Theory to study its solvability in closed form. We obtain closed form solutions for the semiclassical quantum fluctuations around constant velocity solutions for some systems like the classical Hermite/Verhulst, Bessel, Legendre, and Lamé potentials. We remark that some of the systems studied are not integrable, in the Liouville–Arnold sense.

A Lanczos-type procedure for tensors

The solution of linear non-autonomous ordinary differential equation systems (also known as the time-ordered exponential) is a computationally challenging problem arising in a variety of applications. In this work, we present and study a new framework for the computation of bilinear forms involving the time-ordered exponential. Such a framework is based on an extension of the non-Hermitian Lanczos algorithm to 4-mode tensors. Detailed results concerning its theoretical properties are presented. Moreover, computational results performed on real-world problems confirm the effectiveness of our approach.

Identifiability analysis of linear ordinary differential equation systems with a single trajectory

Ordinary differential equations (ODEs) are widely used to model dynamical behavior of systems. It is important to perform identifiability analysis prior to estimating unknown parameters in ODEs (a.k.a. inverse problem), because if a system is unidentifiable, the estimation procedure may fail or produce erroneous and misleading results. Although several qualitative identifiability measures have been proposed, much less effort has been given to developing quantitative (continuous) scores that are robust to uncertainties in the data, especially for those cases in which the data are presented as a single trajectory beginning with one initial value. In this paper, we first derived a closed-form representation of linear ODE systems that are not identifiable based on a single trajectory. This representation helps researchers design practical systems and choose the right prior structural information in practice. Next, we proposed several quantitative scores for identifiability analysis in practice. In simulation studies, the proposed measures outperformed the main competing method significantly, especially when noise was presented in the data. We also discussed the asymptotic properties of practical identifiability for high-dimensional ODE systems and conclude that, without additional prior information, many random ODE systems are practically unidentifiable when the dimension approaches infinity.

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%.

Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity

A time discretization method is called strongly stable (or monotone), if the norm of its numerical solution is nonincreasing. Although this property is desirable in various of contexts, many explicit Runge-Kutta (RK) methods may fail to preserve it. In this paper, we enforce strong stability by modifying the method with superviscosity, which is a numerical technique commonly used in spectral methods. Our main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal operators. We propose two approaches for stabilization: the modified method and the filtering method. The modified method is achieved by modifying the semi-negative operator with a high order superviscosity term; the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small superviscosity. For linear problems, most explicit RK methods can be stabilized with either approach without accuracy degeneration. Furthermore, we prove a sharp bound (up to an equal sign) on diffusive superviscosity for ensuring strong stability. For nonlinear problems, a filtering method is investigated. Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.

Iterative Learning Based Modulating Functions Method for Distributed Solar Source Estimation

Modulating functions method is a non asymptotic estimation method, which provides accurate and robust estimations of states, parameters and inputs for different classes of systems, which include unknown linear ordinary differential systems, fractional systems and linear partial differential equations. In the case of time or space varying unknown, the method requires the decomposition of the unknown into predefined basis functions. However, the estimation performance will depend on the nature of the basis functions which in some cases are not easy to determine. This letter proposes a new iterative learning based modulating functions method, which combines the standard modulating functions with a dictionary learning procedure. The dictionary learning step allows the determination of appropriate set of functions to decompose the unknown, while the modulating function step allows the non-asymptotic and robust estimation of the projection coefficients. The performance of the proposed method is illustrated in a distributed solar collector application, modeled by partial differential equations and where the unknown solar irradiance is estimated.

Sampled-Data Control of Distributed Parameter Systems via an Event-Based Communication Scheme

This paper studies the problem of event-triggered control for a class of networked distributed parameter systems with Markov jump parameters. To reduce the number of packages transmitted over the communication network, an adaptive event-triggered mechanism is introduced. The Galerkin method is employed to obtain the nonlinear ordinary differential equation systems, which can accurately describe the dynamics of the dominant modes of the considered distributed parameter systems. The systems are subsequently parameterized by a multilayer neural network with one-hidden layer and zero bias terms, and the linear ordinary differential equation systems are derived. Then, Lyapunov approach is used to analyze stability of the considered systems, and by employing the strong law of large numbers and Gronwall inequality technique, almost surely exponential stability condition is derived. Moreover, a linear sampled-data-based controller is designed to stabilize the closed-loop systems. Finally, a practical example is shown to demonstrate the effectiveness of the achieved theoretical results.

Checkpoints in searching for rational solutions of linear ordinary difference and differential systems

It is quite common that search algorithms for those solutions of difference and differential equations and systems that belong to a fixed class of functions are designed so that nonexistence of solutions of the desired type is detected only in the last stages of the algorithm. However, performing additional tests on the intermediate results makes it possible to stop the algorithm as soon as these tests imply that no solutions of the desired type exist. This gives an opportunity to save time and other computing resources. So, it makes sense to equip algorithms with checkpoints and some tests. We consider these questions in connection with the search for rational solutions of linear homogeneous difference and differential systems with polynomial coefficients, and propose a scheme equipped with such checkpoints and tests, and also report results of experiments with our implementation of the scheme in Maple.

Row reduction process for matrices of scalar operators

It is well known that if the leading matrix of a linear ordinary differential or difference system is nonsingular, then the determinant of this matrix contains some useful information on solutions of the system. We investigate a kind of non-arithmetic complexity of known algorithms for transforming a matrix of scalar operators to an equivalent matrix which has non-singular frontal, or, leading matrix. In the algorithms under consideration, the differentiation in the differential case and the shift in the difference case play a significant role. We give some analysis of the complexity measured as the number of differentiations or, resp., shifts in the worst case. We not only offer estimates of the complexity written using the O -notation, but we also show that some estimates are sharp and can not be improved.

Levy solution for bending response of FG carbon nanotube reinforced plates under uniform, linear, sinusoidal and exponential distributed loadings

A new analytical approach for bending response of functionally graded single-walled carbon nanotube (FGCNT) reinforced plate resting on double-layered elastic foundations in thermal environments is presented based on Levy and Navier types solutions. The distribution of the carbon nanotubes is varied through the plate thickness in accordance with a modified power law. Four types of carbon nanotube distributions are considered. The four edges of the plate are simply supported for Navier method, whereas for Levy method, two opposite only of them are simply supported and the other ones are arbitrary. The present FGCNT plate is subjected to uniform, linear, sinusoidal or exponential distributed loadings. A refined shear deformation plate theory with four unknown functions is employed to obtain the closed form solution. The four coupled governing partial differential equations are derived utilizing Hamilton’s principle. Applying Levy solution and then the state space concept to the governing equations, a nonhomogeneous first-order linear ordinary differential system with constant coefficients is obtained. The solution of the homogeneous system (homogeneous solution) is obtained by using the matrix method. While, the method of undetermined coefficients is applied to find the particular solution of the nonhomogeneous linear system. The results obtained by Navier and Levy methods are compared with available results in the literature. Several examples are discussed for various values of the foundation stiffness, CNT volume fraction, various types of plate geometries, CNT distributions, external applied loads and boundary conditions.

Applications of linear ordinary differential equations and dynamic system to economics - an example of Taiwan stock index TAIEX

Mean reversion of stock price is an observed phenomenon in finance. It is similar to the concept of Newton's law of cooling. Three models of ordinary differential equation are derived from the concept and solved by dynamic integration. The models are applied to the daily closing price of Taiwan Stock Exchange Capitalisation Weighted Stock Index. Empirical study shows that the models have good fits and forecasts although errors appear during the dynamic forecasting process. The applicability of the models and future modification are also discussed.

Applications of linear ordinary differential equations and dynamic system to economics - an example of Taiwan stock index TAIEX

Applications of linear ordinary differential equations and dynamic system to economics - an example of Taiwan stock index TAIEX

English

The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense ${\rm d} x(t)={\rm d} A_0(t)\cdot x(t)+{\rm d} f_0(t)$, $x(t_{0})=\nobreak c_0$ $(t\in I)$ with a unique solution $x_0$ is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems ${\rm d} x(t)={\rm d} A_k(t)\cdot x(t)+{\rm d} f_k(t)$, $x(t_{k})=c_k$ $(k=1,2,\dots )$ to have a unique solution $x_k$ for any sufficiently large $k$ such that $x_k(t)\to x_0(t)$ uniformly on $I$. Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems. Moreover, efficient sufficient conditions for the problem of well-posedness are given.

Resolving sequences of operators for linear ordinary differential and difference systems of arbitrary order

We introduce the notion of a resolving sequence of (scalar) operators for a given differential or difference system with coefficients in some differential or difference field K. We propose an algorithm to construct, such a sequence, and give some examples of the use of this sequence as a suitable auxiliary tool for finding certain kinds of solutions of differential and difference systems of arbitrary order. Some experiments with our implementation of the algorithm are reported.

On ramification indices of formal solutions of constructive linear ordinary differential systems

We consider full rank linear ordinary differential higher-order systems whose coefficients are computable power series. It is shown that the algorithmic problems connected with the ramification indices of irregular formal solutions of a given system are mostly undecidable even if we fix a conjectural value r of the ramification index. This enables us to obtain a strengthening of the theorem which has been proven earlier and states that we are not able to compute algorithmically the dimension of the space of all formal solutions although we can construct a basis for the subspace of regular solutions. In fact, it is impossible to compute algorithmically this dimension even if, in addition to the system, we know the list of all values of the ramification indices. However, there is nearby an algorithmically decidable problem: if a system S and integers r, d are such that for S the existence of d linearly independent formal solutions of ramification index r is guaranteed then one can compute such d solutions of S.

Procedures for searching local solutions of linear differential systems with infinite power series in the role of coefficients

Construction of Laurent, regular, and formal (exponential---logarithmic) solutions of full-rank linear ordinary differential systems is discussed. The systems may have an arbitrary order, and their coefficients are formal power series given algorithmically. It has been established earlier that the first two problems are algorithmically decidable and the third problem is not decidable. A restricted variant of the third problem was suggested for which the desired algorithm exists. In the paper, a brief survey of algorithms for the abovementioned decidable problems is given. Implementations of these algorithms in the form of Maple procedures with a uniform interface and data representation are suggested.

Resolving sequence of operators for linear ordinary differential and difference systems of arbitrary order

Resolving sequence of operators for linear ordinary differential and difference systems of arbitrary order

Floquet Theory for Stability of Differential Algebraic Equations.

Motivated by a great useful of some types of non autonomous differential algebraic
 equation systems ( so called strangeness free ) and its applied in different scientific
 fields, we present several new results for studying such systems by classical Floquet
 Theory, which we extended from linear periodic ordinary differential equation systems
 into linear periodic differential algebraic equation systems. For both systems we
 investigate that they have the same Floquet exponents. The relation between monodromy
 matrices of both systems is also presented. Classification of solution according to the
 nature of Floquet exponent is established. Then according to these results, we study the
 stability and bifurcation phenomenon of our differential algebraic equation systems.

On Solution Spaces of Products of Linear Differential or Difference Operators

We consider linear ordinary differential or difference systems of the form L(y) = 0 where L is an operator with matrix coefficients, the unknown vector y has m components y1, . . . , ym, m > 1. The matrix coefficients are of size m x m, their entries belong to a differential or difference field K of characteristic 0. For any such a system the solution space VL is considered, and the components of each solution are in a fixed appropriate differential or difference extension of K (e.g., in the universal Picard-Vessiot extension). We prove that dim VLM = dim VL+dim VM for arbitrary operators L and M of the considered form, and discuss some algorithms based on this property of operators. In particular, we propose an algorithm to compute dim VL, as well as a new algorithm having a low complexity for recognizing unimodular operators and constructing the inverse of a unimodular operator.

On full rank differential systems with power series coefficients

We consider the following problem: given a linear ordinary differential system of arbitrary order with formal power series coefficients, decide whether the system has non-zero Laurent series solutions, and find all such solutions if they exist (in a truncated form preserving the space dimension). If the series coefficients of the original systems are represented algorithmically then these problems are algorithmically undecidable. However, it turns out that they are decidable in the case when we know in advance that a given system is of full rank.We define the width of a given full rank system S with formal power series coefficients as the smallest non-negative integer w such that any l-truncation of S with l⩾w is a full rank system. We prove that the value w exists for any full rank system and can be found algorithmically.We propose corresponding algorithms and their Maple implementation, and report some experiments.