General Linear Differential Equations Research Articles

The adjoint map of Euclidean plane curves and curvature problems

The adjoint map of a pair of naturally parametrized curves in the Euclidean plane is studied from the point of view of the curvature. A main interest is when the given curve and its adjoint curve share the same natural parameter and the same curvature. For the general linear second order differential equation we introduce a function expressing the deformation of curvatures induced by the adjoint map.

Time-marching based quantum solvers for time-dependent linear differential equations

The time-marching strategy, which propagates the solution from one time step to the next, is a natural strategy for solving time-dependent differential equations on classical computers, as well as for solving the Hamiltonian simulation problem on quantum computers. For more general homogeneous linear differential equations ddt|ψ(t)⟩=A(t)|ψ(t)⟩,|ψ(0)⟩=|ψ0⟩, a time-marching based quantum solver can suffer from exponentially vanishing success probability with respect to the number of time steps and is thus considered impractical. We solve this problem by repeatedly invoking a technique called the uniform singular value amplification, and the overall success probability can be lower bounded by a quantity that is independent of the number of time steps. The success probability can be further improved using a compression gadget lemma. This provides a path of designing quantum differential equation solvers that is alternative to those based on quantum linear systems algorithms (QLSA). We demonstrate the performance of the time-marching strategy with a high-order integrator based on the truncated Dyson series. The complexity of the algorithm depends linearly on the amplification ratio, which quantifies the deviation from a unitary dynamics. We prove that the linear dependence on the amplification ratio attains the query complexity lower bound and thus cannot be improved in the worst case. This algorithm also surpasses existing QLSA based solvers in three aspects: (1) A(t) does not need to be diagonalizable. (2) A(t) can be non-smooth, and is only of bounded variation. (3) It can use fewer queries to the initial state |ψ0⟩. Finally, we demonstrate the time-marching strategy with a first-order truncated Magnus series, which simplifies the implementation compared to high-order truncated Dyson series approach, while retaining the aforementioned benefits. Our analysis also raises some open questions concerning the differences between time-marching and QLSA based methods for solving differential equations.

Nash Equilibria Strategies and Equivalent Single-Objective Optimization Problems. The Case of Linear Partial Differential Equations

In this paper we study the existence and uniqueness of Nash equilibria (solution to competition-wise problems, with several controls trying to reach possibly different goals) associated to linear partial differential equations and show that, in some cases, they are also the solution of suitable single-objective optimization problems (i.e. cooperative-wise problems, where all the controls cooperate to reach a common goal). We use cost functions associated with a particular linear parabolic partial differential equation and distributed controls, but the results are also valid for more general linear differential equations (including elliptic and hyperbolic cases) and controls (e.g. boundary controls, initial value controls,...).

Stability and robust stability of non-autonomous linear differential equations with infinite delay

By the idea of comparison principle, we first introduce novel explicit criteria for the $ \varrho $-stability of general linear non-autonomous functional differential equations with infinite delay. Some characterizations for the time-independent exponential stability are also investigated. We then present a sufficient condition for the exponential stability of the equations subject to time-varying structured perturbations. These results lead to applications to the linear non-autonomous differential system with unbounded delay and the linear non-autonomous integro-differential equations with delay. Some examples are given to illustrate the obtained results.

Parameter estimation of linear fractional-order system from laplace domain data

A novel parameter estimation method based on the Laplace transform and the response sensitivity method has been presented to recognize the parameters of linear fractional-order systems (FOS) rapidly. The proposed method consumes two orders of magnitude computational resources lower than the traditional time-domain method. Fractional-order operators are increasingly widely used in control and synchronization, epidemiology, viscoelastic material modelling and other emerging disciplines. It is difficult to measure the fractional-order α and system parameters directly in real engineering applications. This paper’s main work include: Firstly, a general linear fractional differential equation is transformed into an algebraic equation by the Laplace transform, and the parameter sensitivity analysis concerning the unknown parameters is also deduced. Then, the parameter estimation problem of the linear FOS is established as a nonlinear least-squares optimization in the Laplace domain, and the enhanced response sensitivity method is adopted to resolve this nonlinear minimum optimization equation iteratively. In addition, the Tikhonov regularization is employed to cope with the potential ill-posed situations, and the trust-region restriction is also introduced to improve the convergence. Finally, taking a differential system with two types of fractional-order operators, a multi-degree-of-freedom FOS with external excitation and an actual piezoelectric actuator model as examples, the specific implementation process is demonstrated in detail to test the robustness and validity of the proposed approach.

A UNIFIED APPROACH TO EXPONENTIAL STABILITY ANALYSIS FOR A GENERAL CLASS OF SWITCHED TIME-DELAY LINEAR SYSTEMS

This paper proposes a unified approach to study global exponential stability for a class of switched time-delay linear systems described by general linear functional differential equations. Several new delay-dependent criteria of exponential stability are established for this class of systems, under arbitrary switching which satisfies some assumptions on the minimum dwell time or the average dwell time. As particular cases, the obtained results are shown to include and improve many previously known results. An example is given to illustrate the proposed method.

Decay rates for second-order linear evolution problems with fractional laplacian operators

In this work we study the asymptotic behavior of solutions for a general linear second-order evolution differential equation in time with fractional Laplace operators in $\mathbb{R}^n$. We obtain improved decay estimates with less demand on the initial data when compared to previous results in the literature. In certain cases, we observe that the dissipative structure of the equation is of regularity-loss type. Due to that special structure, to get decay estimates in high frequency region in the Fourier space it is necessary to impose additional regularity on the initial data to obtain the same decay estimates as in low frequency region. The results obtained in this work can be applied to several initial value problems associated to second-order equations, as for example, wave equation, plate equation, IBq, among others.

Liouvillian solutions for second order linear differential equations with polynomial coefficients

In this paper we present an algebraic study concerning the general second order linear differential equation with polynomial coefficients. By means of Kovacic’s algorithm and asymptotic iteration method we find a degree independent algebraic description of the spectral set: the subset, in the parameter space, of Liouville integrable differential equations. For each fixed degree, we prove that the spectral set is a countable union of non accumulating algebraic varieties. This algebraic description of the spectral set allow us to bound the number of eigenvalues for algebraically quasi-solvable potentials in the Schrödinger equation.

A third integrating factor for indefinite integrals of special functions

ABSTRACT An integrating factor is presented involving the terms in and of the general homogenous second-order linear ordinary differential equation. The new integrating factors obey second-order differential equations, rather than being given by quadrature. The new factors provides old and new integrals for special functions which obey such differential equations. The functions considered here are cylinder functions, parabolic cylinder functions and Whittaker functions. All the integrals given have been checked using Mathematica.

Second order linear differential equations with analytic uncertainties: Stochastic analysis via the computation of the probability density function

This paper concerns the analysis of random second order linear differential equations. Usually, solving these equations consists of computing the first statistics of the response process, and that task has been an essential goal in the literature. A more ambitious objective is the computation of the solution probability density function. We present advances on these two aspects in the case of general random non-autonomous second order linear differential equations with analytic data processes. The Fröbenius method is employed to obtain the stochastic solution in the form of a mean square convergent power series. We demonstrate that the convergence requires the boundedness of the random input coefficients. Further, the mean square error of the Fröbenius method is proved to decrease exponentially with the number of terms in the series, although not uniformly in time. Regarding the probability density function of the solution at a given time, which is the focus of the paper, we rely on the law of total probability to express it in closed-form as an expectation. For the computation of this expectation, a sequence of approximating density functions is constructed by reducing the dimensionality of the problem using the truncated power series of the fundamental set. We prove several theoretical results regarding the pointwise convergence of the sequence of density functions and the convergence in total variation. The pointwise convergence turns out to be exponential under a Lipschitz hypothesis. As the density functions are expressed in terms of expectations, we propose a crude Monte Carlo sampling algorithm for their estimation. This algorithm is implemented and applied on several numerical examples designed to illustrate the theoretical findings of the paper. After that, the efficiency of the algorithm is improved by employing the control variates method. Numerical examples corroborate the variance reduction of the Monte Carlo approach.

Large spectral gap induced by small delay and its application to reduction

We consider general linear neutral differential equations with small delays in the view of pseudo exponential dichotomy. For the autonomous case, we first count the eigenvalues in a certain half plane, which generalized the previous works on serval special retarded differential equations. We next establish the existence of a pseudo exponential dichotomy for the nonautonomous case, and prove that the corresponding spectral gap approaches infinity as the delay tends to zero. The proof for this large spectral gap induced by small delay is owing to exact bounds and exponents for pseudo exponential dichotomy. Then based on above results, we give an invariant manifold reduction theorem for nonlinear neutral differential equations with small delays. Finally, our results are applied to a concrete example.

Ulam stability of linear differential equations using Fourier transform

The purpose of this paper is to study the Hyers-Ulam stability and generalized Hyers-Ulam stability of general linear differential equations of $n$th order with constant coefficients by using the Fourier transform method. Moreover, the Hyers-Ulam stability constants are obtained for these differential equations.

Non parametric estimation for fractional diffusion processes with random effects

ABSTRACTWe propose a nonparametric estimation for a class of fractional stochastic differential equations (FSDE) with random effects. We precisely consider general linear fractional stochastic differential equations with drift depending on random effects and non-random diffusion. We build ordinary kernel estimators and histogram estimators and study their risk (), when . Asymptotic results are evaluated as both and N tend to infinity.

Quantum algorithm for simulating the wave equation

We present a quantum algorithm for simulating the wave equation under Dirichlet and Neumann boundary conditions. The algorithm uses Hamiltonian simulation and quantum linear system algorithms as subroutines. It relies on factorizations of discretized Laplacian operators to allow for improved scaling in truncation errors and improved scaling for state preparation relative to general purpose linear differential equation algorithms. We also consider using Hamiltonian simulation for Klein-Gordon equations and Maxwell's equations.

Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation

AbstractThis paper presents a methodology to quantify computationally the uncertainty in a class of differential equations often met in Mathematical Physics, namely random non-autonomous second-order linear differential equations, via adaptive generalized Polynomial Chaos (gPC) and the stochastic Galerkin projection technique. Unlike the random Fröbenius method, which can only deal with particular random linear differential equations and needs the random inputs (coefficients and forcing term) to be analytic, adaptive gPC allows approximating the expectation and covariance of the solution stochastic process to general random second-order linear differential equations. The random inputs are allowed to functionally depend on random variables that may be independent or dependent, both absolutely continuous or discrete with infinitely many point masses. These hypotheses include a wide variety of particular differential equations, which might not be solvable via the random Fröbenius method, in which the random input coefficients may be expressed via a Karhunen-Loève expansion.

Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling

The objective of this paper is to complete certain issues from our recent contribution (Calatayud, J.; Cortés, J.-C.; Jornet, M.; Villafuerte, L. Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties. Adv. Differ. Equ. 2018, 392, 1–29, doi:10.1186/s13662-018-1848-8). We restate the main theorem therein that deals with the homogeneous case, so that the hypotheses are clearer and also easier to check in applications. Another novelty is that we tackle the non-homogeneous equation with a theorem of existence of mean square analytic solution and a numerical example. We also prove the uniqueness of mean square solution via a habitual Lipschitz condition that extends the classical Picard theorem to mean square calculus. In this manner, the study on general random non-autonomous second order linear differential equations with analytic data processes is completely resolved. Finally, we relate our exposition based on random power series with polynomial chaos expansions and the random differential transform method, the latter being a reformulation of our random Fröbenius method.

Novel Criteria for Exponential Stability of Linear Non-Autonomous Functional Differential Equations

General linear non-autonomous functional differential equations are considered. Explicit criteria for exponential stability are given. Furthermore, the authors present an explicit robust stability bound for systems subject to time-varying perturbations. Two examples are given to illustrate the obtained results.

New indefinite integrals of confluent Heun functions

ABSTRACTWe obtain some new formulae to compute the first derivative of confluent and biconfluent Heun functions under the minimal assumption of fixing only one parameter. These results together with the Lagrangian formulation of a general homogeneous linear ordinary differential equation allow to construct several new indefinite integrals for the confluent, biconfluent, doubly confluent, and triconfluent Heun functions.

New indefinite integrals of Heun functions

ABSTRACTWe present a conspicuous number of indefinite integrals involving Heun functions and their products obtained by means of the Lagrangian formulation of a general homogeneous linear ordinary differential equation. As a by-product we also derive new indefinite integrals involving the Gauss hypergeometric function and products of hypergeometric functions with elliptic functions of the first kind. All integrals we obtained cannot be computed using Maple and Mathematica.

MHD Kelvin-Helmholtz instability in the anisotropic solar wind plasma

We investigated a shear instability of the Kelvin-Helmholtz (KH) type in a plasma with temperature anisotropy under the magnetohydodynamics (MHD) approximation. To solve the problem, a system of 16-moment MHD transport equations are used in a collisionless bi-Maxwellian plasma, including the various components of the heat fluxes along the magnetic field. We consider supersonic flows of two semi-infinite anisotropic and homogeneous plasma layers with different physical parameters and velocities. For the general case, i.e., when the interface between these two flows is a transition layer with a finite thickness, we derived a general linear differential equation framework for determining the eigenmodes in the system. Furthermore, we considered thoroughly the limiting case of a zero thickness transition zone (contact discontinuity). The analysis enabled applying appropriate boundary conditions to derive the dispersion equation for interface waves. The obtained equation analyzes in detail for the case when heat fluxes are absent along the discontinuity in the background state. It is shown that the shear flow excites the KH instability and “couples” the various branches of the free-plasma oscillations to each other. It is found that the region of mode interaction is determined by the resonance regions when the longitudinal phase velocities of the waves match. In the resonance flows with an average speed, the KH instability occurs. The growth rates of the KH instability are calculated as a function of the parameters, including the degree of plasma anisotropy. It is found that in most cases the KH instability is dominant in the considered configuration. The obtained results are applied to the plasma conditions in the bimodal solar wind in the vicinity of the contact discontinuity between different flow patterns (fast and slow wind).