Linear Differential Equations Research Articles

Relative controllability of linear state-delay fractional systems

AbstractIn this paper, our focus is on exploring the relative controllability of systems governed by linear fractional differential equations incorporating state delay. We introduce a novel counterpart to the Cayley-Hamilton theorem. Leveraging a delayed perturbation of the Mittag-Leffler function, along with a determining function and an analog of the Cayley-Hamilton theorem, we establish an algebraic Kalman-type rank criterion for assessing the relative controllability of fractional differential equations with state delay. Moreover, we articulate necessary and sufficient conditions for relative controllability criteria concerning linear fractional time-delay systems, expressed in terms of a new $$\alpha $$ α -Gramian matrix and define a control which transfer the system from any initial state to any final state within a given time. The theoretical findings are exemplified through the presentation of illustrative examples.

A Score-Based Deterministic Diffusion Algorithm with Smooth Scores for General Distributions

Score matching based diffusion has shown to achieve the state of art results in generation modeling. In the original score matching based diffusion algorithm, the forward equation is a differential equation for which the probability density equation evolves according to a linear partial differential equation, the Fokker-Planck equation. A drawback of this approach is that one needs the data distribution to have a Lipschitz logarithmic gradient. This excludes a large class of data distributions that have a compact support. We present a deterministic diffusion process for which the vector fields are always Lipschitz and hence the score does not explode for probability measures with compact support. This deterministic diffusion process can be seen as a regularization of the porous media equation equation, which enables one to guarantee long term convergence of the forward process to the noise distribution. Though the porous media equation is itself not always guaranteed to have a Lipschitz vector field, it can be used to understand the closeness of the output of the algorithm to the data distribution as a function of the the time horizon and score matching error. This analysis enables us to show that the algorithm has better dependence on the score matching error than approaches based on stochastic diffusions. Using numerical experiments we verify our theoretical results on example one and two dimensional data distributions which are compactly supported. Additionally, we validate the approach on a modified MNIST data set for which the distribution is concentrated on a compact set. In each of the experiments, the approach using deterministic diffusion performs better that the diffusion algorithm with stochastic forward process, when considering the FID scores of the generated samples.

Simple Lyapunov spectrum for linear homogeneous differential equations with Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} parameters

In the present paper we prove that densely, with respect to an Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-like topology, the Lyapunov exponents associated to linear continuous-time cocycles Φ:R×M→GL(2,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi :\\mathbb {R}\ imes M\\rightarrow {{\\,\ extrm{GL}\\,}}(2,\\mathbb {R})$$\\end{document} induced by second order linear homogeneous differential equations x¨+α(φt(ω))x˙+β(φt(ω))x=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ddot{x}+\\alpha (\\varphi ^t(\\omega ))\\dot{x}+\\beta (\\varphi ^t(\\omega ))x=0$$\\end{document} are almost everywhere distinct. The coefficients α,β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha ,\\beta $$\\end{document} evolve along the φt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi ^t$$\\end{document}-orbit for ω∈M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega \\in M$$\\end{document} and φt:M→M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi ^t: M\\rightarrow M$$\\end{document} is an ergodic flow defined on a probability space. We also obtain the corresponding version for the frictionless equation x¨+β(φt(ω))x=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ddot{x}+\\beta (\\varphi ^t(\\omega ))x=0$$\\end{document} and for a Schrödinger equation x¨+(E-Q(φt(ω)))x=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ddot{x}+(E-Q(\\varphi ^t(\\omega )))x=0$$\\end{document}, inducing a cocycle Φ:R×M→SL(2,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi :\\mathbb {R}\ imes M\\rightarrow {{\\,\ extrm{SL}\\,}}(2,\\mathbb {R})$$\\end{document}.

Oscillator Simulation with Deep Neural Networks

The motivation behind this study is to overcome the complex mathematical formulation and time-consuming nature of traditional numerical methods used in solving differential equations. It seeks an alternative approach for more efficient and simplified solutions. A Deep Neural Network (DNN) is utilized to understand the intricate correlations between the oscillator’s variables and to precisely capture their dynamics by being trained on a dataset of known oscillator behaviors. In this work, we discuss the main challenge of predicting the behavior of oscillators without depending on complex strategies or time-consuming simulations. The present work proposes a favorable modified form of neural structure to improve the strategy for simulating linear and nonlinear harmonic oscillators from mechanical systems by formulating an ANN as a DNN via an appropriate oscillating activation function. The proposed methodology provides the solutions of linear and nonlinear differential equations (DEs) in differentiable form and is a more accurate approximation as compared to the traditional numerical method. The Van der Pol equation with parametric damping and the Mathieu equation are adopted as illustrations. Experimental analysis shows that our proposed scheme outperforms other numerical methods in terms of accuracy and computational cost. We provide a comparative analysis of the outcomes obtained through our proposed approach and those derived from the LSODA algorithm, utilizing numerical techniques, Adams–Bashforth, and the Backward Differentiation Formula (BDF). The results of this research provide insightful information for engineering applications, facilitating improvements in energy efficiency, and scientific innovation.

Growth of $(\alpha ,\beta ,\gamma )$-order solutions of linear differential equations with entire coefficients

Growth of $(\alpha ,\beta ,\gamma )$-order solutions of linear differential equations with entire coefficients

Solving Nonlinear Time-Fractional Partial Differential Equations Using Conformable Fractional Reduced Differential Transform with Adomian Decomposition Method

In this article, we use a new technique called conformable fractional reduced differential transform (CFRDT) with Adomian decomposition to estimate the solution of one and two-dimensional time-fractional partial linear and nonlinear differential equations with initial values. We explain the convergence analysis of this technique. The obtained results illustrate that the novel method is efficient and easy to use to find approximate solutions for the time-fractional partial differential equations (PDEs). Thus, the suggested method has a significant impact on how engineering, physics, and other disciplines solve fractional PDEs. Furthermore, we analyze the solution of problems with a 2D or 3D graphical representation by using Mathematica software.

A comparative analytical investigation for some linear and nonlinear time-fractional partial differential equations in the framework of the Aboodh transformation

This article discusses two simple, complication-free, and effective methods for solving fractional-order linear and nonlinear partial differential equations analytically: the Aboodh residual power series method (ARPSM) and the Aboodh transform iteration method (ATIM). The Caputo operator is utilized to define fractional order derivatives. In these methods, the analytical approximations are derived in series form. We calculate the first terms of the series and then estimate the absolute error resulting from leaving out the remaining terms to ensure the accuracy of the derived approximations and determine the accuracy and efficiency of the suggested methods. The derived approximations are discussed numerically using some values for the relevant parameters to the subject of the study. Useful examples are thought to illustrate the practical application of current approaches. We also examine the fractional order results that converge to the integer order solutions to ensure the accuracy of the derived approximations. Many researchers, particularly those in plasma physics, are anticipated to gain from modeling evolution equations describing nonlinear events in plasma systems.

Existence of basic solutions of first order linear homogeneous set-valued differential equations

The paper presents various derivatives of set-valued mappings,their main properties and how they are related to each other.Next, we consider Cauchy problems with linear homogeneousset-valued differential equations with different types ofderivatives (Hukuhara derivative, PS-derivative andBG-derivative). It is known that such initial value problems withPS-derivative and BG-derivative have infinitely many solutions.Two of these solutions are called basic. These are solutions suchthat the diameter function of the solution section is amonotonically increasing (the first basic solution) or monotonicallydecreasing (the second basic solution) function. However, the secondbasic solution does not always exist. We provideconditions for the existence of basic solutions of such initialvalue problems. It is shown that their existence depends on thetype of derivative, the matrix of coefficients on the right-handand the type of the initial set. Model examples are considered.

Hyers–Ulam Stability of Linear Quaternion-Valued Differential Equations with Two-Sided Constant Coefficients

Hyers–Ulam Stability of Linear Quaternion-Valued Differential Equations with Two-Sided Constant Coefficients

On the Hill Discriminant of Lamé's Differential Equation

Lamé's differential equation is a linear differential equation of the second order with a periodic coefficient involving the Jacobian elliptic function ${\rm sn}$ depending on the modulus $k$, and two additional parameters $h$ and $\nu$. This differential equation appears in several applications, for example, the motion of coupled particles in a periodic potential. Stability and existence of periodic solutions of Lamé's equations is determined by the value of its Hill discriminant $D(h,\nu,k)$. The Hill discriminant is compared to an explicitly known quantity including explicit error bounds. This result is derived from the observation that Lamé's equation with $k=1$ can be solved by hypergeometric functions because then the elliptic function ${\rm sn}$ reduces to the hyperbolic tangent function. A connection relation between hypergeometric functions then allows the approximation of the Hill discriminant by a simple expression. In particular, one obtains an asymptotic approximation of $D(h,\nu,k)$ when the modulus $k$ tends to $1$.

Formulation of physical fields from systems of first order linear partial differential equations

We present a heuristic approach to formulate physical fields from systems of first order linear partial differential equations by considering the symmetric and antisymmetric properties of the matrices that represent the systems of equations. Particularly, we show that Maxwell and Dirac fields can be established by imposing different types of symmetric and antisymmetric conditions on their associated representing matrices.

Exponential dichotomy and invariant manifolds of semi-linear differential equations on the line

In this paper we investigate the homogeneous linear differential equation vi(t) = A(t)v(t) and the semi-linear differential equation vi(t) = A(t)v(t) + g(t, v(t)) in Banach space X, in which A : R → L(X) is a strongly continuous function, g : R × X → X is continuous and satisfies ϕ-Lipschitz condition. The first we characterize the exponential dichotomy of the associated evolution family with the homogeneous linear differential equation by space pair (E, E∞), this is a Perron type result. Applying the achieved results, we establish the robustness of exponential dichotomy. The next we show the existence of stable and unstable manifolds for the semi-linear differential equation and prove that each a fiber of these manifolds is differentiable submanifold of class C1. Mathematics Subject Classification (2010): 34C45, 34D09, 34D10. Received 14 June 2021; Accepted 09 September 2022

A systematic approach to obtain the analytical solution for linear second order ordinary differential equations: part I

A systematic approach to obtain the analytical solution for linear second order ordinary differential equations: part I

A systematic approach to obtain the analytical solution for coupled linear second-order ordinary differential equations: Part II

A systematic approach to obtain the analytical solution for coupled linear second-order ordinary differential equations: Part II

Well posedness of linear parabolic partial differential equations posed on a star-shaped network with local time Kirchhoff's boundary condition at the vertex

The main purpose of this work is to provide an existence and uniqueness result for the solution of a linear parabolic system posed on a star-shaped network, which presents a new type of Kirchhoff's boundary transmission condition at the junction. This new type of Kirchhoff's condition - that we decide to call here local-time Kirchhoff's condition - induces a dynamical behavior with respect to an external variable that may be interpreted as a local time parameter, designed to drive the system only at the singular point of the network. The seeds of this study point towards a theoretical inquiry of a particular generalization of Walsh's random spider motions, whose spinning measures would select the available directions according to the local time of the motion at the junction of the network.

Unveiling new insights: taming complex local fractional Burger equations with the local fractional Elzaki transform decomposition method

This study aims to address the difficulties in solving coupled generalized non-linear Burger equations using local fractional calculus as a framework. The methodology used in this work, particularly in the area of local fractional calculus, combines the Elzaki transform with the Adomian decomposition method. This combination has proven to be a highly effective strategy for addressing non-linear partial differential equations within the local fractional context, which finds numerous practical applications. The proposed method offers a systematic and easily understandable procedure for tackling both linear and non-linear partial differential equations (PDEs). It provides an easy-to-follow path to solve these problems. We offer a real-world example that exhibits the method's successful use in resolving issues to corroborate its efficacy. The obtained solution is visually represented to illustrate the practical utility of this approach.2010 Mathematics Subject Classification34A34, 65M06, 26A33.

DYNAMIC CHARACTERISTICS OF TIME-VARYING CONTROL SYSTEM OF THE ROCKET’S ROTATIONAL MOVEMENT

The rocket motion control system is time-varying, because its parameters during flight depend on the trajectory point and fuel consumptions. In the available sources, due attention is not paid to the development of a mathematical apparatus of an applied value for the quan- titative assessment of the dynamic characteristics of a time-varying system.
 The purpose of the work is to justify the possibility of an algorithm building for calcu- lating the parameters of a link with constant parameters, which is equivalent to a time- varying system in terms of dynamic characteristics at the selected trajectory section.
 The link’s parameters are found by using the criterion of equivalence of the array of values of the motion model’soutput signal and the analytical solution of the link’s differential equation for a given sequence of input signals. This makes it possible to use the mathematical apparatus of stationary systems to determine the indicators of the disturbances compensation.
 The model of the control system of the rocket’s rotational movement in one plane is taken as a linear differential equation with time-varying parameters without taking into ac- count the executive device inertia and other disturbances. The link with constant parameters is a fractional-rational function of the second order, for the determination of which a se- quence of signals is applied to the input of the system model, the duration of which depends on the desired stability margin on the roots plane of the characteristic polynomial.
 Using the example of a time-varying system for controlling the rocket’srotational movement in the yaw plane, the possibility of determining the parameters of the transition process of compensation for the disturbances characteristic of it is shown for the selected tra- jectory section.
 The obtained results can be used in the design of a motion control system with time- varying parameters.
 The next stage of the research is an assessment of the complexity level of the calcula- tion algorithm when increasing the order of the system’smathematical model.

A quantum algorithm for linear differential equations with layerwise parameterized quantum circuits

Solving linear differential equations is a common problem in almost all fields of science and engineering. Here, we present a variational algorithm with shallow circuits for solving such a problem: given an N×N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N \ imes N$$\\end{document} matrix A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{A}}$$\\end{document}, an N-dimensional vector b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{b}$$\\end{document}, and an initial vector x(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{x}(0)$$\\end{document}, how to obtain the solution vector x(T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{x}(T)$$\\end{document} at time T according to the constraint dx(t)/dt=Ax(t)+b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{d}\\varvec{x}(t)/\ extrm{d} t = {\\varvec{A}}\\varvec{x}(t) + \\varvec{b}$$\\end{document}. The core idea of the algorithm is to encode the equations into a ground state problem of the Hamiltonian, which is solved via hybrid quantum-classical methods with high fidelities. Compared with the previous works, our algorithm requires the least qubit resources and can restore the entire evolutionary process. In particular, we show its application in simulating the evolution of harmonic oscillators and dynamics of non-Hermitian systems with PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{P}\\mathcal{T}$$\\end{document}-symmetry. Our algorithm framework provides a key technique for solving so many important problems whose essence is the solution of linear differential equations.

Challenges in Numerical Solutions of Higher-Dimensional Differential Equations

Differential equations constitute a fundamental tool in modeling various natural phenomena across scientific disciplines such as physics, engineering, and finance. We provide an overview of fractional differential equations, focusing on the computational requirements associated with their numerical solutions from a computer science perspective. We analyze the computational intricacies concerning First-Order Linear ODE, First-Order Nonlinear ODE, Second-Order Linear ODE, Second-Order Nonlinear ODE, Heat Equation (PDE), and Wave Equation (PDE). This comparative assessment delves into the computational demands of solving these equations using differential equation methodologies. While analytical solutions provide deep insights, obtaining numerical solutions, particularly in higher dimensions, remains a persistent challenge. Finite difference methods commonly employed for numerical solutions, In higher-dimensional problems, traditional numerical methods face challenges stemming from an exponential surge in grid points and the consequent demand for substantially decreased time step sizes. This paper explores the challenges posed by higher-dimensional differential equations in numerical solutions. It highlights the infeasibility of finite difference methods in such scenarios and emphasizes the need for innovative numerical techniques capable of efficiently handling the complexities of higher-dimensional differential equations. Overcoming these challenges is crucial for advancing our understanding and modeling capabilities in complex real-world systems governed by differential equations. Continued research efforts strive to develop novel numerical methodologies capable of addressing these challenges, aiming to broaden the scope of solvable higher-dimensional differential equations and expand their application across diverse scientific domains.

Solving second‐order differential equations in terms of confluent Heun's functions

This paper presents an algorithm that checks if a given second‐order homogeneous linear differential equation can be reduced to the confluent Heun's equation by using the change of variables transformation and the exp‐product transformation. The main purpose of this paper is finding solutions in terms of the confluent Heun's functions for the given differential equation.