Linear Differential Equations Research Articles

About the solutions of Modified Lorenz system

A Modified Lorenz system is a system of three differential equations. The third differential equation in the Lorenz system is replaced with a fifth-order linear homogeneous differential equation with constant coefficients. In the paper [B. Zlatanovska and D. Dimovski, A modified Lorenz system: Definition and solution, Asian-Eur. J. Math. 13(08) (2020) 2050164], the Modified Lorenz system is defined by offering a solution, but without solving the fifth-order linear homogeneous differential equation with constant coefficients. This differential equation is solved by solving the characteristic equation of fifth degree. A formula for solving the algebraic equation of the fifth degree does not exist. Therefore, solutions for the special cases of the characteristic equation of fifth degree will be offered. The solution of the Modified Lorenz system will be completed using these forms of the characteristic equation of the fifth degree.

Solving a class of Thomas–Fermi equations: A new solution concept based on physics-informed machine learning

This paper presents a novel physics-informed machine learning approach, designed to approximate solutions to a specific category of Thomas–Fermi differential equations. To tackle the inherent intricacies of solving Thomas–Fermi equations, we employ the quasi-linearization technique, which transforms the original non-linear problem into a series of linear differential equations. Our approach utilizes collocation least-squares support vector regression, leveraging fractional Chebyshev functions for finite interval simulations and fractional rational Chebyshev functions for semi-infinite intervals, enabling precise solutions for linear differential equations across varied spatial domains. These selections facilitate efficient simulations across both finite and semi-infinite intervals. Key contributions of our approach encompass its versatility, demonstrated through successful approximation of solutions for diverse Thomas–Fermi problem types, including those featuring non-local integral terms, Bohr radius boundary conditions, and isolated neutral atom boundary conditions defined on semi-infinite domains. Furthermore, our method exhibits computational efficiency, surpassing classical collocation methods by solving a sequence of positive definite linear equations or quadratic programming problems. Notably, our approach showcases precision, as evidenced by experiments, including the attainment of the initial slope of the renowned Thomas–Fermi equation with an impressive 39-digit precision.

Results about structural stability and the existence of limit cycles for piecewise smooth linear differential equations separated by the unit circle

Results about structural stability and the existence of limit cycles for piecewise smooth linear differential equations separated by the unit circle

Extremization to fine tune physics informed neural networks for solving boundary value problems

We propose a novel method for fast and accurate training of physics-informed neural networks (PINNs) to find solutions to boundary value problems (BVPs) and initial boundary value problems (IBVPs). By combining the methods of training deep neural networks (DNNs) and Extreme Learning Machines (ELMs), we develop a model which has the expressivity of DNNs with the fine-tuning ability of ELMs. We showcase the superiority of our proposed method by solving several BVPs and IBVPs which include linear and non-linear ordinary differential equations (ODEs), partial differential equations (PDEs) and coupled PDEs. The examples we consider include a stiff coupled ODE system where traditional numerical methods fail, a 3+1D non-linear PDE, Kovasznay flow and Taylor–Green vortex solutions to incompressible Navier–Stokes equations and pure advection solution of 1+1 D compressible Euler equation.The Theory of Functional Connections (TFC) is used to exactly impose initial and boundary conditions (IBCs) of (I)BVPs on PINNs. We propose a modification to the TFC framework named Reduced TFC and show a significant improvement in the training and inference time of PINNs compared to IBCs imposed using TFC. Furthermore, Reduced TFC is shown to be able to generalize to more complex boundary geometries which is not possible with TFC. We also introduce a method of applying boundary conditions at infinity for BVPs and numerically solve the pure advection in 1+1 D Euler equations using these boundary conditions.

Numerical Solution of the Linear Fractional Delay Differential Equation Using Gauss–Hermite Quadrature

Fractional order differential equations often possess inherent symmetries that play a crucial role in governing their dynamics in a variety of scientific fields. In this work, we consider numerical solutions for fractional-order linear delay differential equations. The numerical solution is obtained via the Laplace transform technique. The quadrature approximation of the Bromwich integral provides the foundation for several commonly employed strategies for inverting the Laplace transform. The key factor for quadrature approximation is the contour deformation, and numerous contours have been proposed. However, the highly convergent trapezoidal rule has always been the most common quadrature rule. In this work, the Gauss–Hermite quadrature rule is used as a substitute for the trapezoidal rule. Plotting figures of absolute error and comparing results to other methods from the literature illustrate how effectively the suggested approach works. Functional analysis was used to examine the existence of the solution and the Ulam–Hyers (UH) stability of the considered equation.

The transition from equations presented in matrix form for the perturbed Earth’s rotation to excitation equations

A convenient method for studying the influence of disturbing factors on Earth’s rotation, such as influence of oceans, atmosphere and hydrology, is representation of the Earth’s rotation equations in the excitation equations form, which was introduced by W. Munc and G. Macdonald. Traditionally, these factors are taken into account in two stages: first, an exact solution is found for an equation of the unperturbed rotation for given Earth’s model, and then the influence of small perturbation factors is taken into account in the form of corrections to the exact solution of unperturbed Earth’s rotation. The equations of unperturbed rotation for most modern theories of Earth’s rotation (and, possible, futures too) can be written in the form of matrix linear differential equations. The purpose of this study is to take into account the influence of small perturbations such as oceans, atmosphere and hydrology on Earth’s rotation in newly created theories of Earth’s rotation. A fairly general algorithm of converting perturbed equations in matrix form to the form of the excitation equations has been developed and presented. As an example, this method was applied to the relatively simple equations of the Sasao, Okubo and Saito theory based on the original Earth’s model of Molodenskii. The equations for finding corrections to the original solution are reduced to the form of excitation equations. The developed algorithm is proposed to be used to transition to equations in the form of excitation in newly created theories of the Earth’s rotation.

Oscillation criteria for the second-order linear advanced differential equation

New oscillatory criteria are established for the second-order linear advanced differential equation. Riccati’s technique and suitable estimates of non-oscillatory solutions are used for the proof of the results obtained. The presented criteria, in a certain sense, generalize the known ones.

A model ODE for the exponential asymptotics of nonlinear parasitic capillary ripples

Abstract In this work, we develop a linear model ordinary differential equation (ODE) to study the parasitic capillary ripples present on steep Stokes waves when a small amount of surface tension is included in the formulation. Our methodology builds upon the exponential asymptotic theory of Shelton & Trinh (J. Fluid Mech., vol. 939, 2022, A17), who demonstrated that these ripples occur beyond-all-orders of a small-surface-tension expansion. Our model equation, a linear ODE forced by solutions of the Stokes wave equation, forms a convenient tool to calculate numerical and asymptotic solutions. We show analytically that the parasitic capillary ripples that emerge in solutions to this linear model have the same asymptotic scaling and functional behaviour as those in the fully nonlinear problem. It is expected that this work will lead to the study of parasitic capillary ripples that occur in more general formulations involving viscosity or time-dependence.

Deflection of light by a Reissner–Nordström black hole and Painlevé VI equation

We consider the bending angle of the trajectory of a photon incident from and deflected to infinity around a Reissner–Nordström black hole. We treat the bending angle as a function of the squared reciprocal of the impact parameter and the squared electric charge of the background normalized by the mass of the black hole. It is shown that the bending angle satisfies a system of two inhomogeneous linear partial differential equations with polynomial coefficients. This system can be understood as an isomonodromic deformation of the inhomogeneous Picard–Fuchs equation satisfied by the bending angle in the Schwarzschild spacetime, where the deformation parameter is identified as the background electric charge. Furthermore, the integrability condition for these equations is found to be a specific type of the Painlevé VI equation that allows an algebraic solution. We solve the differential equations both at the weak and strong deflection limits. In the weak deflection limit, the bending angle is expressed as a power series expansion in terms of the squared reciprocal of the impact parameter and we obtain the explicit full-order expression for the coefficients. In the strong deflection limit, we obtain the asymptotic form of the bending angle that consists of the divergent logarithmic term and the finite O(1) term supplemented by linear recurrence relations which enable us to straightforwardly derive higher order coefficients. In deriving these results, the isomonodromic property of the differential equations plays an important role. Lastly, we briefly discuss the applicability of our method to other types of spacetimes such as a spinning black hole.

On pole-skipping with gauge-invariant variables in holographic axion theories

We study the pole-skipping phenomenon within holographic axion theories, a common framework for studying strongly coupled systems with chemical potential (μ) and momentum relaxation (β). Considering the backreaction characterized by μ and β, we encounter coupled equations of motion for the metric, gauge, and axion field, which are classified into spin-0, spin-1, and spin-2 channels. Employing gauge-invariant variables, we systematically address these equations and explore pole-skipping points within each sector using the near-horizon method. Our analysis reveals two classes of pole-skipping points: regular and singular pole-skipping points in which the latter is identified when standard linear differential equations exhibit singularity. Notably, pole-skipping points in the lower-half plane are regular, while those elsewhere are singular. This suggests that the pole-skipping point in the spin-0 channel, associated with quantum chaos, corresponds to a singular pole-skipping point. Additionally, we observe that the pole-skipping momentum, if purely real or imaginary for μ = β = 0, retains this characteristic for μ ≠ 0 and β ≠ 0.

An Efficient Algorithm for Basic Elementary Matrix Functions with Specified Accuracy and Application

If the matrix function f(At) posses the properties of f(At)=gf(tkA, then the recurrence formula fi−1=gfi,i=N,N−1,⋯,1,f(tA)=f0, can be established. Here, fN=f(AN)=∑j=0majANj,AN=tkNA. This provides an algorithm for computing the matrix function f(At). By specifying the calculation accuracy p, a method is presented to determine m and N in a way that minimizes the time of the above algorithm, thus providing a fast algorithm for f(At). It is important to note that m only depends on the calculation accuracy p and is independent of the matrix A and t. Therefore, f(AN) has a fixed calculation format that is easily computed. On the other hand, N depends not only on A, but also on t. This provides a means to select t such that N is equal to 0, a property of significance. In summary, the algorithm proposed in this article enables users to establish a desired level of accuracy and then utilize it to select the appropriate values for m and N to minimize computation time. This approach ensures that both accuracy and efficiency are addressed concurrently. We develop a general algorithm, then apply it to the exponential, trigonometric, and logarithmic matrix functions, and compare the performance with that of the internal system functions of Mathematica and Pade approximation. In the last section, an example is provided to illustrate the rapid computation of numerical solutions for linear differential equations.

Z-transform Solution for Partial Difference Equations

The aim of this research is to study the Z transformation to solve linear partial difference equations. This is due to the fact that linear partial difference equations cannot be solved directly using the Z transformation, and the research concluded that direct partial difference conditions cannot be treated directly by the Z transformation except after converting the partial difference equations to linear difference equations using one of the transformation methods. The research recommended using Binomial transformation to solve partial difference equations.

Well-posedness and parabolicity of the boundary-value problem for systems of partial differential equations

The paper examines two-point boundary value problem for systems of linear partial differential equations. Not every system has a well posed two-point boundary value problem. For example, for equation {$$\displaystyle\frac{\partial u(x_1,x_2,t)}{\partial t}=\displaystyle\frac{\partial u(x_1,x_2,t)}{\partial x_1}+i \frac{\partial u(x_1,x_2,t)}{\partial x_2}$$ there are no boundary conditions of the form $au(x_1,x_1,0)+bu(x_1,x_2,T)=\varphi(x_1,x_2)$, under which this boundary value problem will be of a well posed in the Schwartz spaces. In the work, the conditions for the matrix of the system under which exist well posed boundary-value problem are found in the Schwartz spaces, and the form of these boundary conditions is also indicated. So, for systems with a Hermitian matrix, the boundary value problem with conditions of form $$u(x,0)+u(x,T)=\varphi(x)$$ will always be of a well posed in the Schwartz spaces, as well as in the spaces of functions of finite smoothness of power growth. For systems with one spatial variable, it is proved that well posed boundary value problems with the condition $u(x,0)+bu(x,T)=\varphi(x)$ ith positive b always exist. In addition, parabolic boundary value problems with property of increasing smoothness of the solutions are investigated. Similar results hold for linear partial differential equations. The Helmholtz equation $$\displaystyle\frac{\partial^2 u(x,t)}{\partial t^2}+\Delta u(x,t)=ku(x,t),$$ is considered as an example. It is not well posed according to Petrovsky, but for it exist a boundary value problem that is parabolic. For an equation with one spatial variable, sufficient conditions for the second-order equation in time are given, under which parabolic boundary value problems exist.

Two-point boundary-value problems for differential equations with generalized piecewise-constant argument

UDC 517.9 We consider a two-point boundary-value problem for a system of differential equations with generalized piecewise-constant argument. To solve the problem, we propose to use a constructive method based on the Dzhumabaev parametrization method and a new approach to the concept of general solution. The interval is partitioned with regard for the singularities of the argument. The values of the solution at the interior points of the partition are regarded as additional parameters, and the differential equation is transformed into a system of Cauchy problems with parameters on subintervals of the partition. By using the solutions of these problems, we obtain a new general solution of the differential equation with piecewise-constant argument and establish its properties. The new general solution, boundary conditions, and the conditions of continuity of the solution at the interior points of the partition are used to construct a linear system of algebraic equations for the introduced parameters. The coefficients and the right-hand side of the system are found as a result of the solution of Cauchy problems for linear ordinary differential equations on the subintervals of the partition. It is shown that the solvability of the boundary-value problem is equivalent to the solvability of the constructed system. We propose algorithms of the parametrization method for solving the analyzed boundary-value problem and establish necessary and sufficient conditions for the well-posedness of this problem.

Exact solutions to differential equations with different arguments

Various linear and non-linear first-order differential equations with different arguments are considered. Exact solutions to these equations are provided. Systems of two coupled linear first-order differential equations are also solved explicitly and exactly.

Solving Higher Orders Linear Complex Partial Differential Equations via Two Dimensional Differential Transform Method

Solving Higher Orders Linear Complex Partial Differential Equations via Two Dimensional Differential Transform Method

Unveiling the underlying drivers of Phanerozoic marine diversification.

In investigating global patterns of biodiversity through deep time, many large-scale drivers of diversification have been proposed, both biotic and abiotic. However, few robust conclusions about these hypothesized effectors or their roles have been drawn. Here, we use a linear stochastic differential equation (SDE) framework to test for the presence of underlying drivers of diversification patterns before examining specific hypothesized drivers. Using a global dataset of observations of skeletonized marine fossils, we infer origination, extinction and sampling rates (collectively called fossil time series) throughout the Phanerozoic using a capture-mark-recapture approach. Using linear SDEs, we then compare models including and excluding hidden (i.e. unmeasured) drivers of these fossil time series. We find evidence of large-scale underlying drivers of marine Phanerozoic diversification rates and present quantitative characterizations of these. We then test whether changing global temperature, sea-level, marine sediment area or continental fragmentation could act as drivers of the fossil time series. We show that it is unlikely any of these four abiotic factors are the hidden drivers we identified, though there is evidence for correlative links between sediment area and origination/extinction rates. Our characterization of the hidden drivers of Phanerozoic diversification and sampling will aid in the search for their ultimate identities.

Use of GeoGebra as a didactic resource for the solution of linear ordinary differential equations

Use of GeoGebra as a didactic resource for the solution of linear ordinary differential equations

On the generalized Apostol-Kolodner differential equation of the second-order

This paper concerns another approach for solving the matrix differential equations of the second order $X''(t)=AX'(t)+BX(t)$, where $A$, $B$ are square matrices of order ${d\times d}$. Such approach is based on some properties of matrix square root and the linear difference equation. We establish various new results and explicit formulas for the solutions of this type of matrix differential equations. Moreover, because of the non-commutativity condition between $A$ and $B$, the matrix fundamental system will play an important role. Finally, our results and their robustness, are validated by providing some examples and applications.

Solving Second-Order Homogeneous Linear Differential Equations in Terms of the Tri-Confluent Heun’s Function

In this paper, we state an algorithm that checks whether a given second-order linear differential equation can be reduced to the tri-confluent Heun’s equation. The algorithm provides a method for finding solutions of the form exp(∫r(x)dx)·HeunT(q,α,γ,δ,ϵ,f(x)), where the parameters α,β,λ∈C, the functions r,f∈C(x), and f are not constant.