Linear Differential Equations Research Articles

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.

Neural Network Methods Based on Efficient Optimization Algorithms for Solving Impulsive Differential Equations

In view of the fact that impulsive differential equations have the discreteness due to the impulse phenomenon, this paper proposes a single hidden layer neural network method based extreme learning machine and a physicsinformed neural network method combined with learning rate attenuation strategy to solve linear impulsive differential equations and nonlinear impulsive differential equations respectively. For the linear impulsive differential equations, firstly, the interval is segmented according to the impulse points, and a single hidden layer neural network model is constructed, the weight parameters of hidden layer are randomly set, the optimal output parameters and solution of the first segment are obtained by the extreme learning machine algorithm, then we calculate the initial value of the second segment according to the jumping equation, and the remaining segments are solved in turn in the same way. Although the single hidden layer neural network method proposed can solve linear equations with high accuracy, it is not suitable for solving nonlinear equations. Therefore, we propose the physics-informed neural network combined with learning rate attenuation strategy to solve the nonlinear impulsive differential equations, then the Adam algorithm and L-BFGS algorithm are combined to find the optimal approximate solution of each segment. Numerical examples show that the single hidden layer neural network method with Legendre polynomials as the activation function and the physics-informed neural network method combined with learning rate attenuation strategy can solve linear and nonlinear impulsive differential equations with higher accuracy.

Influence of yeast size on the kinetics of cell disruption in an ultrasonic homogenizer

The results of studies on the disintegration kinetics of the yeast Saccharomyces cerevisiae are presented. The process was carried out in a 500 W ultrasonic homogenizer equipped with a spherical working chamber with a volume of 100 cm 3. The concentration of the suspension of microorganisms was 0.05 g d.m./cm 3. The continuous phase was water solution containing 0.15 M NaCl and 4 mM K 2HPO 4. The kinetics of cell disruption were studied by the direct method. The theory of random transformation of dispersed matter was used to analyze the process. There was significant variation in the size of yeast cells. The range of changes in the values of parameters describing the size of microorganisms was divided into size classes. The kinetics of cell disruption in individual classes was described by a first-order linear differential equation. During the implosion of cavitation bubbles, the transformation volume of individual microorganisms is generated. It has been shown that as the volume of cells in subsequent size classes increases, their transformation volumes do not increase significantly. The safe volume for cells remains unchanged. As the size of the microorganisms increased, there was no increase in the constant rate of cell disruption.

Estimation of the Impact of Vaccination Intervention on Recovered Coronavirus Patients

This work estimated the impact of vaccination intervention on coronavirus patients who have recovered from the disease and the vulnerability index of the recovered population due to the impact of vaccine was also investigated. This work adopted a numerical solution to study the continuous dynamical system of linear first order differential equations describing a SEIR (Susceptible, Exposed, Infected, Recovered) model on the spread of Coronavirus Disease – 2019 (COVID-19). To tackle this problem, MATLAB ordinary differential equation of order 45 (ODE45) numerical method was adopted for the analysis. The vulnerability index of the recovered population was low due to the impact of vaccine meaning that the recovered population will gain immunity and they will not be re-infected. The study recommended that coronavirus patients who have recovered from the disease should ensure that they have vaccination administered to them to avoid re-occurrence of the virus attack as an intervention strategy.

Analysis of the chemical diffusion master equation for creation and mutual annihilation reactions

We propose an infinite dimensional generating function method for finding the analytical solution of the so-called chemical diffusion master equation (CDME) for creation and mutual annihilation chemical reactions. CDMEs model by means of an infinite system of coupled Fokker–Planck equations the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles; here, we focus an creation and mutual annihilation chemical reactions combined with Brownian diffusion of the single particles. Using our method we are able to link certain finite dimensional projections of the solution of the CDME to the solution of a single linear fourth order partial differential equation containing as many variables as the dimension of the aforementioned projection space. Our technique extends the one presented in Lanconelli [J. Math. Anal. Appl. 526, 127352 (2023)] and Lanconelli et al. [arXiv:2302.10700 [math.PR] (2023)] which allowed for an explicit representation for the solution of birth-death type CDMEs.

Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical methods to solve linear Partial Differential Equations

Objectives: This paper aims to address the limitations of the Crank-Nicolson Finite Difference method and propose an improved version called the modified Crank-Nicolson method. Methods: Utilized implicit discretization in time and space, with parameters k = 0.001, h = 0.1, and γ = 0.1. Conducted extensive testing on various partial differential equations. Findings: Results, displayed in Table 1, showcase the method's stability and accuracy. Comparative analysis in Table 2 demonstrates the Modified Crank-Nicolson method consistently outperforming the traditional approach, reaffirming its superiority in accuracy. Novelty: The modified Crank-Nicolson method offers a significant enhancement to the traditional Crank-Nicolson finite difference method, making it a valuable tool for effectively solving partial differential equations. Keywords: Crank­Nicolson Method, Modified Crank­Nicolson Method, Finite Difference, Partial Differential Equations, Parabolic Equations, Python Software

Approximate Solution of Linear Fuzzy Random Ordinary Differential Equations Using Laplace Variational Iteration Method

In this article, the Laplace transformation method in connection with the variational iteration method will be used to solve approximately fuzzy random ordinary differential equations. After that, the sequence of approximated closed form iterated solutions is derived based on the general Lagrange multiplier evaluated using the well-known convolution theorem of the Laplace transformation method. In addition, two examples are given and solved to illustrate the reliability, efficiency and applicability of the proposed method, they are simulated using computer programs with two different generations of stochastic processes, namely the Wiener process or Brownian motion, which are 1000 and 10000, respectively.

Existence results and stabilization of homogeneous conformable fractional order systems

In this paper, we give results on the stability of homogeneous conformable fractional order systems using assumptions on a family of compact sets and provide the stabilization of an affine control system and given an explicit homogeneous feedback control with the requirement that a control Lyapunov function exists and satisfying a homogeneous condition and we present existence and uniqueness theorems for sequential linear conformable fractional differential equations.

A note on Laplace Adomian decomposition method on a linear problem

In last two decades, Laplace Adomian Decomposition Method (LADM) is vastly used to solve non-linear (or even fractional order) differential equations. The method approaches the solution with the partial sums of function series. However, it is not easy to show that the limit of the function series is the exact solution of the problem. In this article, we consider a simple problem such as homogenous second-order linear ordinary differential equation with constant coefficients. We prove analytically that the LADM gives the right exact solution to the considered problem.

Growth of Solutions to Complex Linear Differential Equations with Analytic or Meromorphic Coefficients in ℂ̅ − {z0} of Finite Logarithmic Order

Growth of Solutions to Complex Linear Differential Equations with Analytic or Meromorphic Coefficients in ℂ̅ − {z0} of Finite Logarithmic Order

Simulation of exchange processes in multi-component environments with account of data uncertainty

The goal of the work. Proposals for methods of solving systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients that defined in interval form and intended for modeling exchange processes in multicomponent environments. Research subject: systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients defined in interval form. Research method: interval analysis. The obtained results. Systems of linear homogeneous and non-homogeneous differential equations, which are used in modeling exchange processes in multicomponent environments, are considered. Such systems can be considered, for example, in problems of chemical kinetics, materials science, and the theory of Markov processes. To obtain the solution of these equations, specialized calculators of analytical transformations were used and tested. The Matlab system (ode15s solver) was used for numerical analysis of systems of differential equations. It is shown that the application of interval methods of numerical analysis at the initial stage of system modeling has some advantages over probabilistic methods because they do not require knowledge of the laws of distribution of the results of the system state parameter measurements and their errors. It is shown that existing methods of solving systems of linear differential equations can be divided into two groups. Common to these groups is the use of interval expansion of classical methods for solving differential equations given in interval form. The difference between these two groups of methods is as follows. The methods of the first group can be used for all types of differential equations but require the creation of special software. The peculiarity of the methods of the second group is that they can be used to solve equations analytically or using numerical analysis packages. The application of the methods of the second group is shown on the example of solving a system of differential equations, the coefficients of which are determined in interval form. The system of these equations is intended for modeling the processes of exchange with the external environment of the elements of the model of a specific physical system. In the case when the coefficients of these equations are variables, their piecewise-constant approximation is applied and a criterion that determines the possibility of its application is given. The technique proposed in the paper can be applied to solve systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients if they are given by slowly varying functions. In the case when the coefficients of the equations are determined in the interval form, the technique allows obtaining their solution also in the interval form and does not require the creation of special software.

A simple model for the prediction of CO2 concentrations in the atmosphere, depending on global CO2 emissions

We present a very simple model for estimating time dependent atmospheric CO2 concentrations c(t) from global carbon emission scenarios, serving as single input data. We derive a single linear differential equation of 1st order, based on parameters which are estimated from quantitative data of the global carbon project and Mauna Loa data for CO2 concentrations. The model is tested first by comparing it to the 1960–2021 period with reasonably good quantitative agreement and, second to two of the typical current IPCC scenarios with good qualitative agreement. Finally, some new emission scenarios are modelled. Despite several drawbacks concerning absolute quantitative predictions, there are two important advantages of the model. First, it can be easily executed by students already with simple programmable spreadsheet programs such as Excel. Second input emission scenarios can be changed easily and expected changes are immediately seen for discussion during undergraduate and graduate courses on the carbon cycle and climate change.

On the structure of the infinitesimal generators of scalar one-dimensional semigroups with discrete Lyapunov functionals

AbstractDynamical systems generated by scalar reaction-diffusion equations on an interval enjoy special properties that lead to a very simple structure for the semiflow. Among these properties, the monotone behavior of the number of zeros of the solutions plays an essential role. This discrete Lyapunov functional contains important information on the spectral behavior of the linearization and leads to a Morse-Smale description of the dynamical system. Other systems, like the linear scalar delay differential equations under monotone feedback conditions, possess similar kinds of discrete Lyapunov functions. Here we discuss and characterize classes of linear equations that generate semiflows acting on $$C^0[0,1]$$ C 0 [ 0 , 1 ] or on $$C^1[0,1]$$ C 1 [ 0 , 1 ] which admit discrete Lyapunov functions related to the zero number. We show that, if the space is $$C^1[0,1]$$ C 1 [ 0 , 1 ] , the corresponding equations are essentially parabolic partial differential equations. In contrast, if the space is $$C^0[0,1]$$ C 0 [ 0 , 1 ] , the corresponding equations are generalizations of monotone feedback delay differential equations.

Degeneration phenomenon in linear ordinary differential equations

Abstract Given a linear constant coefficient ODE depending on a parameter, when this parameter approaches zero, the solution set converges to the solution set of the limit differential equation if the leading coefficient does not vanish. The situation is very subtle in the singular case, i.e., in the case when this coefficient becomes zero. The solution set then may even collapse completely. In this note, a formalism is developed in which the solution set of a linear constant coefficient ODE always depends continuously on the equation coefficients.

Arbitrary coefficient assignment to the characteristic function of the scalar type for linear systems with state delays by linear static delayed output feedback

We consider a control system defined by a system of linear differential equations in real or complex space with multiple lumped and distributed delays in the state variable. For this system, we study the problem of arbitrary coefficient assignment for the characteristic function of the scalar type by linear static delayed output feedback with lumped and distributed delays. We obtain necessary and sufficient conditions for solvability of the arbitrary coefficient assignment problem by the static delayed output feedback controller when the system coefficients have a special form. Corollaries on arbitrary finite spectrum assignment and on stabilization of the system are obtained. We prove that the obtained results generalize the corresponding theorem proved earlier for a control system defined by a linear delay-differential equation of nth order. Examples are presented illustrating the results obtained.

Painlevé analysis, Painlevé–Bäcklund, multiple regular and singular kink solutions of dynamical thermopherotic equation drafting wrinkle propagation

The thermophoretic motion (TM) system with a variable heat transmission factor, based on the Korteweg-de Vries (KdV) equation, is used to model soliton-like thermophoresis of creases in graphene sheets. Painlevé test is employed to discover that the equation is Painlevé integrable. Then an auto-Bäcklund transformation using the truncated Painlevé expansion is obtained. Concerning the additional variables, the auto-Bäcklund transformations convert the nonlinear model to a set of linear partial differential equations. Finally, various explicit precise solutions based on the acquired auto-Bäcklund transformations are investigated and the researched solutions are illustrated in 3D, 2D and contour plots. Furthermore, the Cole-Hopf transformation is used in conjunction with Hirota’s bilinear technique to get multiple regular and singular kink solutions.

Symplectic solutions of the plane annular sectors in micropolar elasticity

The symplectic approach is utilized to derive the solutions of the plane annular sector in micropolar elasticity. According to the Hellinger-Reissner variational principle, the Hamiltonian canonical equations are obtained for the plane problem in an annular sectorial region. Through the method of separation of variables, we obtain an eigenvalue equation of the Hamiltonian operator matrix which is a system of linear differential equations with variable coefficients. By the method of series solutions, its general solutions can be expressed as the linear combination of Bessel functions and power functions. The corresponding eigensolutions for three types of homogeneous boundary conditions are derived. According to the adjoint symplectic orthogonality of the eigensolutions, the solution of the micropolar plane annular sector is expressed as a series expansion of eigensolutions. Numerical analyses are presented to demonstrate the validity of the proposed method and confirm its ability to accurately capture the size effect.

The new notion of Bohl dichotomy for non-autonomous difference equations and its relation to exponential dichotomy

In A. Czornik et al. [Spectra based on Bohl exponents and Bohl dichotomy for non-autonomous difference equations, J. Dynam. Differ. Equ. (2023)] the concept of Bohl dichotomy is introduced which is a notion of hyperbolicity for linear non-autonomous difference equations that is weaker than the classical concept of exponential dichotomy. In the class of systems with bounded invertible coefficient matrices which have bounded inverses, we study the relation between the set BD of systems with Bohl dichotomy and the set ED of systems with exponential dichotomy. It can be easily seen from the definition of Bohl dichotomy that ED ⊆ BD . Using a counterexample we show that the closure of ED is not contained in BD . The main result of this paper is the characterization int ⁡ BD = ED . The proof uses upper triangular normal forms of systems which are dynamically equivalent and utilizes a diagonal argument to choose subsequences of perturbations each of which is constructed with the Millionshikov Rotation Method. An Appendix describes the Millionshikov Rotation Method in the context of non-autonomous difference equations as a universal tool.

A posteriori error estimates of the weak Galerkin finite element methods for parabolic problems

In this paper, We propose the residual-based a posteriori error estimator of the weak Galerkin finite element method with the backward Euler time discretization for the linear parabolic partial differential equation. For the a posteriori error estimator, we introduce the Helmholtz decomposition technique to prove its reliability. We mainly study WG element (Pj(K),Pℓ(∂K),V(K,r)=RTj(K)). Numerical experiments based on the lowest order case, i.e. (P0(K),P0(∂K),RT0(K)), are provided to verify the theoretical research.