Solutions Of Ordinary Differential Equations Research Articles

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

In this paper, we introduce a special system of linear equations with a symmetric, tridiagonal matrix, whose solution vector contains the values of the analytical solution of the original ordinary differential equation (ODE) in grid points. Further, we present the derivation of an exact scheme for an arbitrary mesh grid and prove that its application can completely avoid other errors in discretization and numerical methods. The presented method is constructed on the basis of special local green functions, whose special properties provide the possibility to invert the differential operator of the ODE. Thus, the newly obtained results provide a general, exact solution method for the second-order ODE, which is also effective for obtaining the arbitrary grid, Dirichlet, and/or Neumann boundary conditions. Both the results obtained and the short case study confirm that the use of the exact scheme is efficient and straightforward even for ODEs with discontinuity functions.

Symmetries and exact solution of certain nonlinear fractional ordinary differential equations

Symmetries and exact solution of certain nonlinear fractional ordinary differential equations

Analysis of and Decision-Making on the Operational Reliability of Cyber-Physical Systems

Matters concerned with analyzing the operational reliability and recovery process of cyber-physical systems (CPS) are addressed. The proposed approach is based on dividing the CPS life cycle into time intervals with linking them to pre-emergency, emergency, restored and predicted states. The CPS serviceability is restored with respect to extremely critical and mission-critical equipment. In carrying out the analysis, graphs representing the states of cybernetic and physical components are introduced. Differential equations linking the probability of returning into a serviceable state with failure rates and restorations of various types are derived. The solution of the appropriate ordinary differential equations yields a clear graphical interpretation in the time domain and shows the changes in the probability with which the system components transfer from one state to another at different CPS life cycle time slices. By using the developed flow chart of making decisions on restoring or replacing the corresponding component, a reasonable choice can be made among possible options of actions aimed at restoring the serviceability of a multicomponent system. An algorithm for implementing the method has been developed and tested on a numerical example.

Сингулярности квазилинейных дифференциальных уравнений

We study solutions of quasi-linear ordinary differential equations of the second order at their singular points, where the coefficient of the second-order derivative vanishes. Either solutions entering a singular point with definite tangential direction (proper solutions) or those without definite tangential direction (oscillating solutions) are considered. It is shown that oscillating solutions generically do not exist, and proper solutions enter a singular point in strictly definite tangential directions. A local representation for proper solutions in a form similar to Newton-Puiseux series is obtained.

Solving Differential Equations and Systems of Differential Equations with Inverse Laplace Transform

The inverse Laplace transform enables the solution of ordinary linear differential equations as well as systems of ordinary linear differentials with applications in the physical and engineering sciences. The Laplace transform is essentially an integral transform which is introduced with the help of a suitable generalized integral. The ultimate goal of this work is to introduce the reader to some of the basic ideas and applications for solving initially ordinary differential equations and then systems of ordinary linear differential equations.

Exact Solutions for Some Singular Linear Ordinary Differential Equations of High Orders via NB1 Polynomials

In this study, we numerically solve the singular linear ordinary differential equations (SLODEs) of higher order using the collocation method based on the NB1 polynomial. An operational matrix form of the given ordinary differential equations (ODEs), the relations of various solutions and the derivatives are obtained from NB1 polynomials. The proposed method reduces the given problem to a linear algebraic equation system, which removes the singularity of ordinary differential equations. The inverse matrix method is used to solve the resulting system to obtain the coefficients of NB1 polynomials. The obtained exact solutions to different problems of high orders show the reliability and accuracy of the presented method.

On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces

ABSTRACT This article discusses differential equations in infinite-dimensional spaces. We introduce a general version of the Lipschitz-type continuity criterion without the notion of a distance and prove an existence and uniqueness result for ordinary differential equations in locally convex topological linear spaces. It is shown that remarkable techniques of classical analysis, such as the method of successive approximations, are still adaptable tools in the study of the abstract models.

Oscillating Solutions of a Second-Order Ordinary Differential Equation with a Three-Position Hysteresis Relay without Entering the Saturation Zones

Oscillating Solutions of a Second-Order Ordinary Differential Equation with a Three-Position Hysteresis Relay without Entering the Saturation Zones

Rate-limiting recovery processes in neurotransmission under sustained stimulation

At active zones of chemical synapses, an arriving electric signal induces the fusion of vesicles with the presynaptic membrane, thereby releasing neurotransmitters into the synaptic cleft. After a fusion event, both the release site and the vesicle undergo a recovery process before becoming available for reuse again. Of central interest is the question which of the two restoration steps acts as the limiting factor during neurotransmission under high-frequency sustained stimulation. In order to investigate this problem, we introduce a non-linear reaction network which involves explicit recovery steps for both the vesicles and the release sites, and includes the induced time-dependent output current. The associated reaction dynamics are formulated by means of ordinary differential equations (ODEs), as well as via the associated stochastic jump process. While the stochastic jump model describes the dynamics at a single active zone, the average over many active zones is close to the ODE solution and shares its periodic structure. The reason for this can be traced back to the insight that recovery dynamics of vesicles and release sites are statistically almost independent. A sensitivity analysis on the recovery rates based on the ODE formulation reveals that neither the vesicle nor the release site recovery step can be identified as the essential rate-limiting step but that the rate-limiting feature changes over the course of stimulation. Under sustained stimulation, the dynamics given by the ODEs exhibit transient changes leading from an initial depression of the postsynaptic response to an asymptotic periodic orbit, while the individual trajectories of the stochastic jump model lack the oscillatory behavior and asymptotic periodicity of the ODE-solution.

Ordinary fractional differential equations to show results of oscillations, using the graphical and numerical approximation in the Julia programming language

The present study developed at a documentary and explanatory level, which aims to assume through an illustrative approach in the Julia programming language, the representation of oscillating results in solutions of fractional ordinary differential equations, among these topics the solutions stand out in: Systems of differential equations equations, differential equations of partial fractions and the differential equations of fractional lags. In this sense, this scientific article assumes the recent foundations in this field and serves as foundations for future research contributions.

Reproducing Kernel Method with Global Derivative

Ordinary differential equations describe several phenomena in different fields of engineering and physics. Our aim is to use the reproducing kernel Hilbert space method (RKHSM) to find a solution to some ordinary differential equations (ODEs) that are described by using the global derivative. In this research, we used the RKHSM to construct new numerical solutions for nonlinear ODEs with global derivative. The used method systematically produces analytic and approximate solutions in the series’s form. We tested three applications for showing the performance of the RKHSM.

On the zeros of solutions of ordinary and fractional differential equations

This paper is devoted to studying on the locations of zeros of related integral operators and the solutions of some ordinary and fractional differential equations. We generalize Sturm and Picone's theorems and Leighton and Levin's criteria. Moreover, we share some oscillation and disconjugacy criteria for the solutions of ordinary second‐order Sturm–Liouville and fractional differential equations. Finally, we introduce some properties of the solutions of fractional differential equations.

Data simulation of optimal model for numerical solution of differential equations based on deep learning and genetic algorithm

Calculus equation is an important tool for mathematical research and plays an important role in most natural science research. Since the beginning of the eighteenth century, people have gradually used differential and integral equations to solve physical problems. In general, several different aspects of differential equations in the field of mathematics are concerned and studied by most scholars. However, this paper studies and establishes the optimal model for numerical solution of differential equations through deep learning and genetic algorithm. In this paper, the solution of ordinary differential equations is solved through the use of polynomial function space, while the linear combination of simple function x and its product nx can obtain multinomial function space. The space function form of polynomial is very simple, and the operation ability is very strong. Almost all functions can be approximated, and the function space can be transformed by a simple function. Through data simulation test results, it can be found that the oscillation of neural network output is stronger and stronger with the increase in depth, that is to say, the deeper depth endows the neural network with stronger oscillation properties, so for the oscillation function, the depth neural network fitting effect is better than the shallow neural network. Therefore, by combining deep learning and genetic algorithm, this paper studies and establishes the optimal model for numerical solution of differential equations, and finds that the deep neural network can largely complete data simulation.

Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations

We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schrödinger framework to solve the Liouville equation for the HJE with the Hamiltonian formulation of classical mechanics, since it can be recast as the semiclassical limit of the Wigner transform of the Schrödinger equation. Comparison between the Schrödinger and the Liouville framework will also be made.

Generalized Solutions of Ordinary Differential Equations Related to the Chebyshev Polynomial of the Second Kind

In this work, we employed the Laplace transform of right-sided distributions in conjunction with the power series method to obtain distributional solutions to the modified Bessel equation and its related equation, whose coefficients contain the parameters ν and γ. We demonstrated that the solutions can be expressed as finite linear combinations of the Dirac delta function and its derivatives, with the specific form depending on the values of ν and γ.

Numerical Solution of Second Order Fuzzy Ordinary Differential Equations using Two-Step Block Method with Third and Fourth Derivatives

Fuzzy differential equation models are suitable where uncertainty exists for real-world phenomena. Numerical techniques are used to provide an approximate solution to these models in the absence of an exact solution. However, existing studies that have developed numerical techniques for solving second-order fuzzy ordinary differential equations (FODEs) possess an absolute error accuracy that could be improved. Therefore, this article developed a more accurate higher derivative self-starting block scheme for the numerical solution of second-order FODEs with fuzzy initial and boundary conditions imposed. Linear block approach using Taylor series expansion is adopted for the derivation of the proposed method and the basic properties are established using the definitions of stability and consistency for block methods. According to the numerical results, when compared to the exact solution in terms of absolute error, the new method proposed in this article outperformed existing numerical methods. It is thus concluded that the proposed method is effective for solving second-order FODEs directly.

Direct Solution of Second Order Ordinary Differential Equations with a OneStep Hybrid Numerical Model

This study shows how to solve general second order ordinary differential equations directly using a one-step hybrid block technique. By means of collocation and interpolation of power series approximation, a continuous linear multistep method was developed. The developed method was evaluated at several grid and off-grid points to generate a discrete block approach. Order, error-constant, zero stability, consistency, and convergence were all investigated as basic features of the approach. The method was found to be more efficient and produce better approximations than previous approaches for solving some second order ordinary differential equations with initial value problems.

Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero

Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero

The New Integral Transform: “NE Transform” and Its Applications

Abstract This work introduces a new integral transform for functions of exponential order called “NE integral transform”. We prove some properties of NE -transform. Also, some applications of the NE- transform to find the solution to ordinary linear equation are given. The relationships of the new transform with well-known transforms are characterized by integral identities. We study the properties of this transform. Then we compare it with few exiting integral transforms in the Laplace family such as Laplace, Sumudu, Elzaki, Aboodh and etc. As well, the NE integral transform is applied and used to find the solution of linear ordinary differential equations.

High accuracy breaking point detection for stiff DDEs

We have recently introduced a very high order Discontinuous Galerkin (DG) method for the solution of stiff ordinary differential equations (ODEs) and delay differential equations (DDEs). In this communication we focus on the detection of breaking points frequently arising in the solution of stiff DDEs. Breaking points are discontinuities appearing in the solution of DDEs and/or in some of its derivatives. Breaking points affect the efficiency of the numerical method by forcing very small time steps in their vicinity. They also locally affect the solution accuracy. We therefore introduce a very simple breaking point detection algorithm in order to further improve the efficiency our DG method.