Solutions Of Ordinary Differential Equations Research Articles

Analysis of Roll Decay for Surface-Ship Model Experiments With Uncertainty Estimates

Abstract Roll decay of David Taylor model basin (DTMB) Model 5720, a 23rd-scale free-running model of the research vessel (R/V) Melville, is evaluated with uncertainty estimates. Experimental roll-decay time series was accurately modeled as an exponentially decaying cosine function, which is the solution of a second-order ordinary differential equation for damping coefficient of less than one (N < 1). The curve-fit provides damping coefficient (N), period (T), and offset. Roll period in calm water was dependent on Froude number (Fr) and initial roll angle (a). Roll decay data are from 76 runs for three nominal Froude numbers, Fr = 0, 0.15, and 0.22. The initial roll angle variation was 3 deg–25 deg. The natural roll period was 2.139 ± 0.041 s (±1.9%). The decay coefficient data were approximated by a plane in three dimensions with Fr and initial roll amplitudes (a) as the independent variables. Curve-fit results are compared to decay coefficient by log decrement and period from time between zero crossings. Examples demonstrate average values for a single roll decay event from log decrement are the same as values by the curve-fitting method within uncertainty estimates. The uncertainty estimate for the decay coefficient is significantly less by curve-fit method in comparison to log-decrement method. By log decrement, the relative uncertainty increases with decreasing roll amplitude peak; consequently, focus should be on the damping coefficient at the largest peaks, where the uncertainty is the smallest.

Common Fixed Point Theorems for Novel Admissible Contraction with Applications in Fractional and Ordinary Differential Equations

In our work, we offer a novel idea of contractions, namely an (α,β,γ)P−contraction, to prove the existence of a coincidence point and a common fixed point in complete metric spaces. This leads us to an extension of previous results in the literature. Furthermore, we applied our acquired results to prove the existence of a solution for ordinary and fractional differential equations with integral-type boundary conditions.

Quadratized Taylor series methods for ODE numerical integration

We focus on Taylor Series Methods (TSM) and Automatic Differentiation (AD) for the numerical solution of Ordinary Differential Equations (ODE) characterized by a vector field given by a finite composition of elementary and standard functions. We show that computational advantages are achieved if a kind of pre-processing said Exact Quadratization (EQ) is applied to the ODE before applying the TSM and the AD. In particular, when the ODE function is given by a formal polynomial (i.e. with real powers) of n variables and m monomials, the computational complexity required by our EQ based method for the calculation of the k-th order Taylor coefficient is O(k) whereas by using the existing AD methods it amounts to O(k2).

Reliability Assessment of Safety-Critical Systems of Nuclear Power Plant using Ordinary Differential Equations and Reachability Graph

This study proposes a novel and integrated approach incorporating reachability graph (RG) and ordinary differential equations (ODE) to assess the reliability of safety–critical systems (SCS). The suggested framework is divided into nine steps. The system is first modeled using the Petri net (PN) to generate the ODE and RG. The ODE solution can be used to compute the marking probabilities of PN model. The marking probability values can be utilized to measure the state probabilities of RG. The resulting RG contains various states of the system. The failure states of RG are identified to assess the system's reliability. The suggested approach is implemented on the Shutdown System of Nuclear Power Plant, and is validated using Brown & Lipow model. The obtained accuracy of 99.53905% in the evaluation of reliability validates the technique. The state-space explosion is one of the major problems associated with conventional stochastic approaches like Markov chains, which use assumed transition probabilities to quantify reliability between different states. The proposed strategy is capable of avoiding the problem of state-space explosion.

Finding Solutions to Undamped and Damped Simple Harmonic Oscillations via Kashuri Fundo Transform

Differential equations are expressions that are frequently encountered in mathematical modeling of laws or problems in many different fields of science. It can find its place in many fields such as applied mathematics, physics, chemistry, finance, economics, engineering, etc. They make them more understandable and easier to interpret, by modeling laws or problems mathematically. Therefore, solutions of differential equations are very important. Many methods have been developed that can be used to reach solutions of differential equations. One of these methods is integral transforms. Studies have shown that the use of integral transforms in the solutions of differential equations is a very effective method to reach solutions. In this study, we are looking for a solution to damped and undamped simple harmonic oscillations modeled by linear ordinary differential equations by using Kashuri Fundo transform, which is one of the integral transforms. From the solutions, it can be concluded that the Kashuri Fundo transform is an effective method for reaching the solutions of ordinary differential equations.

Asymptotic forms of solutions of half-linear ordinary differential equations with integrable perturbations

Asymptotic forms of solutions of half-linear ordinary differential equations with integrable perturbations

A novel numerical scheme for fractional differential equations using extreme learning machine

In this paper, we propose a neural network-based approach with an Extreme Learning Machine (ELM) for solving fractional differential equations. The solution procedure for the linear and nonlinear fractional differential equations has been derived. Also the convergence and stability of the proposed method is provided. Then we examine the numerical solution of several fractional-order ordinary and partial differential equations. As a last example the Burgers equation without an explicit exact solution. The effect of changing the number of neurons on the accuracy of the solution is obtained graphically.

Dhage iteration method for an algorithmic approach to local solution of the nonlinear second order ordinary hybrid differential equations

Dhage iteration method for an algorithmic approach to local solution of the nonlinear second order ordinary hybrid differential equations

Resolviendo ecuaciones diferenciales ordinarias con Symbolic Math Toolbox™ (Matlab) y SymPy (Python)

This paper shows solutions of ordinary differential equations (EDOS) obtained by using two symbolic packages: Symbolic Math Toolbox™ (Matlab) and SymPy (Python). The basic instructions to obtain solutions of both packages are explained step by step, through a group of examples from a traditional ordinary differential equations course. Differential equations that are solved with methods such as: separable variables, linear equations, indeterminate coefficients, variation of parameters, power series, Laplace transform, and numerical solutions are included. By means of the symbolic computation carried out with these packages it is possible to obtain the solution of linear systems, as well as the visualization of the direction field of a differential equation or of a non-linear system of differential equations. The main contribution of this work consists in providing the reader with a practical guide that allows him to start the study of differential equations assisted by Symbolic Math Toolbox™ or SymPy. Among the benefits of using these computational tools in teaching and/or learning practices, it is shown how the use of symbolic or numerical computation saves us effort in the computation of tedious calculations; focusing attention on important ideas and concepts such as: the relationship between the mathematical model and its physical counterpart, asymptotic behavior and qualitative analysis of the solutions.

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.