Order Ordinary Differential Equations Research Articles

Development of a fully implicit ODE-solver for containment analysis code

The thermal–hydraulic dynamics in containment are governed by a system of stiff ordinary differential equations (ODEs). A fully implicit discretization scheme is adopted to discretize these ODEs in order to mitigate the effects of stiffness. In comparison with explicit or semi-implicit discretization schemes that are subject to Courant limits on time steps, the fully implicit discretization scheme is more suitable for a containment analysis code that focuses on predicting both short-term and long-term thermal–hydraulic parameters after an accident. This study introduces a general-purpose ODE solver for the containment analysis code. The outline of the solver is as follows: The fully implicit discrete equations lead to a large set of nonlinear equations that need to be solved using Newton’s iterative method. The partial derivative components in the Jacobi matrix are calculated by the perturbation method using finite difference approximation, which avoids the complicated derivation of partial derivatives. The scaling modification technique is incorporated into this ODE solver to deal with significant differences in unknown variable magnitudes, and the line search method is introduced to address the difficulty of obtaining an accurate root estimate with Newton’s method when the initial guess is far from the actual root. This proposed ODE solver was applied to two typical stiff ODE problems to test its stiffness-suppressed ability and to demonstrate that this proposed solver can perform calculations with a very large time step. Then, the CASSIA code, a containment analysis code developed by China Nuclear Power Technology Research Institute Co., Ltd (CNPRI), equipped with this ODE solver, was applied to the CSNI (Committee on the Safety of Nuclear Installations) benchmark problem and the Carolinas Virginia Tube Reactor (CVTR) test 3 problem to preliminarily demonstrate that the proposed ODE solver can perform containment thermal–hydraulic analysis correctly. This study could provide references for the development of a home-made containment analysis code.

Review of Adams-Bashforth method for numerical solution of first order ordinary differential equations

This project work presents a comprehensive review of the Adams-Bashforth method for the numerical solution of explicit first-order ordinary differential equations (ODEs). The study begins with a historical overview of the development of ordinary differential equations (ODEs), tracing back to the seminal works of Newton, Leibniz, and their contemporaries. The evolution of differential equations as a distinct mathematical discipline and their wide-ranging applications across various fields are discussed. Furthermore, the paper provides an in-depth analysis of the Adams-Bashforth method, a prominent numerical technique for solving ODEs. The method is derived and analyzed using Mathematica Software, demonstrating its efficacy and accuracy in approximating solutions to ODEs. A comparison with existing methods such as the Euler method and the Runge-Kutta method highlights the advantages of the Adams-Bashforth method, particularly in terms of computational efficiency and accuracy. Moreover, fundamental concepts related to ODEs, including the initial value problem, existence, and uniqueness of solutions, are explored in the context of numerical solution methods. The properties of linear multistep methods and their relevance to solving ODEs efficiently are also discussed. In conclusion, this project work provides valuable insights into the Adams-Bashforth method as a powerful tool for the numerical solution of first-order ODEs. By leveraging modern computational tools such as Mathematica Software, the method offers a practical and efficient approach to solving complex differential equations encountered in various scientific and engineering applications.

On Period Annuli and Induced Chaos

Nontrivial period annuli in the second order ordinary differential equation are continua of periodic trajectories that contain inside more than one critical point. They can appear in conservative equations, which are known to have no attractors. Nevertheless, according to some authors, their behavior may be done chaotic by adding a periodic external force. Is the period of the external force correlated with periods of solutions in period annuli? Is the chaotic behavior of a solution dependent on the initial value and, in turn, on a certain periodic annulus? These, and related questions are studied in the article.

Investigating the new perspectives of Caudrey–Dodd–Gibbon equation arising in quantum field theory

The main purpose of the paper is obtaining the analytical results for beta fractional Caudrey–Dodd–Gibbon equation which is used to resolve complex problems in fluid dynamics, chemical kinetics, plasma physics, quantum field theory, crystal dislocations, and nonlinear optics by using auxiliary method. Beta derivative is a useful fractional operator due to satisfying basic properties of integer order derivative and also, allows us using chain rule and wave transform to turn nonlinear fractional partial differential equations into integer order ordinary differential equations. By the way many analytical methods can be applied to these equations. In order to understand the physical features of the solutions, 3D and 2D graphical illustrations are given. Finally, authors expect that the obtained solutions may give a deep insight for the explanation of physical phenomena in the fluid dynamics, chemical kinetics, plasma physics, quantum field theory, crystal dislocations, and nonlinear optics.

A Chebyshev polynomial based block integrator for the direct numerical solution of fourth order ordinary differential equations

This paper introduces an innovative method for numerically integrating fourth-order initial value problems by utilizing Chebyshev polynomials as the fundamental basis function. The block integrator base on Chebyshev polynomial demonstrates significant improvements in accuracy and stability, rendering it a valuable tool across various scientific and engineering fields. By leveraging the characteristics of Chebyshev polynomials, this approach accurately estimates solutions for fourth-order differential equations without reducing it to a system of first order Ordinary Differential Equations while at the same time effectively managing error accumulation within a block integration framework and thereby enhancing its accuracy over extended intervals. Through rigorous numerical experiments, the effectiveness and reliability of the new integrator are demonstrated and compared with existing methods. The new method is consistent, zero stable and convergent. The method also shows an appreciable error constants. The new method performed better in terms of accuracy than the existing methods in the literature in both linear and nonlinear problems.

Explicit Runge-Kutta Method for Evaluating Ordinary Differential Equations of type v^vi = f(u, v, v′)

The initiative of this paper is to present the Runge Kutta Type technique for the development of mathematical solutions to the problems concerning to ordinary differential equation of order six of structure v vi = f(u, v, v′ ) denoted as RKSD with initial conditions. The three and four stage Runge-Kutta methods with order conditions up to order seven (RKSD7) have been designed to evaluate global and local truncated errors for the ordinary differential equation of order six. The framework and evaluation of equations with their results are well established to obtain the effectiveness of RK method towards implicit function satisfying the required initial conditions and for obtaining zero-stability of RKSD7 in terms of their accuracy with maximum precision under minimal processing.

Model order reduction, a novel method using krylov sub-spaces and genetic algorithm

Model Order Reduction (MOR) of complex and large systems in Electrical engineering, continuous to be an attractive field for Engineers and Scientists over the last few decades, this complexity of models makes the control designs and simulation using Computer Aided Design (CAD) more and more difficult and consuming a lot of time. There for, accurate, robust and fast algorithms for simulation are needed. The goal of MOR is to replace the original system by an appropriate reduced system which preserves the main properties of the original one such that stability and passivity. Several analytical MOR techniques have been proposed in the literature over the past few decades, to approximate high order linear dynamic systems like Krylov sub-space techniques and SVD (Singular Value Decomposition) techniques. However, most of these techniques lead to computationally demanding, time consuming, iterative procedures that usually result in non-robustly stable models with poor frequency response resemblance to the original high order model in some frequency ranges. Recently a set of new techniques based on Artificial Intelligence (AI) were proposed in [1] for MOR. This article considers the problem of model order reduction of Linear Time In varying (LTI) systems. It is described by first and second order ordinary differential equations model. A tow steps method for model order reduction of LTI systems is proposed here, which combined features of an analytic technique (Krylov approach) and an AI technique (Genetic Algorithm). In the first step, the size of the original model is reduced to an intermediate order, using an analytical technique based on Krylov sub-spaces. In the final step of the reduction process, an AI approach based on Genetic Algorithm (GA) is applied to obtain an optimized nominal model.

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

The analysis of stability problems of beams on two-parameter foundations (Bo2PFs) is an important part of their design for compressive loads. This work presents novel first principles derivation of the governing differential equations of elastic stability (GDES) of thin beams resting on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models. The requirements of translational and rotational equilibrium of all the applied, reactive and internal forces on an infinitesimal segment of the Bo2PF and the laws of infinitesimal calculus were used to formulate the GDES as a fourth order ordinary differential equation (ODE) in terms of the transverse displacement function ux. The GDES is non-homogeneous in the presence of applied transverse load qx but homogeneous when qx vanishes. This study presents the Fourier series method (FSM) for solving the governing differential equation of stability (GDES) for the case of Dirichlet boundary conditions. The FSM has the advantage of amenability to differentiation, and integration due to the orthogonality properties of the sinusoidal functions. Implementation of the FSM by assuming the unknown function in the GDES as a Fourier series of infinite terms and the exploitation of orthogonalization simplifies the problem to an algebraic eigenvalue problem which is the characteristic buckling equation. The exact eigenvalues are found by algebraic solution of the buckling equation. The exact eigenvalues were used to find the exact buckling loads and the exact buckling load coefficients. The critical buckling load was found to correspond to the first buckling mode (n= 1), and is identical with previous solutions in the literature. Numerical calculations for the critical buckling load parameters Kcr were presented for the Bo2PF problem for values of the dimensionless foundation parameters k-1= 0, k-2= 0; k-1= 100, k-2= 0; k-1= 0, k-2= 1; k-1= 100, k-2= 100; k-1= 0, k-2= 2.5; k-1= 100, k-2= 2.5. The present solutions were compared with previous solutions for Kcr in the literature. The comparison shows that the present FSM results are identical with previous results obtained using various other methods such as Recursive Differentiation Method, Finite Element Method, Generalized Integral Transform Method (GITM) and Stodola-Vianello Iteration Method. The study has illustrated the effectiveness of the FSM for solving Bo2PFs.

Penyelesaian khusus persamaan diferensial biasa ordo dua linier tak homogen dengan koefisien konstan untuk fungsi bagian demi bagian

Spring mechanical vibration motion system with a damped degree of freedom and influenced by external forces, is mathematically expressed as an ordinary differential equation of order of two linear constant coefficients that are not homogeneous. If an external force acts on a stationary system expressed as a continuous function f(t) for any time t, then the system will experience mechanical vibrational motion which mathematically the equation of motion can be expressed as a superposition. The equation consists of  as a solution to a homogeneous form with mechanical vibrations  as a solution to a particular form. In terms of the particular solution  this article will show a mathematical way when f(t) is a continuous function section by part which is defined at an interval, such that the mechanical vibration motion equation is at the same time a special solution of the equation the mechanical vibration system.
 Keywords: Vibration, Impulse Functions, ConvolutionMSC2020: 34A37

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

C PROGRAM AS A TOOL FOR THE TEACHING OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATION

Second order ordinary differential equation (ODE) has many applications in science and engineering. Undergraduate students in science and engineering departments must study this subject in order to understand other subjects which are related to real applications they will encounter later. There are many excellent textbooks on differential equations where the students can study the theory and solve the problems. However, a textbook cannot give a quick answer for a problem particularly when the problem is quite difficult. A good choice is to use software such as Maple or Mathematica. However, this software is not always available for the teaching and purchasing it is usually beyond the ability of a student or even a lecturer. On the other hand, lecturers who want to create problems by themselves will follow the theory of the ODE. While creating the problems may not be difficult, answering them are harder. Problems which are very easy to answer are less worthless because they will not increase the students’ knowledge. Here comes the solution. A C program has been created to help lecturers create problems and solve them quickly. The program is interactive and can be easily understood by anyone who has basic theory of ODE. No knowledge of programming is needed; a user just runs it and follows the instruction. Students can also use the program to sharpen their knowledge. They can compare the solution of a problem they have solved with the answer given by the program. While commercial software such as Maple and Mathematica is very powerful, they cannot be used without writing necessary commands to solve a problem.

Boundedness of Vector Linéard Equation with Multiple Variable Delays

In this article, we consider a system of ordinary differential equations (ODEs) of second order with two variable time delays. We obtain new conditions for uniform ultimate bounded (UUB) solutions of the considered system. The technique of the proof is based on the Lyapunov–Krasovskii functional (LKF) method using a new LKF. The main result of this article extends and improves a recent result for ODEs of second order with a constant delay to a more general system of ODEs of second order with two variable time delays. In this particular case, we also give a numerical example to verify the application of the main result of this article.

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. 

Initial Value Solvers for Direct Solution of Fourth Order Ordinary Differential Equations in a Block from Using Chebyshev Polynomial as Basis Function

The numerical computation of fourth order ordinary differential equations cannot be gloss over easily due to its significant and importance. There have been glowing needs to find an appropriate numerical method that will handle effectively fourth order ordinary differential equations without resolving such an equation to a system of first order ordinary differential equations. To this end, this presentation focuses on direct numerical computation to fourth order ordinary differential equations without resolving such equations to a system of first order ordinary differential equations. The method is not predictor – corrector one due to its limitation in the level of accuracy. The method is order wise christened “Block Method” which is a self-starting method. In order to achieve this objective, Chebyshev polynomial is hereby used as basis function.

Modeling the effect of shape deformation induced by gravity on the evaporation of pendant and sessile drops

Pendant and sessile drops form a spherical cap only in the absence of gravity. The effect of gravity on drop shape is often neglected on the basis of the assumption that the drop size is smaller than the capillary length [Lc=(σ/gρ)1/2], although the deformation may not be fully negligible even in those cases. This paper focuses on evaluation of the effect that deformation due to gravity has on the evaporation characteristics of pendant and sessile drops. The drop shape is described by the Bashforth–Adams equation, a non-linear second order ordinary differential equation, which is solved numerically using a Runge–Kutta method with variable time steps. Under quasi-steady approximation, the species and energy conservation equations in the gas phase have analytical solutions, even for temperature-dependent gas thermophysical properties, once the solution of a basic Laplace problem is known. The Laplace equation is solved in axial symmetric geometry by using COMSOL Multiphysics®, for a wide range of drop sizes and contact angles, yielding vapor distribution, vapor fluxes, and evaporation rates. Comparison with the results from drops of same size in microgravity (i.e., having a spherical cap shape) shows that the effect is also perceptible for drops with a size smaller than the capillary length and that it can become quite important for those with larger sizes. Complementary results are found for sessile and pendant drops with respect to wall wettability, suggesting that the phenomenon can be analyzed using a unitary approach.

On polynomial solutions of certain finite order ordinary differential equations

Given a finite order differential operator, some properties and relations satisfied by its polynomial eigenfunctions are studied. Under certain restrictions, such eigenfunctions are explicitly obtained, as well as the corresponding eigenvalues. Also, some linear transformations are applied to sequences of eigenfunctions and a necessary condition for this to be a sequence of eigenfunctions of a new differential operator is obtained. These results are applied to the particular case of classical Hermite polynomials.

On the Dynamic Characteristics of Orthotropic Rectangular Plates under the Influence of Moving Distributed Masses and Resting on a Variable Elastic Pasternak Foundation

This work studies the dynamic characteristics of orthotropic rectangular plates under the influence of moving distributed masses and resting on a variable elastic Pasternak foundation. The governing equation is a fourth order partial differential equation with variable and singular coefficients. The solutions to the problem are obtained by transforming the fourth order partial differential equation for the problem to a set of coupled second order ordinary differential equations using the technique of Shadnam et al, then simplified using asymptotic method of Struble [11]. The closed form solution is analyzed, resonance conditions are obtained and the results are depicted graphically for both cases of moving distributed mass and moving distributed force

An efficient way to determine Liouvillian first integrals of rational second order ordinary differential equations

The Lie and Darboux methods to obtain Liouvillian first integrals of rational second order ordinary differential equations (rational 2ODEs) are very powerful. Nevertheless, there are cases where these procedures may encounter difficulties. Namely, when the Darboux polynomials present in the integrating factor of the 2ODE have a very high degree and/or when the 2ODE does not admit point symmetries. In [3,4] we developed a method (S-function method) that is successful in treating certain classes of rational second order ordinary differential equations that are particularly ‘resistant’ to canonical Lie methods and to Darbouxian approaches. However, although determining the S-function is in general much more efficient than determining Darboux polynomials or nonlocal symmetries, for very complicated rational 2ODEs, even finding the S-function itself can be quite hard. In this work we present a simple way of (in almost all cases) computing the S-function with a very efficient procedure. New version program summaryProgram Title: InSyDE – Invariants and Symmetries of (rational second order ordinary) Differential Equations.CPC Library link to program files:https://doi.org/10.17632/4ytft6zgk7.2Licensing provisions: CC by NC 3.0Programming language: Maple 17Journal reference of previous version: Comput. Phys. Commun. 234 (2019) 302–314Does the new version supersede the previous version?: Yes.Nature of problem: Determining Liouvillian first integrals of rational second order ordinary differential equations.Solution method: The method is explained in the body of the paper.Reasons for new version: The InSyDE package depends, in order to work, on determining the S-function. The problem is that, for very complicated 2ODEs, this can be very costly (computationally) or even unrealizable (within time and memory limits). We have developed a computationally cheaper way of determining the S-function that, in addition to being very efficient, is also surprisingly broad.Summary of revisions: We have, based on important new theoretical results presented on Appendix A: Supplementary Material, modified and improved the command Sfunction.

Solution of First Order Ordinary Differential Equations Using Fourth Order Runge-Kutta Method with MATLAB.

Differential Equations are used in developing models in the physical sciences, engineering, mathematics, social science, environmental sciences, medical sciences and other numerous fields. This article examined solution of first ordinary differential equation using fourth order Runge-Kutta method with MATLAB. The fourth order Runge-Kutta method for modelling differential equations improves upon the Euler’s method to obtain a greater accuracy without the necessity for higher-order derivatives of the given function. A first order differential equation was solved using fourth order Runge-Kutta method with MATLAB and the same problem was solved analytically in order to obtain the exact solution. The MATLAB commands match up quickly with the steps of the fourth order Runge-Kutta algorithm. Slight variation of the MATLAB code was used to show the effect of the size of h on the accuracy of the solution (see figure 4.1, 4.2, 4.3).The MATLAB and exact solutions are approximately equal though the MATLAB approach is easier and faster. The obtained results are in agreement with those in existing literature and improved the results obtained by [1]

A basic theory for initial value problems of first order ordinary differential equations with Lp-Carathéodory functions and applications

A basic theory on the existence and uniqueness of solutions and local solutions for initial value problems (IVPs) of first order ordinary differential equations is established, where the nonlinearities are Lp-Carathéodory functions. The basic theory contains three parts: (i) the existence and uniqueness of solutions in C([a,b],[c,d]); (ii) the existence and uniqueness of local solutions in C([a,a+h1(p)];[c,d]), where the expression of the number h1(p) will be given and is a computable constant, and (iii) the existence and uniqueness of solutions and nonnegative solutions by establishing a new Bihari inequality. The classical results dealt with the above part (ii) when f is continuous or an L1-Carathéodory function, and only obtained the existence of a number h such that the IVPs had local solutions in C[a,a+h]. Our methodology is to use the results obtained on (i) to study the results mentioned in (ii) when p∈[1,p] while the classical results used the Schauder fixed point theorems directly to obtain the existence and uniqueness of local solutions when f is continuous or p=1. Our results in parts (i) and (iii) are new. The results in part (ii) are new when p∈(1,∞] and improve the classical results when p=1. Our results will enrich the classical basic theory. As applications of the basic theory, the existence of solutions and nonnegative local solutions for logistic type population models with heterogeneous environments is studied, where the local rates of changes in the population density are Lp-Carathéodory functions.