Solutions Of Ordinary Differential Equations Research Articles

Active learning modules for a numerical methods course

Education research has suggested that students who receive a wider variety of sensory inputs during the learning process will be able to make more connections and more deeply understand course content. To this end, three physical hands-on laboratory learning modules were designed for a computational numerical methods course to help students develop intuition for physical systems, to allow for a comparison of real data against numerical solutions of physics equations, and to increase student engagement. These modules were designed to reinforce specific topics covered in the numerical methods curriculum, including numerical differentiation and numerical solution of ordinary differential equations. Pre- and post-tests showed an increase in student knowledge after module completion. In addition, end-of-term surveys showed that a strong majority of students believed modules helped them visualize and understand key numerical methods concepts, and that most would be interested in learning new material with similar modules in the future.

Model discovery of compartmental models with Graph-Supported Neural Networks

In this proposal, our objective is to create a neural network for the discovery of models using compartmental models as systems of Ordinary Differential Equations (ODEs), employing Graph-Supported Neural Networks (GSNN). We design the GSNN as a graph of transition functions of the structure of the model to ensure that the three properties of the ODE solution are satisfied: additivity to one, positivity, and boundness. Rather than directly estimating the solution, these neural networks approximate the transition functions. We present theoretical evidence substantiating that our GSNN maintains these properties, along with approximation outcomes from both simulated and real-world data. These outcomes demonstrate that our GSNN outperforms the non-graph-supported approach for these issues, with instances where it even surpasses full model-based solutions and Recurrent Neural Networks. Furthermore, an evolutionary algorithm successfully generated a consistent model using the data, offering a comprehensive framework for neural network-based model discovery in these scenarios.

Remarks on the Solution of Fractional Ordinary Differential Equations Using Laplace Transform Method

In this work we used the Laplace transform method to solve linear fractional-order differential equation, fractional ordinary differential equations with constant and variable coefficients. The solutions were expressed in terms of Mittag-Leffler functions, and then written in a compact simplified form. As a special case for simplicity, the order of the derivative determined the order of the solution that was obtained. This paper presented several case studies involving the implementation of Fractional Order calculus-based models, whose results demonstrate the importance of Fractional Order Calculus.

Exact soliton solutions and soliton diffusion of two kinds of stochastic KdV equations with variable coefficients

In this paper, we consider the exact solutions and soliton diffusion phenomenon of two kinds of stochastic KdV equations with variable coefficients. Firstly, according to symmetric reduction, stochastic KdV equations with variable coefficients are transformed into a coupled system of a deterministic KdV-type equation with variable coefficients and a solvable stochastic ordinary differential equation. Then, by the generalized wave transformation and the Clarkson-Kruskal direct method, we obtain the exact solutions of the deterministic KdV-type equation with variable coefficients. By coupling with the exact solutions of the stochastic ordinary differential equation, the exact solutions of stochastic KdV equations with variable coefficients are obtained. Compared to Wick-type stochastic KdV equations, our research work does not require additional inverse transformations, but solves stochastic partial differential equations more concisely, systematically, and directly. Secondly, two examples are given to verify the correctness of the theoretical analysis, and the soliton diffusion phenomenon of the system is discussed. Finally, by the Zabusky-Kruskal finite difference scheme, numerical simulations are provided to demonstrate the effectiveness of the analytic methods. The results indicate that the soliton diffusion phenomenon is subject to noise influence. In particular, the wave speed accelerates the soliton diffusion over time in the multiplicative noise background, and the wave speed slows down the soliton diffusion over time in the additive noise background.

Analytical solutions of the generalized Kaup–Newell equation

The generalized Kaup–Newell equation for describing the pulse propagation in a nonlinear optical medium is considered. The traveling wave reduction of this equation is taken into account. The method of direct transformations is used to construct a general solution of a nonlinear ordinary differential equation. Analytical solutions are found, expressed in terms of elliptic integrals in canonical form. The implicit function method is applied to find some degenerate solutions under constraints on the model parameters. Among the solutions obtained there are doubly periodic solutions on the complex plane, as well as optical solitons.

A note on the asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations

A note on the asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations

Optimal interatomic potentials using modified method of least squares: Optimal form of interatomic potentials

Abstract The problem of optimization of interatomic potentials is formulated and solved by means of generalization of the Morse, Kaxiras–Pandey, and Rydberg potentials. The interatomic potentials are treated as solutions of some second-order ordinary differential equations which will be classified and analyzed. The most appropriate analytic form of the understudied potentials will be proposed based on a one-dimensional search for the parameter, γ \gamma , which is the power of the interatomic distance, r r . The optimal analytic form will also be proposed for metals such as gold, copper, aluminium, titanium, and the silver–copper alloy. The method of least squares will be used to estimate the potential parameters. Phenomenological potentials such as the classical Rydberg, classical Morse, generalized Morse, Kaxiras–Pandey, and classical Lennard–Jones will be studied, and new potentials based on the combination of some of the aforementioned potentials will also be proposed. Metrics such as the goal function values, will be used to identify which optimal value of the parameter, γ \gamma , is most appropriate to introduce into the preferred interatomic potential for interaction between atoms.

THE USE OF MECHANICAL FILTER MODELS IN THE ANALYSIS OF FORMING AND COMPACTION PROCESSES OF FORMATION AND COMPACTION OF BUILDING/CONCRETE MIXTURES BY VIBRATING FIELD

The paper describes the use of various types of mechanical filter models, which are used for the analysis of the processes of formation and compaction of the construction/concrete mixtures of building/concrete mixtures by means of vibrating fields. The values of resonant frequencies and equivalent masses for different resonators modeling the propagation in the latter of vibrating-wave formations have been established. The analysis of the influence of a vibrating field on the processes of formation and compaction of concrete/concrete mixtures in this study is based on the methods of mathematical physics, classical variation calculus, physics of oscillations and waves and methodology of solution of ordinary differential equations and partial differential equations. The conditions and main integral characteristics of resonance phenomena, the possibility of occurrence of which is conditioned by: 1) the geometry of the initial boundary-edge problem (it is The so-called "geometric resonances" of the considered system with distributed parameters simulating the mixture to be processed); 2) the working rheological model of the mixture involved in the study (these are the so-called "rheological resonances"). The approach developed and scientifically substantiated in this work allows us to establish the main parameters and opportunities for the use of energy-saving modes of operation of vibration systems intended for the formation and vibration compaction of the above mixtures. The results obtained in the work The results obtained can be further used to clarify and Improvement of existing engineering methods of calculation of vibration systems for the formation and compaction of concrete/concrete mixtures in order to optimize the operating modes of their functioning both at the design stage and in the modes of real operation.

Numerical Solutions of High-Order Differential Equations with Polynomial Coefficients Using a Bernstein Polynomial Basis

The paper presents a novel method that allows one to establish numerical solutions of linear and nonlinear ordinary differential equations—with polynomial coefficients—that contain any finite products of the unknown functions and/or their general derivatives. The presented algorithm provides numerical solutions of these differential equations subject to initial or boundary conditions. This algorithm proposes the desired solution in terms of B-polynomials (Bernstein polynomial basis) and then uses the orthonormal relation of B-polynomials with its weighted dual basis with respect to the Jacobi weight function to construct a linear/nonlinear system in the unknown expansion coefficients which can be solved using a suitable solver. The properties of B-polynomials provide greater flexibility in which to impose the initial or boundary conditions at the end points of the interval [0, R] and enable us to obtain exactly and explicitly some of the unknown expansion coefficients in the form of a suggested numerical solution. Consequently, the presented algorithm leads to a linear or nonlinear algebraic system in the unknown expansion coefficients that has a simpler form than that was obtained by the other algorithms. So that, this procedure is a powerful tool that we may utilize to overcome the difficulties associated with boundary and initial value problems with less computational effort than the other techniques. An accepted agreement is obtained between the exact and approximate solutions for the given examples. The error analysis was also studied, and the obtained numerical results clarified the validity of the theoretical results.

TrapezoidalNet: A new network architecture inspired from the numerical solution of ordinary differential equations

The architecture of deep neural networks is commonly determined via trial and error, resulting in inefficiency and a lack of architecture interpretability. Recent research shows that numerical solutions of ordinary differential equations have demonstrated great potential for designing network architectures. Implicit methods are generally more stable and accurate than explicit methods but are yet to be applied to neural network design. Unlike explicit methods that directly calculate the current state based on the previous state, implicit methods have to solve a nonlinear equation to get the approximate solution, which discourages using implicit methods in network design. In this paper, we propose replacing Euler's method with the trapezoidal rule, a scheme that is a second-order implicit method with much lower local truncation errors. The crux of our approach lies in first altering the trapezoidal rule from its implicit form to the explicit form and then designing the corresponding neural network. By introducing the multi-channel convolution and layer-sharing mechanism, we propose a new architecture called TrapezoidalNet. We expect TrapezoidalNet can outperform other deep neural networks based on explicit methods such as ResNet, FitResNet, FractalNet, and LM-ResNet on image recognition tasks. Experimental results on CIFAR10/100 datasets verify the effectiveness of TrapezoidalNet.

Development of high-order adaptive multi-step Runge–Kutta–Nyström method for solving special second-order ODEs

Runge–Kutta–Nyström (RKN) methods are extensively used to obtain approximate solutions of ordinary differential equations (ODEs). Specifically, they are widely used to directly solve second-order ODEs of the special form. Although the derivation of new higher-order methods with fewer numbers of function evaluations is of great importance in increasing the precision and effectiveness of the methods, however, this is rarely done due to the difficulty or complexity of some derivations. This study focuses on constructing a 7(5) pair of embedded multi-step Runge–Kutta–Nyström (EMSN) method with lower stages for the numerical solutions of special second-order ODEs. An adaptive step size formulation using an embedded procedure is considered, and the numerical findings reveal that the new embedded pair outperforms existing Runge–Kutta (RK) pairs in terms of the minimum number of functions evaluations.

On Implementation of Numerical Methods for Solving Ordinary Differential Equations in Computer Algebra Systems

This paper presents an original package for investigating numerical solutions of ordinary differential equations, which is built in the Sage computer algebra system. This project is focused on a closer integration of numerical and symbolic methods while primarily aiming to create a convenient tool for working with numerical solutions in Sage. The package defines two new classes: initial problems and approximate solutions. The first class defines tools for symbolic computations related to initial problems, while the second class defines tools for interpolating values of symbolic expressions on an approximate solution and estimating the error with the use of the Richardson method. An implementation of the Runge–Kutta method is briefly described, with its main feature being the possibility of working with arbitrary Butcher tableaux and arbitrary numeric fields.

3D MHD Radiation Flow of Unsteady Casson Fluid with Viscous Dissipation Effect

A numerical evaluation of the crucial physical properties of a 3D unsteady MHD flow along a stretching sheet for a Casson fluid in the presence of radiation and viscous dissipation has been carried out. Meanwhile, by applying similarity transformations, the nonlinear partial differential equations (PDEs) are transformed into a system of ordinary differential equations (ODEs). Furthermore, in the numerical solution of nonlinear ODEs, the shooting method along with Adams Moulton method of order four has been used. The obtained numerical results are computed with the help of FORTRAN. The tables and graphs describe the numerical results for different physical parameters which affect the velocity and temperature profiles.

Exploiting the Abstract Calculus Pattern for the Integration of Ordinary Differential Equations for Dynamics Systems: An Object-Oriented Programming Approach in Modern Fortran

This manuscript relates to the exploiting of the abstract calculus pattern (ACP) for the (numerical) solution of ordinary differential equation (ODEs) systems, which are ubiquitous mathematical formulations of many physical (dynamical) phenomena. We present FOODIE, a software suite aimed to numerically solve ODE problems by means of a clear, concise, and efficient abstract interface. The results presented prove manifold findings, in particular that our ACP approach enables ease of code development, clearness and robustness, maximization of code re-usability, and conciseness comparable with computer algebra system (CAS) programming (interpreted) but with the computational performance of compiled programming. The proposed programming model is also proven to be agnostic with respect to the parallel paradigm of the computational architecture: the results show that FOODIE applications have good speedup with both shared (OpenMP) and distributed (MPI, CAF) memory architectures. The present paper is the first announcement of the FOODIE project: the current implementation is extensively discussed, and its capabilities are proved by means of tests and examples.

On the solution of fractional differential equations using Atangana’s beta derivative and its applications in chaotic systems

In this research, we examine the solution of ordinary fractional differential equations using Atangana’s beta derivative. Our approach is divided into two parts. First, we establish conditions under which the fractional differential equation has a unique solution. Next, we develop a numerical technique based on the Adams–Bashforth method and demonstrate its convergence. We then apply the numerical method to a test example to evaluate its efficacy. Finally, we analyze three nonlinear chaotic and hyperchaotic fractional dynamical systems, calculating the strange attractors for various fractional orders. We analyzed the fractional chaotic systems and showed that by changing the order of the fractional derivative, some properties of the system change, such as the area of the chaos attracting region.

Computational Examination of Heat and Mass Transfer Induced by Ternary Nanofluid Flow across Convergent/Divergent Channels with Pollutant Concentration

Studying waste discharge concentration across a convergent/divergent channel is essential in environmental-related applications. Successful environmental administration must understand the behavior and concentration of waste contaminants released into these channels. Analyzing waste discharge concentrations aids in determining the efficacy of treatment techniques and regulatory controls in lowering pollutant scales. Because of this, the current analysis examines the ternary-based nanofluid flow across convergent/divergent channels, including non-uniform heat source/sink and concentration pollutants. The study also concentrates on understanding the movement and heat transmission characteristics in ternary-based nano-liquid systems with divergent and convergent channels and maximizing the ternary nanofluid flow’s effectiveness. The equations representing the flow, temperature, and concentrations are transformed into a system of ODEs (ordinary differential equations) and are obtained by proper similarity variables. Further, solutions of ODEs are gathered by using the Runge Kutta Fehlberg 4-5 (RKF-45) method and shooting procedure. The significant dimensionless constraints and their impacts are discussed using plots. The results mainly focus on improving local and external pollutant source variation will enhance the concentration for the divergent channel while declining for the convergent channel. Adding a solid fraction of nanoparticles will escalate the surface drag force. These findings may enhance heat management, lessen pollutant dispersion, and enhance the circulation of nanofluid systems.

Exact Solutions of Nonlinear Second-Order Autonomous Ordinary Differential Equations: Application to Mechanical Systems

Many physical processes can be described via nonlinear second-order ordinary differential equations and so, exact solutions to these equations are of interest as, aside from their accuracy, they may reveal beforehand key properties of the system’s response. This work presents a method for computing exact solutions of second-order nonlinear autonomous undamped ordinary differential equations. The solutions are divided into nine cases, each depending on the initial conditions and the system’s first integral. The exact solutions are constructed via a suitable parametrization of the unknown function into a class of functions capable of representing its behavior. The solution is shown to exist and be well-defined in all cases for a general nonlinear form of the differential equation. Practical properties of the solution, such as its period, time to reach an extreme value or long-term behavior, are obtained without the need of computing the solution in advance. Illustrative examples considering different types of nonlinearity present in classical physical systems are used to further validate the obtained exact solutions.

Radiative MHD Casson nanofluid flow through a porous medium with heat generation and slip conditions

This paper presents an investigation of magnetohydrodynamics (MHD) Casson nanofluid flow along a stretchable surface through a permeable medium. The modeling of the physical phenomena is considered with impact of thermal radiation, heat generation, slip conditions and suction. Transformations of the governing set of mathematical equations for the physical model are carried out into nonlinear ordinary differential equations (ODEs) with appropriate similarity variables. The nonlinear ODE solutions are carried out using the optimal homotopy analysis technique (OHAM), and the findings are presented for determining the influences of the emerging important parameters. The results indicate that velocity field increases in respect of porosity parameter, Casson fluid parameter and magnetic parameter while it declines for enhancing velocity slip and suction parameters. The temperature profile shows rising behavior for heat source, Prandtl number, thermophoresis, radiation and Brownian motion parameters while it declines for enhancing thermal slip parameter. Moreover, the concentration profile enhances for rise in Brownian motion parameter while it reduces for Schmidt number and nanoparticle parameter. We also showed the accuracy of the present results by indicating that skin friction values for varied magnetic parameters agree with earlier findings in the literature.

A simple alternative in approximation and asymptotic expansion by exponential/trigonometric functions

Function approximation plays a crucial role in applied mathematics and mathematical physics, involving tasks such as interpolation, extrapolation, and studying asymptotic properties. Over the past two centuries, several approximation methods have been developed, but no universal solution has emerged. Each method has its own strengths and weaknesses. The most commonly used approach, rational Padé approximants, has limitations, performing well only for arguments x < 1 and often containing spurious poles. This report introduces a new and straightforward procedure for exponential/trigonometric approximation that addresses these limitations. The method demonstrates accurate fitting capabilities for various functions and solutions of second-order ordinary differential equations, whether linear or nonlinear. Moreover, it surpasses the performance of Padé approximants. Notably, the proposed algorithm is remarkably simple, requiring only four values of approximating functions. The provided examples show case the potential of this method to offer a straightforward and reliable approach for a wide range of tasks in applied mathematics and mathematical physics.

Plant Community Changes Based on The Lotka-Volterra Model

Globally, ecosystem functions are threatened by repeated droughts and declining plant diversity. Due to frequent droughts, the species in the plant community show different adaptions towards the extreme weather conditions. The aim of this report is to explore how the plant community changes over time. Considering the growth law of population, establish the model of precipitation and plant growth rate based on the Logistic function. Moreover, considering the competition and interaction between populations, this paper chooses the Lotka-Volterra model and adapt it, then use the improved Euler method to obtain numerical solutions of these nonlinear ordinary differential equations of plant population over time under the interaction of multiple populations. The observations are obtained: For the population of a plant community, when the rainy season intersects with droughts, the decline rate of the population at the inflection point slows down, the continuity of population change is interrupted. And the plant population shows an increasing trend in the rainy season, and a decreasing trend in droughts.