Solutions Of Ordinary Differential Equations Research Articles

Adaptive time step selection for spectral deferred correction

AbstractSpectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.

A Six-Point y-Function Hybrid Block Method for Direct Solution of Third Order Ordinary Differential Equations

In this article, a new continuous numerical method was developed for the numerical integration of general third-order initial value problems (IVPs) of ordinary differential equations (ODEs). The method was derived by adopting interpolation and collocation approach using a combination of power series and exponential functions as the basis function for the approximate solution to the ODEs. The approximate solution was interpolated at both nodal and off-nodal points while the differential systems from the approximate solution was collocated at all nodal points to obtain the set of methods with given step-numbers. This derived approach ensured that the hybrid points are obtained at the y-values of the methods. This class of hybrid linear multi-step method satisfied the conditions for usability and accuracy of any given method. These conditions were confirmed by testing the consistency, stability and convergence of the developed method. The method was applied to solve directly linear and non-linear third order ODEs without reduction to system of first-order ordinary differential equations. Results from the computation compared favorably with some existing numerical methods in literature with comparable properties to confirm the superiority of the newly developed method.

Solution of Third-Order Ordinary Linear Differential Equations in Discriminant Domains Implementing the Variational Iteration Method

Solution of Third-Order Ordinary Linear Differential Equations in Discriminant Domains Implementing the Variational Iteration Method

First Derivative Approximations and Applications

In this paper, we consider constructions of first derivative approximations using the generating function. The weights of the approximations contain the powers of a parameter whose modulus is less than one. The values of the initial weights are determined, and the convergence and order of the approximations are proved. The paper discusses applications of approximations of the first derivative for the numerical solution of ordinary and partial differential equations and proposes an algorithm for fast computation of the numerical solution. Proofs of the convergence and accuracy of the numerical solutions are presented and the performance of the numerical methods considered is compared with the Euler method. The main goal of constructing approximations for integer-order derivatives of this type is their application in deriving high-order approximations for fractional derivatives, whose weights have specific properties. The paper proposes the construction of an approximation for the fractional derivative and its application for numerically solving fractional differential equations. The theoretical results for the accuracy and order of the numerical methods are confirmed by the experimental results presented in the paper.

Enhancing Accuracy and Efficiency in Stiff ODE Integration Using Variable Step Diagonal BBDF Approaches

Recent advancements in mathematical modelling have uncovered a growing number of systems exhibiting stiffness, a phenomenon that challenges the effectiveness of traditional numerical methods. Motivated by the need for more robust numerical techniques to address this issue, this paper presents an enhanced version of the Diagonally Block Backward Differentiation Formula (BBDF) that incorporates intermediate points, known as off-step points, to improve the accuracy and efficiency of solutions for stiff ordinary differential equations (ODEs). The new scheme leverages an adaptive step-size strategy to refine accuracy and efficiency between regular and off-grid integration steps. Theoretical analysis confirms that the proposed scheme is an A-stable and convergent method, as it satisfies the fundamental criteria of consistency, zero-stability, and A-stability. Numerical experiments on single and multivariable systems across varying time scales demonstrate significant improvements in solving stiff ODEs compared to existing techniques. Therefore, the new proposed method is an effective solver for stiff ODEs.

Classification of the non-null electrovacuum solution of Einstein–Maxwell equations with three-parameter abelian group of motions

The classification of the Stackel spaces of the electrovacuum of the type (3.0) has been done. These spaces are invariant under the action of the three-parameter abelian group of motions and belong to the first type Bianchi spaces. In the case of a non-zero cosmological term, the metrics and potentials contain solutions of a nonlinear ordinary differential equation of the second order. When the cosmological term equals zero, the metrics and the components of the electromagnetic field tensor are expressed through elementary functions. Thus the classification of the electrovacuum stackel spaces of all types is completed and complete list of these spaces is constructed.

Uncovering Hidden Patterns: Approximate Resurgent Resummation from Truncated Series

We analyze truncated series generated as divergent formal solutions of non-linear ordinary differential equations. Motivating the study is a specific non-linear, first-order differential equation, which is the basis of the resurgent formulation of renormalized perturbation theory in quantum field theory. We use the Borel–Padé approximant and classical analysis to determine the analytic structure of the solution using the first few terms of its asymptotic series. Afterward, we build an approximant, consistent with the resurgent properties of the equation. The procedure gives an approximate expression for the Borel–Ecalle resummation of the solution useful for practical applications. Connections with other physical applications are also discussed.

Investigation of solution behavior of Differential Equations by Sumudu methods of random complex Partial Differential Equations

In this study, solutions of random complex partial differential equations were found using the two-dimensional Sumudu transformation method(STM). The initial conditions of a deterministic equation or the non-homogeneous part of the equation are transformed into random variables to obtain a random complex partial differential equation. With the help of the properties of two-dimensional Sumudu and inverse Sumudu transformation, an approximate analytical solution of a complex partial differential equation with random constant coefficients was obtained by selecting a random variable with an initial condition of Normal and Gamma distribution. The probability characteristics of the resulting solutions, such as expected value and variance, were obtained and graphically shown with the help of the Maple package program.

Leveraging feed-forward neural networks to enhance the hybrid block derivative methods for system of second-order ordinary differential equations

This study introduces an innovative method combining discrete hybrid block techniques and artificial intelligence to enhance the solution of second-order Ordinary Differential Equations (ODEs). By integrating feed-forward neural networks (FFNN) into the hybrid block derivative method (HBDM), the modified approach shows improved accuracy and efficiency compared to traditional methods. Through comprehensive comparisons with exact and existing solutions, the study demonstrates the effectiveness of the proposed approach. The evaluation, utilizing root mean square error (RMSE), confirms its superior performance, robustness, and applicability in diverse scenarios. This research sets a new standard for solving complex ODE systems, offering promising avenues for future research and practical implementations.

Symmetric structures and dynamic analysis of a (2+1)-dimensional generalized Benny-Luke equation

We study the symmetric structures and dynamic analysis of a (2 + 1)-dimensional generalized Benny-Luke (GBL) equation based on the Lie symmetry method, the GBL equation is an important non-integrable model of water waves. Specifically, we construct multiple exact solutions of the GBL equation and obtain its nonlocally related systems. Firstly, the Lie point symmetries and conservation laws of the GBL equation are computed, and then we get the reduced ordinary differential equation from one of the conservation laws. Multiple methods, for example, the dynamical systems method, the power series method, the homogeneous balancing method and generalized variable separation method, are used to solve the ordinary differential equation and abundant exact solutions of the GBL equation are got. Finally, we extend these exact solutions by discrete symmetries, and give three-dimensional graphs of partial exact solutions. In addition, we construct the nonlocally related PDE systems, which contains the potential systems from the conservation laws and an inverse system from a Lie point symmetry of the GBL equation. These findings reveal the dynamical behavior behind the GBL equation and broaden the range of nonlinear water wave model solutions.

New Two Parameter Integral Transform "MAHA Transform" and its Applications in Real Life

In this paper, an integral transform with two parameters was proposed, and this transform was applied to obtain the exact solutions for the linear ordinary differential equations with constant coefficients of higher orders. This integral transform was also applied and we called it Maha transform in nuclear physics and for medical applications.

Analysis of Solutions for Nonlinear ψ-Caputo Fractional Differential Equations with Fractional Derivative Boundary Conditions in Banach Algebra

This article explores the solutions of nonlinear implicit ψ-Caputo fractional-order ordinary differential equations (NLIFDEs) with two-point fractional derivatives boundary conditions in Banach algebra. The research aims to establish the existence and uniqueness of solutions for this complex class of differential equations. Utilizing Banach’s and Krasnoselskii’s fixed point theorems, the study conducts a rigorous analysis of the solutions, ensuring their existence and uniqueness. This comprehensive investigation contributes to enhancing the understanding of the behavior of solutions of nonlinear fractional differentials within a challenging mathematical framework.

Vibration Characteristics of Magnetostrictive Composite Cantilever Resonator with Nonlocal Effect.

Taking the nonlocal effect into account, the vibration governing differential equation and boundary conditions of a magnetostrictive composite cantilever resonator were established based on the Euler magnetoelastic beam theory. The frequency equation and vibration mode function of the composite cantilever were obtained by means of the separation of variables method and the analytic solution of ordinary differential equations. The lateral deflection, vibration governing equations, and boundary conditions were nondimensionalized. Furthermore, the natural frequency and modal function of the composite beam were quantitatively analyzed with different nonlocal parameters and transverse geometry dimensions using numerical examples. Compared with the results without considering the nonlocal effect, the influence of the nonlocal effect on the vibration characteristics was analyzed. The numerical results show that the frequency shift and frequency band narrowing of the magnetostrictive cantilever resonator are induced by nonlocal effects. In particular, the high-frequency vibration characteristics, such as vibration amplitude and modal node of the composite beam, are significantly affected. These analysis results can provide a reference for the functional design and optimization of magnetostrictive resonators.

A Dynamic Thickening Strategy for High-Fidelity CFD Analyses of Multi-Regime Combustion

Abstract In this study, a dynamic thickening strategy for DTFLES application to multi-regime combustion is proposed. The main idea lies in using the numerical solution of an Ordinary Differential Equation (ODE) as a thickening factor. The equation relates the time derivative of the local thickening factor to its production and destruction rates, which are proportional to the gap between the instantaneous value and optimal target values. The smoothness of the thickening factor in time is ensured by the ODE solution, while in space it is achieved through a mathematical function defined in a continuous flame index space. The equation is numerically integrated with a semi-implicit scheme by making use of the backward Euler formula. The strategy has been implemented in a commercial Computational Fluid Dynamics (CFD) solver and it has been tested by performing Large Eddy Simulations of the hydrogen/air flame produced by the HYLON injector, which has been individuated as an interesting test case for the proposed dynamic strategy. Turbulence-chemistry interactions are recovered by means of a well-assessed subgrid efficiency model. Numerical results are compared with the experimental ones obtained at Institut de Mécanique des Fluides de Toulouse (IMFT).

New improvement of the [formula omitted]-model expansion method and its applications to the new [formula omitted]-dimensional integrable Kadomtsev–Petviashvili equation

In this paper, an improvement for the ϕ6-model expansion method is presented. In this approach, contrary to the classical ϕ6-model expansion method, obtaining explicit solutions for nonlinear ordinary and partial differential equations is congenial and undemanding of any constraint conditions, where the method can be applied and used for obtaining solutions without having any conditions on them. Moreover, the new approach is used to obtain new solutions for the new (3+1)-dimensional integrable Kadomtsev–Petviashvili equation. We demonstrated that for the same equation, the classical ϕ6-model expansion and the improved ϕ6-model expansion approaches produce the same family of solutions. However, the improved ϕ6-model expansion method is found to be more efficient and convenient.

On the differential equations of frozen Calogero-Moser-Sutherland particle models

Multivariate Bessel and Jacobi processes describe Calogero-Moser-Sutherland particle models. They depend on a parameter k and are related to time-dependent classical random matrix models like Dysom Brownian motions, where k has the interpretation of an inverse temperature. There are several stochastic limit theorems for k→∞ were the limits depend on the solutions of associated ordinary differential equations (ODEs) where these ODEs admit particular simple solutions which are connected with the zeros of the classical orthogonal polynomials. In this paper we show that these solutions attract all solutions. Moreover we present a connection between the solutions of these multivariate ODEs with associated one-dimensional inverse heat equations. These inverse heat equations are used to compute the expectations of some determinantal formulas for the multivariate Bessel and Jacobi processes.

Solution of Linear Ordinary Differential Equations with Polynomial Coefficients

Solution of Linear Ordinary Differential Equations with Polynomial Coefficients

Truncated Taylor Solvers for Simple First-Order Ordinary Differential Equations

Certain exact solutions of first-order ordinary differential equations naturally become corresponding solvers for simple firstorder ordinary differential equations whose form is the equality of a twice continuously differentiable function of the dependent variable to the derivative with respect the independent variable of the dependent variable. When the twice continuously differentiable function of the dependent variable is replaced by its truncated Taylor expansions through second order about its initial value, the resulting first-order ordinary differential equations have exact solutions that naturally become corresponding solvers for those simple first-order ordinary differential equations

Stability and numerical solutions for second-order ordinary differential equations with application in mechanical systems

This study undertakes a comprehensive analysis of second-order Ordinary Differential Equations (ODEs) to examine animal avoidance behaviors, specifically emphasizing analytical and computational aspects. By using the Picard–Lindelöf and fixed-point theorems, we prove the existence of unique solutions and examine their stability according to the Ulam-Hyers criterion. We also investigate the effect of external forces and the system’s sensitivity to initial conditions. This investigation applies Euler and Runge–Kutta fourth-order (RK4) methods to a mass-spring-damper system for numerical approximation. A detailed analysis of the numerical approaches, including a rigorous evaluation of both absolute and relative errors, demonstrates the efficacy of these techniques compared to the exact solutions. This robust examination enhances the theoretical foundations and practical use of such ODEs in understanding complex behavioral patterns, showcasing the connection between theoretical understanding and real-world applications.

A Study of Movable Singularities in Non-Algebraic First-Order Autonomous Ordinary Differential Equations

We studied the movable singularities of solutions of autonomous non-algebraic first-order ordinary differential equations in the form of y′=I(y(t)) and y′=I1(y(t))+I2(y(t))+⋯+In(y(t)), aiming to prove that all movable singularities of all complex solutions of these equations are at most algebraic branch points. This study explores the use of the constructing triangle method to analyze complex solutions of autonomous non-algebraic first-order ordinary differential equations. For complex solutions in the form of y=w+iv, we treat the constructing triangle method as a way to construct a right-angled triangle in the complex plane, with the lengths of the adjacent sides being w and v. We use the definitions of the trigonometric functions sin and cos (the ratio of the adjacent side to the hypotenuse) to represent the trigonometric functions of complex solutions y=w+iv. Since the movable singularities of the inverse functions of trigonometric functions are easy to analyze, the properties of the movable singularities of the complex solutions are then easy to deal with.