Constant Coefficient Ordinary Differential Equations Research Articles

Bifurcation analysis, and exact solutions of the two-mode Cahn–Allen equation by a novel variable coefficient auxiliary equation method

This document elaborates on a newly introduced analytical method known as the “Variable Coefficient Generalized Abel Equation Method,” as proposed by Hashemi in Hashemi (2024), designed specifically for addressing the two-mode Cahn–Allen equation. Diverging from conventional techniques that heavily rely on constant coefficient ordinary differential equations and auxiliary ordinary differential equations, our method innovatively incorporates variable coefficient ordinary differential equations within a sub-equation framework. Demonstrating its versatility, we apply this innovative technique to the two-mode Cahn–Allen equation, showcasing its effectiveness and efficiency through the derivation of analytical solutions. Notably, this method emerges as a promising tool for tackling complex nonlinear partial differential equations prevalent in fluid dynamics and wave propagation scenarios. Beyond merely expanding the repertoire of available analytical tools, our approach contributes to advancing solutions for various models within the realm of mathematical physics. Various forms of exact solutions, including exponential-type solutions, Kink solitons, dark solitons, and bright soliton solutions, are obtained for the model under consideration. Moreover, we delve into the analysis of bifurcation, chaotic behavior, and sensitivity within the context of the two-mode Cahn–Allen model, further enhancing the depth and breadth of our study. Three equilibria are analyzed across various classifications, including center point, focus point, saddle point, and node point. Chaotic behavior of the corresponding dynamical system is considered by adding the function ω1sin(ω2ζ). Lastly, sensitivity analysis of the system is conducted by examining different parameters of the model and imposing noise to the initial conditions.

Calibration of low-fidelity ship motion programs through regressions of high-fidelity forces

Lower-fidelity or reduced-order models based on constant coefficient ordinary differential equations provide computationally efficient means to generate ship motions. In these models, forces acting on a ship are typically approximated through closed-form expressions involving fixed coefficients that need to be set for the wave and operational conditions of interest. Here, this problem is addressed through the regressions of forces from higher-fidelity, for example, potential flow-based, solvers, that the lower-fidelity models aim to emulate. Numerical simulations are provided to illustrate the considered approach.

A Novel Method for Solving Ordinary Differential Equations with Artificial Neural Networks

This research work investigates the use of Artificial Neural Network (ANN) based on models for solving first and second order linear constant coefficient ordinary differential equations with initial conditions. In particular, we employ a feed-forward Multilayer Perceptron Neural Network (MLPNN), but bypass the standard back-propagation algorithm for updating the intrinsic weights. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial or boundary conditions and contains no adjustable parameters. The second part involves a feed-forward neural network to be trained to satisfy the differential equation. Numerous works have appeared in recent times regarding the solution of differential equations using ANN, however majority of these employed a single hidden layer perceptron model, incorporating a back-propagation algorithm for weight updation. For the homogeneous case, we assume a solution in exponential form and compute a polynomial approximation using statistical regression. From here we pick the unknown coefficients as the weights from input layer to hidden layer of the associated neural network trial solution. To get the weights from hidden layer to the output layer, we form algebraic equations incorporating the default sign of the differential equations. We then apply the Gaussian Radial Basis function (GRBF) approximation model to achieve our objective. The weights obtained in this manner need not be adjusted. We proceed to develop a Neural Network algorithm using MathCAD software, which enables us to slightly adjust the intrinsic biases. We compare the convergence and the accuracy of our results with analytic solutions, as well as well-known numerical methods and obtain satisfactory results for our example ODE problems.

A fresh look at linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations of any order based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and variation of parameters. The approach presented here can be used in a first course on differential equations for science and engineering majors.

On the particular solution of constant coefficient fractional differential equations

The eigenfunction approach to compute the particular solution of constant coefficient ordinary differential equations is extended to the fractional case. It is shown that the exponentials are also the eigenfunctions of such equations. Solutions corresponding to products of powers and exponentials are presented. The singular case is studied and a matricial algorithm for its implementation is presented.

A simple approach to the particular solution of constant coefficient ordinary differential equations

An eigenfunction approach to compute the particular solution of constant coefficient ordinary differential equations is presented. It is shown that the exponentials are the eigenfunctions of such equations. Solutions corresponding to products of powers and exponentials are obtained. The singular case is studied and a fast algorithm for its implementation is presented.

Linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and the variation of constants method. The approach presented here can be used in a first course on differential equations for science and engineering majors.

Solving ay″ + by′ + cy = 0 with a Simple Product Rule Approach

When elementary ordinary differential equations (ODEs) of first and second order are included in the calculus curriculum, second-order linear constant coefficient ODEs are typically solved by a method more appropriate to differential equations courses. This method involves the characteristic equation and its roots, complex-valued solutions, and some method (often mysterious, unnatural, or simply left unjustified) for establishing te rt as a solution when r is the only real root. Here we outline a seldom used but more natural approach better suited to the calculus curriculum. Complex numbers are not needed, and the solution te rt arises constructively.

The simulation problem for high order linear differential systems

In this paper we consider linear constant coefficient ordinary differential equations of arbitrary order of the form R( d dt )w=M( d dt )f with R and M given, but otherwise arbitrary, polynomial matrices. To these equations we associate a behavior B defined as the set of all vector-valued distributions w, f that solve the equations themselves. We think of f as a given distribution, while w is to be found in such a way that the equations defining the behavior are satisfied. Alongside R( d dt )w=M( d dt )f we also consider initial conditions obtained by imposing the value at time t=0 of a linear combination of the variables w and their derivatives, yielding S( d dt )w(0)=Ta with S a polynomial matrix, T a fixed real matrix, and a a real vector. The three main issues we address are: (i) Solvability of the behavior, meaning the problem of finding conditions that assure that corresponding to a (or any) given vector distribution f, a trajectory w exists such that ( w, f) belong to B . This question is formalized and answered in Section 4. (ii) The index of the behavior, which means investigating how the smoothness of the given distribution f is related to smoothness of the solution w. Section 5 is dedicated to this issue. (iii) Compatibility of initial conditions, in which case the initial conditions S( d dt )w(0)=Ta are considered alongside the behavior B . We first check that S( d dt )w(0) is well defined. If it is, we provide conditions under which there exists a (unique) distribution w such that w, f belong to B and such that S( d dt )w(0)=Ta are also satisfied for a (or any) given a. This problem is addressed in Section 6. We show how our results generalize to behaviors defined by equations of arbitrary order classical properties of behaviors defined by first order systems such as E d dt w+Aw=f .

Fluid Transmission Line Modeling Using a Variational Method

A variational method is used to derive numerical models for transient flow simulation in fluid transmission lines. These are generalizations of models derived using the more traditional modal method. Three different transient compressible laminar pipe flow models are considered (inviscous, one-dimensional linear viscous, and two-dimensional dissipative viscous flow), and a model for transient turbulent pipe flow is given. The (model) equations in the laminar case are given in the form of a set of constant coefficient ordinary differential equations, and for the turbulent case (model) in the form of a set of nonlinear ordinary differential equations. Explicit equations are given for various end conditions. Attenuation factors, similar to the window functions used in spectral analysis, are used to attenuate Gibbs phenomenon oscillations. [S0022-0434(00)03201-9]

A method for estimating intensity and impulse response functions of filtered Poisson processes

This paper presents a method for estimating the intensity and impulse response (IR) function of filtered Poisson processes. The case is considered where the filtered Poisson process is modeled as an output of the linear constant-coefficient ordinary differential equations having poles and zeros driven by Poisson impulse processes. It is shown that an explicit formula for estimating the intensity is derived by combining second-and third-order cumulants of the residual time series generated from the discretized filtered Poisson process. It is also shown that the IR function can be estimated from the parameters of the discretized filtered Poisson process. Then, Monte Carlo simulations demonstrate the validity of the proposed method in some specific examples. It is concluded that the proposed method can be extensively applied to actual phenomena appearing in engineering and science.

Computer implementation of a series solution for constant coefficient ordinary differential equations

Solving constant coefficient ordinary differential equations (CCODEs) with degree greater than four has long been a problem without precise theoretical or numerical solutions except in special cases. We now propose a new accurate theoretical and economic computer method using Taylor's series to solve CCODEs of any degree.

Addition theorems for solutions to linear homogeneous constant coefficient ordinary differential equations

Addition theorems for solutions to linear homogeneous constant coefficient ordinary differential equations

Addition theorems for solutions to linear homogeneous constant coefficient ordinary differential equations

Addition theorems for solutions to linear homogeneous constant coefficient ordinary differential equations

SPECTRAL THEORY IN SPACES OF ANALYTIC FUNCTIONALS FOR OPERATORS GENERATED BY MULTIPLICATION BY THE INDEPENDENT VARIABLE

This paper is devoted to the spectral theory for the adjoint of the operator of multiplication by the independent variable in weight spaces of entire functions of one complex variable, and is closely connected with the theory of constant coefficient ordinary differential equations of infinite order and of convolution type equations , with the theory of mean-periodic functions, and with the general theory of subspaces which are invariant under differentiation. Bibliography: 62 titles.

A fundamental theorem on lambda-matrices with applications —I. Ordinary differential equations with constant coefficients

The fundamental theorem of the title refers to a spectral resolution for the inverse of a lambda-matrix L(λ) = ∑ i=0 l A iλ i where the A i are n× n complex matrices and det A l ≠ 0. The idea of a Jordan normal form associated with such a lambda-matrixis developed in proving the theorem. Applications are made to the study of initial value problems and two-point boundary value problems for systems of constant-coefficient ordinary differential equations.