General Nonlinear Differential Equations Research Articles

A new solution of the nonlinear fractional logistic differential equations utilizing efficient techniques

The formulation of models and solutions for various physical problems are the primary goals of scientific achievements in engineering and physics. Our paper focuses on using the Caputo fractional derivative operator to solve nonlinear fractional logistic differential equations. In order to solve general nonlinear fractional differential equations, we first introduce a novel numerical methodology termed the Homotopy perturbation transform method. The perturbation approach and the Yang transform method are combined to create the suggested strategy. Second, we introduce a new hybrid method that uses the time-fractional Caputo derivative to approximate and analytically solve nonlinear fractional logistic differential equations. This method combines the Yang transform with the decomposition method. To validate the analysis, we offer three numerical cases of nonlinear fractional logistic differential equations employing the Caputo fractional derivative operator. The resulting solutions exhibit rapid convergence and are presented in series form. In order to verify the efficacy and relevance of the suggested methodologies, the investigated issues were assessed through the implementation of different fractional orders. We examine and show that, under the specified initial conditions, the solution approaches under evaluation are accurate and effective. Graphs in two and three dimensions show the results that were obtained. Numerical simulations are presented to confirm the efficacy of the strategies. The numerical results show that an accurate, reliable, and efficient approximation can be obtained with a minimal number of terms. The results obtained demonstrate that the new analytical solution method is easy to apply and very successful in solving difficult fractional problems that occur in relevant engineering and scientific domains.

Seq-SVF: An unsupervised data-driven method for automatically identifying hidden governing equations

In this work, an unsupervised data-driven method based on sequential singular value filtering (Seq-SVF) is proposed to simultaneously identify multiple partial differential equations from observed data considering potential noises. This method is aimed to extend the Sparse Identification of Nonlinear Dynamics (SINDy) to the identification of general nonlinear partial differential equations by transforming the paradigm based on regression to an unsupervised paradigm. To discover the complex coupled equations of vector or tensor forms without prior knowledge, the techniques of singular value decomposition (SVD) and strong rank-revealing QR factorization (sRRQR) are applied to the data matrix, which ensures that the method can automatically identify the number and the corresponding linearly independent terms as the left-hand terms of governing equations. To balance the complexity and the precision of modeling, a strategy for filtering singular values is designed to determine the sparse structure of governing equations from a large number of nonlinear basis functions. We show the success of the method to extract explicit and succinct models from many complex linear, nonlinear, and multiphysics mechanical systems, and the examples show more accuracy compared with using traditional sparse learning methods.

Extremum seeking for infinite-dimensional systems

The last year was the centennial of the 1922 invention, in the context of maximizing the power transfer to an electric tram car, of the method for real-time model-free optimization called extremum seeking (ES). Forgotten after a few decades of practical use and attempts at theoretical study, ES returned to life when the stability of its operation for not only static input–output maps but for dynamic systems modeled by general nonlinear ordinary differential equations (ODEs) was proven in 1997. Then it is natural to ask “why limit the use and the theoretical advances of ES to ODE systems?” So many—if not the majority of physical systems—involve delays or are modeled by partial differential equations (PDEs); why not pursue the application of ES in the presence of delays and for PDE systems? With this tutorial paper, we show a small portion of the vast space of possibilities of designing ES algorithms for infinite-dimensional systems governed by transport hyperbolic PDEs as well as diffusion parabolic PDEs.

Finite line method for solving high-order partial differential equations in science and engineering

In this paper, a completely new numerical method, named Finite Line Method (FLM), is proposed for solving general linear and non-linear high-order partial differential equations (PDEs) in science as well as engineering problems in heat conduction and mechanics. In this method, the computational domain is discretized into a number of collocation points as in the free element method (Gao et al., 2019; Liu et al., 2019), and at each collocation point a set of straight or curved lines crossing the point is formed, which is represented by a number of nodes distributed over lines. The shape functions for each line are constructed using the Lagrange interpolation formulation and the first and high order spatial partial derivatives of the shape functions with respect to global coordinates are derived through an ingenious technique. The derived partial derivatives can be directly substituted into the governing differential equations and related boundary conditions to set up the discretized system of equations.FLM is a type of collocation method, not needing any integration to establish the solution scheme. And since the Lagrange interpolation formulation is used to construct the shape functions, arbitrarily high order lines can be easily formulated for solving any types of linear and non-linear high order PDEs and engineering problems. A number of numerical examples for PDEs, heat conduction and solid mechanics problems are given to demonstrate the correctness and stability of the proposed method.

Physics-informed neural networks for modelling power transformer’s dynamic thermal behaviour

This paper focuses on the thermal modelling of power transformers using physics-informed neural networks (PINNs). PINNs are neural networks trained to consider the physical laws provided by the general nonlinear partial differential equations (PDEs). The PDE considered for the study of power transformer’s thermal behaviour is the heat diffusion equation provided with boundary conditions given by the ambient temperature at the bottom and the top-oil temperature at the top. The model is one dimensional along the transformer height. The top-oil temperature and the transformer’s temperature distribution are estimated using field measurements of ambient temperature, top-oil temperature and the load factor. The measurements from a real transformer provide more realistic solution, but also an additional challenge. The Finite Volume Method (FVM) is used to calculate the solution of the equation and further to benchmark the predictions obtained by PINNs. The results obtained by PINNs for estimating the top-oil temperature and the transformer’s thermal distribution show high accuracy and almost exactly mimic FVM solution.

On the convergence rate of Fletcher‐Reeves nonlinear conjugate gradient methods satisfying strong Wolfe conditions: Application to parameter identification in problems governed by general dynamics

Over the last decades, many efforts have been made toward the understanding of the convergence rate of the gradient‐based method for both constrained and unconstrained optimization. The cases of the strongly convex and weakly convex payoff function have been extensively studied and are nowadays fully understood. Despite the impressive advances made in the convex optimization context, the nonlinear non‐convex optimization problems are still not fully exploited. In this paper, we are concerned with the nonlinear, non‐convex optimization problem under system dynamic constraints. We apply our analysis to parameter identification of systems governed by general nonlinear differential equations. The considered inverse problem is presented using optimal control tools. We tackle the optimization through the use of Fletcher‐Reeves nonlinear conjugate gradient method satisfying strong Wolfe conditions with inexact line search. We rigorously establish a convergence analysis of the method and report a new linear convergence rate which forms the main contribution of this work. The theoretical result reported in our analysis requires that the second derivative of the payoff functional be continuous and bounded. Numerical evidence on a selection of popular nonlinear models is presented as a direct application of parameter identification to support the theoretical findings.

The existence of solutions for Sturm\u2013Liouville differential equation with random impulses and boundary value problems

In this article, we consider the existence of solutions to the Sturm–Liouville differential equation with random impulses and boundary value problems. We first study the Green function of the Sturm–Liouville differential equation with random impulses. Then, we get the equivalent integral equation of the random impulsive differential equation. Based on this integral equation, we use Dhage’s fixed point theorem to prove the existence of solutions to the equation, and the theorem is extended to the general second order nonlinear random impulsive differential equations. Then we use the upper and lower solution method to give a monotonic iterative sequence of the generalized random impulsive Sturm–Liouville differential equations and prove that it is convergent. Finally, we give two concrete examples to verify the correctness of the results.

Existence, uniqueness, and approximate solutions for the general nonlinear distributed-order fractional differential equations in a Banach space

The purpose of this paper is to provide sufficient conditions for the local and global existence of solutions for the general nonlinear distributed-order fractional differential equations in the time domain. Also, we provide sufficient conditions for the uniqueness of the solutions. Furthermore, we use operational matrices for the fractional integral operator of the second kind Chebyshev wavelets and shifted fractional-order Jacobi polynomials via Gauss–Legendre quadrature formula and collocation methods to reduce the proposed equations into systems of nonlinear equations. Also, error bounds and convergence of the presented methods are investigated. In addition, the presented methods are implemented for two test problems and some famous distributed-order models, such as the model that describes the motion of the oscillator, the distributed-order fractional relaxation equation, and the Bagley–Torvik equation, to demonstrate the desired efficiency and accuracy of the proposed approaches. Comparisons between the methods proposed in this paper and the existing methods are given, which show that our numerical schemes exhibit better performances than the existing ones.

Matrix computational collocation approach based on rational Chebyshev functions for nonlinear differential equations

In this work, a numerical technique for solving general nonlinear ordinary differential equations (ODEs) with variable coefficients and given conditions is introduced. The collocation method is used with rational Chebyshev (RC) functions as a matrix discretization to treat the nonlinear ODEs. Rational Chebyshev collocation (RCC) method is used to transform the problem to a system of nonlinear algebraic equations. The discussion of the order of convergence for RC functions is introduced. The proposed base is specified by its ability to deal with boundary conditions with independent variable that may tend to infinity with easy manner without divergence. The technique is tested and verified by two examples, then applied to four real life and applications models. Also, the comparison of our results with other methods is introduced to study the applicability and accuracy.

Lidstone\u2013Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

General nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

Which image-based phenotypes are most promising for using AI to understand cellular functions and why?

Which image-based phenotypes are most promising for using AI to understand cellular functions and why?

The discontinuous Galerkin method for general nonlinear third-order ordinary differential equations

In this paper, we propose an optimally convergent discontinuous Galerkin (DG) method for nonlinear third-order ordinary differential equations. Convergence properties for the solution and for the two auxiliary variables that approximate the first and second derivatives of the solution are established. More specifically, we prove that the method is L2-stable and provides the optimal (p+1)-th order of accuracy for smooth solutions when using piecewise p-th degree polynomials. Moreover, we prove that the derivative of the DG solution is superclose with order p+1 toward the derivative of Gauss-Radau projection of the exact solution. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise Pp polynomials with arbitrary p≥1. Several numerical results are provided to confirm the convergence of the proposed scheme.

Applying physics informed neural network for flow data assimilation

Data assimilation (DA) refers to methodologies which combine data and underlying governing equations to provide an estimation of a complex system. Physics informed neural network (PINN) provides an innovative machine learning technique for solving and discovering the physics in nature. By encoding general nonlinear partial differential equations, which govern different physical systems such as fluid flows, to the deep neural network, PINN can be used as a tool for DA. Due to its nature that neither numerical differential operation nor temporal and spatial discretization is needed, PINN is straightforward for implementation and getting more and more attention in the academia. In this paper, we apply the PINN to several flow problems and explore its potential in fluid physics. Both the mesoscopic Boltzmann equation and the macroscopic Navier-Stokes are considered as physics constraints. We first introduce a discrete Boltzmann equation informed neural network and evaluate it with a one-dimensional propagating wave and two-dimensional lid-driven cavity flow. Such laminar cavity flow is also considered as an example in an incompressible Navier-Stokes equation informed neural network. With parameterized Navier-Stokes, two turbulent flows, one within a C-shape duct and one passing a bump, are studied and accompanying pressure field is obtained. Those examples end with a flow passing through a porous media. Applications in this paper show that PINN provides a new way for intelligent flow inference and identification, ranging from mesoscopic scale to macroscopic scale, and from laminar flow to turbulent flow.

New conditionally oscillatory class of equations with coefficients containing slowly varying and periodic functions

The main result of this paper is a straightforward oscillation test for linear differential equations whose coefficients consist of products of periodic and slowly varying continuous functions. The attention is paid to the case when the slowly varying parts of the coefficients are unbounded. In particular, using the presented oscillation test, we identify a very general type of conditionally oscillatory equations together with the threshold value of the coefficients (the critical setting on the border of oscillation and non-oscillation). Hence, our results supply a class of linear equations with solved oscillation behaviour which can be used as testing equations and as a starting point to the oscillation theory of the corresponding half-linear and more general non-linear and partial differential equations with unbounded coefficients.

A Novel Approach to Mean Square Exponential Stability of Stochastic Delay Differential Equations

We present a novel approach to the mean square exponential stability of stochastic delay differential equations. Consequently, some new explicit criteria for the mean square exponential stability of general nonlinear stochastic delay differential equations are derived. Both delay-independent and delay-dependent stability criteria are given, and illustrative examples are also provided.

On stability analysis and existence of positive solutions for a general non-linear fractional differential equations

In this article, we deals with the existence and uniqueness of positive solutions of general non-linear fractional differential equations (FDEs) having fractional derivative of different orders involving p-Laplacian operator. Also we investigate the Hyers–Ulam (HU) stability of solutions. For the existence result, we establish the integral form of the FDE by using the Green function and then the existence of a solution is obtained by applying Guo–Krasnoselskii’s fixed point theorem. For our purpose, we also check the properties of the Green function. The uniqueness of the result is established by applying the Banach contraction mapping principle. An example is offered to ensure the validity of our results.

New criteria for mean square exponential stability of stochastic delay differential equations

By a novel approach, we present several new criteria for the mean square exponential stability of general non-linear stochastic delay differential equations. Both delay-independent and delay-dependent stability criteria are provided. A discussion of the obtained results is given.

The Magnus Expansion for Stochastic Differential Equations

In this paper, all the terms in the stochastic Magnus expansion are presented by rooted trees. First, stochastic Magnus methods for linear stochastic differential equations are constructed by truncating the stochastic Magnus expansion. Then, explicit stochastic Magnus methods are constructed by Picard’s iteration for nonlinear stochastic differential equations on matrix Lie group. Furthermore, general nonlinear stochastic differential equations are transformed into linear operator stochastic differential equations by using the Lie derivative. Finally, numerical methods for general nonlinear stochastic differential equations are constructed by using the theory of the stochastic Magnus expansion for the linear case. In particular, for the commutative case, it is shown that the stochastic Magnus expansion provides a novel way to construct computationally inexpensive and arbitrarily high-order numerical methods while avoiding the simulation of multiple stochastic integrals. Moreover, the proposed methods are shown to preserve the intrinsic properties of the original system well and the numerical experiments agree with the theoretical results.

Continuous changes of variables and the Magnus expansion

In this paper, we are concerned with a formulation of Magnus and Floquet-Magnus expansions for general nonlinear differential equations. To this aim, we introduce suitable continuous variable transformations generated by operators. As an application of the simple formulas so-obtained, we explicitly compute the first terms of the Floquet-Magnus expansion for the Van der Pol oscillator and the nonlinear Schrödinger equation on the torus.

Mapping of non-autonomous dynamical systems to autonomous ones

Using a proper choice of the dynamical variables, we show that a non-autonomous dynamical system transforming to an autonomous dynamical system with a certain coordinate transformations can be obtained by solving a general nonlinear first-order partial differential equations. We examine some special cases and provide particular physical examples. Our framework constitutes general machineries in investigating other non-autonomous dynamical systems.