General Nonlinear Differential Equations Research Articles

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.

A highly-accurate finite element method with exponentially compressed meshes for the solution of the Dirichlet problem of the generalized Helmholtz equation with corner singularities

In this study, a highly-accurate, conforming finite element method is developed and justified for the solution of the Dirichlet problem of the generalized Helmholtz equation on domains with re-entrant corners. The k − t h order Lagrange elements are used for the discretization of the variational form of the problem on exponentially compressed polar meshes employed in the neighbourhood of the corners whose interior angle is α π , α ≠ 1 ∕ 2 , and on the triangular and curved mesh formed in the remainder of the polygon. The exponentially compressed polar meshes are constructed such that they are transformed to square meshes using the Log-Polar transformation, simplifying the realization of the method significantly. For the error bound between the exact and the approximate solution obtained by the proposed method, an accuracy of O ( h k ) , h mesh size and k ≥ 1 an integer, is obtained in the H 1 -norm. Numerical experiments are conducted to support the theoretical analysis made. The proposed method can be applied for dealing with the corner singularities of general nonlinear parabolic partial differential equations with semi-implicit time discretization.

An Incremental Harmonic Balance Method With a General Formula of Jacobian Matrix and a Direct Construction Method in Stability Analysis of Periodic Responses of General Nonlinear Delay Differential Equations

A general formula of Jacobian matrix is derived in an incremental harmonic balance (IHB) method for general nonlinear delay differential equations (DDEs) with multiple discrete delays, where the fast Fourier transform is used to calculate Fourier coefficients of partial derivatives of residuals. It can be efficiently and automatically implemented in a computer program, and the only manual work is to derive the partial derivatives, which can be a much easier task than derivation of Jacobian matrix. An advantage of the IHB method in stability analysis is also revealed here. A direct construction method is developed for stability analysis of nonlinear differential equations with use of a relationship between Jacobian matrix in the IHB method and the system matrix of linearized equations. Toeplitz form of the system matrix can be directly constructed, and Hill’s method is used to calculate Floquet multipliers for stability analysis. Efficiency of stability analysis can be improved since no integration is needed to calculate the system matrix. Period-doubling bifurcations and period-p solutions of a delayed Mathieu–Duffing equation are studied to demonstrate use of the general formula of Jacobian matrix in the IHB method and the direct construction method in stability analysis. Its solution is the same as that from the numerical integration method using the spectral element method in the DDE toolbox in matlab, and it has a high convergence rate for solving a delayed Van der Pol equation.

Explicit criteria for mean square exponential stability of stochastic differential equations

Using a novel approach, we present new explicit criteria for the mean square exponential stability of general non-linear stochastic differential equations. An application to stochastic neural networks is given.

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

We introduce physics-informed neural networks – neural networks that are trained to solve supervised learning tasks while respecting any given laws of physics described by general nonlinear partial differential equations. In this work, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct types of algorithms, namely continuous time and discrete time models. The first type of models forms a new family of data-efficient spatio-temporal function approximators, while the latter type allows the use of arbitrarily accurate implicit Runge–Kutta time stepping schemes with unlimited number of stages. The effectiveness of the proposed framework is demonstrated through a collection of classical problems in fluids, quantum mechanics, reaction–diffusion systems, and the propagation of nonlinear shallow-water waves.

On Contraction of Functional Differential Equations

General nonlinear functional differential equations are considered. Explicit criteria for contraction of such equations are presented. Then explicit criteria for global exponential stability of equ...

A Generalized Definition of Caputo Derivatives and Its Application to Fractional ODEs

We propose a generalized definition of Caputo derivatives from $t=0$ of order $\gamma\in(0,1)$ using a convolution group, and we build a convenient framework for studying initial value problems of general nonlinear time fractional differential equations. Our strategy is to define a modified Riemann--Liouville fractional calculus which agrees with the traditional Riemann--Liouville definition for $t>0$ but includes some singularities at $t=0$ so that the group property holds. Then, making use of this fractional calculus, we introduce the generalized definition of Caputo derivatives. The new definition is consistent with various definitions in the literature while revealing the underlying group structure. The underlying group property makes many properties of Caputo derivatives natural. In particular, it allows us to deconvolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general comparison principle with the most general conditions. This then allows for a priori energy estimates of fractional PDEs. Since the new definition is valid for locally integrable functions that can blow up in finite time, it provides a framework for solutions to fractional ODEs and fractional PDEs. Many fundamental results for fractional ODEs are revisited within this framework under very weak conditions.

Synchronization and Impulsive Control of Some Parabolic Partial Differential Equations

Novel equi-attractivity in large generalized non-linear partial differential equations were performed for the impulsive control of spatiotemporal chaotic. Attractive solutions of these general partial differential equations were determined. A proof for existence of a certain kind of impulses for synchronization such that the small error dynamics that is equi-attractive in the large is established. A comparative study between these general non-linear partial differential equations and the existent reported numerical theoretical models was developed. Several boundary conditions were given to confirm the theoretical results of the general non-linear partial differential equations. Moreover, the equations were applied to Kuramoto–Sivashinsky PDE′s equation; Grey–Scott models, and Lyapunov exponents for stabilization of the large chaotic systems with elimination of the dynamic error.

Uniqueness and error estimates for solutions to higher-order boundary value problems

Little work has been done for the error estimates of the homotopy analysis method. For general 2nth-order linear and nonlinear differential equations with Lidstone boundary conditions, we obtain sharp upper bounds for the absolute error of the approximations given by the homotopy analysis method. To achieve this goal, the existence and uniqueness of solutions to these differential equations are considered. Numerical results confirm the theoretical predictions.

A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications

A novel parameterized non-intrusive reduced order model (P-NIROM) based on proper orthogonal decomposition (POD) has been developed. This P-NIROM is a generic and efficient approach for model reduction of parameterized partial differential equations (P-PDEs). Over existing parameterized reduced order models (P-ROM) (most of them are based on the reduced basis method), it is non-intrusive and independent on partial differential equations and computational codes. During the training process, the Smolyak sparse grid method is used to select a set of parameters over a specific parameterized space (Ωp∈RP). For each selected parameter, the reduced basis functions are generated from the snapshots derived from a run of the high fidelity model. More generally, the snapshots and basis function sets for any parameters over Ωp can be obtained using an interpolation method. The P-NIROM can then be constructed by using our recently developed technique (Xiao et al., 2015 [41,42]) where either the Smolyak or radial basis function (RBF) methods are used to generate a set of hyper-surfaces representing the underlying dynamical system over the reduced space.The new P-NIROM technique has been applied to parameterized Navier–Stokes equations and implemented with an unstructured mesh finite element model. The capability of this P-NIROM has been illustrated numerically by two test cases: flow past a cylinder and lock exchange case. The prediction capabilities of the P-NIROM have been evaluated by varying the viscosity, initial and boundary conditions. The results show that this P-NIROM has captured the quasi-totality of the details of the flow with CPU speedup of three orders of magnitude. An error analysis for the P-NIROM has been carried out.

Stability Analysis of Nonlinear Neutral Functional Differential Equations

We present a novel approach to analyze the exponential stability of nonlinear neutral functional differential equations. Our approach is simple and based on a system transformation, the comparison principle, and the spectral properties of Metzler matrices. Consequently, we derive some new explicit criteria for the exponential stability of general nonlinear neutral functional differential equations. It is important to note that our approach leads to both delay-dependent and delay-independent stability criteria. Several illustrative examples with simulations and a brief comparison between the obtained results with some existing results in the literature are presented.

HAAR WAVELETS BASED TIME DISCRETIZATION TECHNIQUE FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

Abstract: A new numerical technique is developed to find the solutions of general nonlinear partial differential equations. The technique is based on the time discretization of Haar wavelet series approximations with quasilinearization process. In order to test the efficiency of the proposed technique, it is applied on well known nonlinear partial differential equations such as the generalized regularized long wave equation, the Benjamin Bona-Mahony equation and the Fitzhugh-Nagumo equation. Numerical results are obtained by preparing MATLAB codes of proposed techniques. The beautiful concentration profiles of u and v are shown by figures at different time level and error norms L2 and L∞ are calculated.

Vibration Analysis of Flexible Structural Components

This article deals with the vibration analysis of flexible structural or mechanical components. The purpose of this article is to derive the general nonlinear differential equations of motion regarding the free vibration of flexible members of uniform and/or variable cross section along their length. Unique solution methodologies are developed that simplify the solution of very complex problems.

Non-linear analysis of FGM composites by finite block method in cylindrical coordinates

In this paper, the Finite Block Method (FBM) is developed to solve two-dimensional general non-linear partial differential equations in the cylindrical/polar coordinate systems. The first order one-dimensional differential matrix, by using the Lagrange series with uniformly distributed nodes is established and the higher order derivative matrices for multi-dimensional problems are obtained afterwards in the normalised space. By introducing the mapping technique, a block of quadratic type in the cylindrical/polar coordinate systems is transformed into the normalised space with eight seeds. The differential matrices in the physical domain for linear and non-linear problems are then determined in the normalised transformed system. Several examples including the static/dynamic and linear/non-linear heat transfer, the elasticity and the plate bending problems in the polar and cylindrical coordinates are given and comparisons are made with the analytical solutions, such as the Finite Element Method (FEM) and Finite Difference Method (FDM), to demonstrate the degree of accuracy and the convergence of the FBM.

On p th moment exponential stability of stochastic differential equations with Markovian switching and time-varying delay

In this paper we discuss the problem of pth moment exponential stability for general nonlinear stochastic differential equations with Markovian switching and time-varying delay. By using the Lyapunov function, the stochastic analysis technique and the generalized Halanay inequality, some novel sufficient conditions on pth moment exponential stability of stochastic differential equations with Markovian switching and time-varying delay are derived. The results obtained in this paper are completely new and modify and improve some known results. Moreover, two numerical examples are also provided to demonstrate the effectiveness and applicability of the theoretical results. The aim of this paper is to investigate pth moment exponential stability of general nonlinear stochastic differential equations with Markovian switching and time-varying delay.

Efficient low-error analytical-numerical approximations for radial solutions of nonlinear Laplace equations

We study radial solution of nonlinear elliptic partial differential equations of the form ??u=f(u) (a nonlinear Laplace equation) by means of an analytical-numerical method, namely optimal homotopy analysis. In this method, one obtain approximate analytical solutions which contain a free control parameter. This control parameter can be adjusted in order to improve the convergence or accuracy of the approximations. We outline the general technique for obtaining radial solutions of the general nonlinear elliptic partial differential equations of the form ??u=f(u), before focusing our attention on several specific equations, namely, the modified Liouville equation (with general positive nonlinearity), the Yamabe equation, and a generalized Lane-Emden equation of second kind. For the general case, we outline the method by which one may control the residual errors of these analytical-numerical approximations. One benefit to this method is that one can obtain solutions with rather low residual errors after only a few terms in the analytical expansion are calculated. This makes the method rather efficient compared to a standard homotopy approach, where many terms may need to be computed to guarantee the accuracy of the solution. By studying the modified Liouville equation (with general positive nonlinearity), the Yamabe equation, and a generalized Lane-Emden equation of second kind, we demonstrate that the benefits of the method are often related to the form of the nonlinearity inherent in the problem. For certain forms of nonlinearity, the method gives very accurate solutions after relatively few terms, while for other forms of nonlinearity this is not the case.

Numerical Solutions of General Fourth Order Two Point Boundary Value Problems by Galerkin Method with Legendre Polynomials

In this paper, Galerkin weighted residual method is presented to find the numerical solutions of the general fourth order linear and nonlinear differential equations with essential boundary conditions. For this, the given differential equations and the boundary conditions over arbitrary finite domain [a, b] are converted into its equivalent form over the interval [0, 1]. Here the Legendre polynomials, over the interval [0, 1], are chosen as trial functions satisfying the corresponding homogeneous form of the Dirichlet boundary conditions. Details matrix formulations are derived for solving linear and nonlinear boundary value problems (BVPs). Numerical examples for both linear and nonlinear BVPs are considered to verify the proposed formulation and the results obtained are compared. DOI: http://dx.doi.org/10.3329/dujs.v62i2.21973 Dhaka Univ. J. Sci. 62(2): 103-108, 2014 (July)

Oscillation Criteria of First Order Nonlinear Neutral Differential Equations

A class of general nonlinear neutral differential equations for its special cases are discussed. sufficient conditionsfor all solutionsof the equations to be oscillatory are obtained. It can be treated in the extreme limite theorem.

Linearized Oscillations in Autonomous Delay Impulsive Differential Equations

In recent years, there have been intensive efforts to establish linearised oscillation results for onedimensional delay, neutral delay and advanced impulsive differential equations. An impressive number of these efforts have yielded fruitful results in many analytical and applied areas. This is particularly obvious in the areas of applied disciplines such as the linear delay impulsive differential equations. However, there still remains a lot more to be explored in this direction, especially, in the area of non-linear autonomous differential equations. In this paper, we are proposing the development of linearised oscillation techniques for some general non-linear autonomous impulsive differential equations with several delays.