Analysis Of Nonlinear Differential Equations Research Articles

Mathematical Analysis of Nonlinear Differential Equations in Polymer Coated Microelectrodes

A theoretical analysis is conducted on the electrochemical behaviour of micro disk electrodes coated with thin coatings of electroactive polymers. The main focus of efforts to characterize the planar diffusion chemical reaction within polymer-modified ultra-microelectrodes is the creation of a theoretical model that explicitly takes into account the potential for the polymer film to cover the inlaid micro disc support surface. Approximate formulae for the steady-state amperometric response and formulation of the boundary value problem are given. The impact of substrate concentration, mediator concentration and current responsiveness in the solution besides the polymer film is also investigated. Akbari-Ganji method (AGM) and differential transformation method (DTM), two adequate and widely available analytical methods, were utilized to determine the steady-state non-linear diffusion equation. The approximate analytical solution for the substrate concentration and mediator concentration and the current for the small experimental kinetic values and diffusion coefficients are presented. We additionally determine the problem’s numerical solution using the MATLAB tool. A satisfactory agreement can be seen when the numerical outcomes verify with the analytical findings.

Oscillation Analysis Algorithm for Nonlinear Second-Order Neutral Differential Equations

Differential equations are useful mathematical tools for solving complex problems. Differential equations include ordinary and partial differential equations. Nonlinear equations can express the nonlinear relationship between dependent and independent variables. The nonlinear second-order neutral differential equations studied in this paper are a class of quadratic differentiable equations that include delay terms. According to the t-value interval in the differential equation function, a basis is needed for selecting the initial values of the differential equations. The initial value of the differential equation is calculated with the initial value calculation formula, and the existence of the solution of the nonlinear second-order neutral differential equation is determined using the condensation mapping fixed-point theorem. Thus, the oscillation analysis of nonlinear differential equations is realized. The experimental results indicate that the nonlinear neutral differential equation can analyze the oscillation behavior of the circuit in the Colpitts oscillator by constructing a solution equation for the oscillation frequency and optimizing the circuit design. It provides a more accurate control for practical applications.

Paradigm for the creation of scales and phases in nonlinear evolution equations

The transition from regular to apparently chaotic motions is often observed in nonlinear flows. The purpose of this article is to describe a deterministic mechanism by which several smaller scales (or higher frequencies) and new phases can arise suddenly under the impact of a forcing term. This phenomenon is derived from a multiscale and multiphase analysis of nonlinear differential equations involving stiff oscillating source terms. Under integrability conditions, we show that the blow-up procedure (a type of normal form method) and the Wentzel-Kramers-Brillouin approximation (of supercritical type) introduced in [7,8] still apply. This allows to obtain the existence of solutions during long times, as well as asymptotic descriptions and reduced models. Then, by exploiting transparency conditions (coming from the integrability conditions), by implementing the Hadamard's global inverse function theorem and by involving some specific WKB analysis, we can justify in the context of Hamilton-Jacobi equations the onset of smaller scales and new phases.

Lie symmetries, Painlevé analysis, and global dynamics for the temporal equation of radiating stars

We study the temporal equation of radiating stars by using three powerful methods for the analysis of nonlinear differential equations. Specifically, we investigate the global dynamics for the given master ordinary differential equation to understand the evolution of solutions for various initial conditions as also to investigate the existence of asymptotic solutions. Moreover, with the application of Lie's theory, we can reduce the order of the master differential equation, while an exact similarity solution is determined. Finally, the master equation possesses the Painlevé property, which means that the analytic solution can be expressed in terms of a Laurent expansion.

THE TECHNOLOGY OF WINTER CONCRETING OF MONOLITHIC FRAME STRUCTURES WITH SUBSTANTIATION OF HEAT TREATMENT MODES BY SOLUTIONS OF THE DIFFERENTIAL EQUATION OF THERMAL CONDUCTIVITY OBTAINED BY THE METHOD OF GROUP ANALYSIS

An innovative method for calculating thermal fields inside monolithic structures has been developed, based on the use and analysis of nonlinear differential equations. The innovativeness of the method lies in the approach to the analysis of nonlinear physical processes using nonlinear differential equations. Thanks to the method of group analysis, 13 expressions are obtained from complex mathematical equations, which are easy to use and depend on several empirical coefficients. It is assumed that this calculation method is a priori more accurate than the existing ones, as well as available to people at a construction site without higher mathematical education, which makes it a priority for research. The applicability of this method must be proven by linking empirical coefficients and variables to the conditions of the experiments, while obtaining reliable data that will turn out to be more accurate than the existing calculation methods. This article demonstrates a systematic approach to establishing the suitability of using the method of group analysis of differential equations for problems of winter concreting on the basis of laboratory experiments under stationary conditions. The equations were subject to verification, which, according to the physical description, correspond to the real conditions of the course of thermal processes inside monolithic structures. Based on the obtained processing results, it was decided that it was necessary to further study the innovative method in the conditions of the construction site, but only for some expressions that showed the best results at the stage of laboratory tests.

Stability analysis of nonlinear differential equations depending on a parameter with conformable derivative

Stability analysis of nonlinear differential equations depending on a parameter with conformable derivative

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.

BEHAVIOR OF THE PHARMACOKINETICS OF ENROFLOXACIN IN THE PHASE SPACE

Phase space is an approach for analysis of nonlinear differential equations. The graphical solutions that are obtained are convenient for qualitative assessment of the behavior of systems and processes. A comparative analysis of the pharmacokinetics of the antibiotic enrofloxacin administered intravenously in dogs and cats has been performed in the present study. The mathematical models that represent the change in blood plasma concentration of the two groups of animals are described by second-order differential equations. For the graphical representation of phase trajectories using the fluoroquinolone, the Mathcad program tools are used. The properties of the peculiar points are determined based on the received images.

ПОВЕДЕНИЕ НА ФАРМАКОКИНЕТИКАТА НА ЕНРОФЛОКСАЦИН ВЪВ ФАЗОВОТО ПРОСТРАНСТВО

Phase space is an approach for analysis of nonlinear differential equations. The graphical solutions that are obtained are convenient for qualitative assessment of the behavior of systems and processes. A comparative analysis of the pharmacokinetics of the antibiotic enrofloxacin administered intravenously in dogs and cats has been performed in the present study. The mathematical models that represent the change in blood plasma concentration of the two groups of animals are described by second-order differential equations. For the graphical representation of phase trajectories using the fluoroquinolone, the Mathcad program tools are used. The properties of the peculiar points are determined based on the received images.

Recent Advances in Numerical Methods and Analysis for Nonlinear Differential Equations

Recent Advances in Numerical Methods and Analysis for Nonlinear Differential Equations

Generalized Lie symmetry analysis for non-linear differential equations: A purely viscous fluid model

Abstract Prime objective of present study is to provide procedure in obtaining symmetries for mixed convection flow of viscous fluid due to an inclined porous stretching sheet. Two cases of stretching velocity namely power law and exponential are utilized. The obtained analysis has been tested for verification of five limiting flow cases. An excellent agreement through verification is observed. Further numerical treatment is carried out for the reduced problems solutions to velocity, temperature and concentration. Interestingly the observations for temperature and concentration with respect to Prandtl and Schmidt numbers are similar in a qualitative sense.

The stability analysis of rolling motion of hypersonic vehicles and its validations

The stability of the rolling motion of near space hypersonic vehicles with rudder control is studied using method of qualitative analysis of nonlinear differential equations, and the stability criteria of the deflected rolling motions are improved. The outcomes can serve as the basis for further study regarding the influence of pitching and lateral motion on the stability of rolling motion. To validate the theoretical results, numerical simulations were done for the rolling motion of two hypersonic vehicles with typical configurations. Also, wind tunnel experiments for four aircraft models with typical configurations have been done. The results show that: 1) there exist two dynamic patterns of the rolling motion under statically stable condition. The first one is point attractor, for which the motion of aircraft returns to the original state. The second is periodic attractor, for which the aircraft rolls periodically. 2) Under statically unstable condition, there exist three dynamic patterns of rolling motion, namely, the point attractor, periodic attractor around deflected state of rolling motion, and double periodic attractors or chaotic attractors.

Application of Normal Forms to a Laminated Composite Piezoelectric Rectangular Plate with One-to-Three Internal Resonance

Normal form theory is very useful for direct bifurcation and stability analysis of nonlinear differential equations modeled in real life. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. In the theoretical model for the nonlinear oscillation of a composite laminated piezoelectric plate under the parametrically and externally excitations, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues.

Further Reduction of Normal Forms for High Dimensional Nonlinear Systems and Application to a Composite Laminated Piezoelectric Plate

Normal form theory is robust and useful for direct bifurcation and stability analysis of nonlinear differential equations in real engineering problems. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. In the theoretical model for the nonlinear oscillation of a composite laminated piezoelectric plate, the computation method is applied to compute the coefficients of the normal forms for the case of one double zero and a pair of pure imaginary eigenvalues. The algorithm is implemented in Maple V and the normal forms of the averaged equations and their coefficients for nonlinear oscillations of the composite laminated piezoelectric plate under combined parametric and transverse excitations are calculated.

Invariant solutions as internal singularities of nonlinear differential equations and their use for qualitative analysis of implicit and numerical solutions

Lie group analysis of nonlinear differential equations reveals existence of singularities provided by invariant solutions and invisible from the form of the equation in question. We call them internal singularities in contrast with external singularities manifested by the form of the equation. It is illustrated by way of examples that internal singularities are useful for analyzing a behaviour of solutions of nonlinear differential equations near external singularities.

Global asymptotic stability analysis of nonlinear differential equations in hybrid bidirectional associative memory neural networks with distributed time-varying delays

In this paper, the problem on an unique equilibrium solution of the bidirectional associative memory neural networks with distributed time-varying delays is investigated via Lyapunov stability theory. By using topological degree theory and M-matrix theory, several sufficient conditions ensuring the existence, uniqueness and global asymptotic stability of equilibrium solution are derived. An illustrative example is given to demonstrate the effectiveness of the obtained results.

Symmetry analysis of nonlinear differential equations with the “Mathematica” program SYMMAN

The symmetry analysis is discussed for some nonlinear partial differential equations with important applications. It is demonstrated that significant progress both in theory and in the construction of explicit solutions is achieved with the aid of the computer algebra system “Mathematica.”

Non-linear discrete time systems analysis by multiple time perturbation techniques

This paper presents the application of the two variable expansion method for the study of a class of non-linear difference equations arising in discrete time systems. The two variable expansion method, a multi-time perturbational technique which has been used to a considerable extent for analysis of non-linear differential equations, has been adopted for the investigation of the response characteristics of non-linear difference equations. Examples are provided to illustrate the theoretical framework.

Regularity Properties of Flows Through Porous Media: A Counterexample

Regularity Properties of Flows Through Porous Media: A Counterexample

Analysis of Non-linear Differential Equations by the Volterra Series†

ABSTRACT The Volterra series is used to investigate aperiodic solutions of certain second-order differential equations with non-linear damping and non-linear restoring force. Solutions are obtained in the phase plane for trajectories with zero initial conditions. These are obtained by approximating the Volterra series, and it is shown that the approximation involved is small over a wide range of values of linear damping.