Solutions Of Nonlinear Differential Equations Research Articles

Existence and asymptotic behavior of strongly monotone solutions of nonlinear differential equations

Existence and asymptotic behavior of strongly monotone solutions of nonlinear differential equations

Doubly periodic meromorphic solutions of autonomous nonlinear differential equations

The problem of constructing and classifying elliptic solutions of nonlinear differential equations is studied. An effective method enabling one to find any elliptic solution of an autonomous nonlinear ordinary differential equation is described. The method does not require integrating additional differential equations. Much attention is being paid to the case of elliptic solutions with several poles in a parallelogram of periods. With the help of the method we find elliptic solutions up to the fourth order inclusively of an ordinary differential equation with a number of physical applications. The method admits a natural generalization and can be used to find elliptic solutions satisfying systems of ordinary differential equations.

An efficient method for nonlinear fractional differential equations: combination of the Adomian decomposition method and spectral method

In this paper, a novel iterative method is employed to give approximate solutions of nonlinear differential equations of fractional order. This approach is based on combination of two different methods which are the Adomian decomposition method (ADM) and the spectral Adomian decomposition method (SADM). The method reduces the nonlinear differential equations to systems of linear algebraic equations and then the resulting systems are solved by a numerical method. Investigating some illustrations, we demonstrate that the obtained numerical results are in a very good agreement with the exact solutions.

Quasi-Exact Solutions of the Equation for Description of Nonlinear Waves in a Liquid with Gas Bubbles

Nonlinear partial differential equation derived by Kudryashov and Sinelshchikov for description of waves in a liquid with gas bubbles is considered. The quasi-exact solutions of this equation are found with the first, second and third order poles. Values of the residual function norm corresponding to all quasi-exact solutions are presented. It is shown that most quasi-exact solutions are transformed into exact solutions of nonlinear differential equation under some additional conditions. The found solutions can be used for description of nonlinear waves in a liquid with gas bubbles.

Difference equations with absolutely unstable trivial solutions

We establish conditions for the absolute instability of the trivial solutions of nonlinear difference equations.

Scrutiny of underdeveloped nanofluid MHD flow and heat conduction in a channel with porous walls

In this paper, laminar fluid flow and heat transfer in channel with permeable walls in the presence of a transverse magnetic field is investigated. Least square method (LSM) for computing approximate solutions of nonlinear differential equations governing the problem. We have tried to show reliability and performance of the present method compared with the numerical method (Runge–Kutta fourth-rate) to solve this problem. The influence of the four dimensionless numbers: the Hartmann number, Reynolds number, Prandtl number and Eckert number on non-dimensional velocity and temperature profiles are considered. The results show analytical present method is very close to numerically method. In general, increasing the Reynolds and Hartman number is reduces the nanofluid flow velocity in the channel and the maximum amount of temperature increase and increasing the Prandtl and Eckert number will increase the maximum amount of theta.

Existence of pseudo-almost automorphic solutions for nonlinear differential equations

In this paper, we discuss the existence of pseudo-almost automorphic solutions to linear differential equation which has an exponential trichotomy, and the results also hold for some nonlinear equations with the form x′(t) = f(t, x(t)) + λ g (t, x(t)), where f, g are pseudo-almost automorphic functions. We prove our main result by the application of Leray-Schauder fixed point theorem.

Direct application of Padé approximant for solving nonlinear differential equations.

This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem. The high accurate handy approximations obtained by the direct application of Padé method shows the high potential if the proposed scheme to approximate a wide variety of problems. What is more, the direct application of the Padé approximant aids to avoid the previous application of an approximative method like Taylor series method, homotopy perturbation method, Adomian Decomposition method, homotopy analysis method, variational iteration method, among others, as tools to obtain a power series solutions to post-treat with the Padé approximant.AMS Subject Classification34L30

An optimized and scalable eigensolver for sequences of eigenvalue problems

SummaryIn many scientific applications, the solution of nonlinear differential equations are obtained through the setup and solution of a number of successive eigenproblems. These eigenproblems can be regarded as a sequence whenever the solution of one problem fosters the initialization of the next. In addition, in some eigenproblem sequences, there is a connection between the solutions of adjacent eigenproblems. Whenever it is possible to unravel the existence of such a connection, the eigenproblem sequence is said to be correlated. When facing with a sequence of correlated eigenproblems, the current strategy amounts to solving each eigenproblem in isolation. We propose an alternative approach that exploits such correlation through the use of an eigensolver based on subspace iteration and accelerated with Chebyshev polynomials (Chebyshev filtered subspace iteration (ChFSI)). The resulting eigensolver is optimized by minimizing the number of matrix–vector multiplications and parallelized using the Elemental library framework. Numerical results show that ChFSI achieves excellent scalability and is competitive with current dense linear algebra parallel eigensolvers. Copyright © 2014 John Wiley & Sons, Ltd.

Conditions for the Existence of Almost Periodic Solutions of Nonlinear Difference Equations with Discrete Argument

We establish conditions for the existence of almost periodic solutions of nonlinear almost periodic difference equations with discrete argument in a Banach space without using the H -classes of these equations.

Conditions for Almost Periodicity of Bounded Solutions of Nonlinear Differential Equations Unsolved with Respect to the Derivative

We establish conditions for the existence of almost periodic solutions of nonlinear almost periodic differential equations in Banach spaces without using the H-classes of these equations.

English

In this article, a technique called Haar wavelet-Picard technique is proposed to get the numerical solutions of nonlinear differential equations of fractional order. Picard iteration is used to linearize the nonlinear fractional order differential equations and then Haar wavelet method is applied to linearized fractional ordinary differential equations. In each iteration of Picard iteration, solution is updated by the Haar wavelet method. The results are compared with the exact solution.   Key words: Fractional differential equations, Wavelet analysis, Caputo derivative, Haar wavelets, Picard iteration.

Особенности инерционных характеристик плота, включающего сплоточные единицы стабилизированной плавучести, при свободном движении

Nonlinear differential equations are presented that fully describe the process of free acceleration of the raft in the river flow and free braking of the raft in a stationary liquid, and take into account all the main forces of the natural character acting on the raft during movement. Solution of nonlinear differential equations presented at a different direction of external forces on the raft, allowed to establish a number of key dependencies to determine the duration and the desired path during acceleration of the raft in the river flow to the speed of the river and braking of the raft in a stationary liquid until it stops.

Some results in Floquet theory, with application to periodic epidemic models

In this paper, we present several new results to the classical Floquet theory on the study of differential equations with periodic coefficients. For linear periodic systems, the Floquet exponents can be directly calculated when the coefficient matrices are triangular. Meanwhile, the Floquet exponents are eigenvalues of the integral average of the coefficient matrices when they commute with their antiderivative matrices. For the stability analysis of constant and nontrivial periodic solutions of nonlinear differential equations, we derive a few results based on linearization. We also briefly discuss the properties of Floquet exponents for delay linear periodic systems. To demonstrate the application of these analytical results, we consider a new cholera epidemic model with phage dynamics and seasonality incorporated. We conduct mathematical analysis and numerical simulation to the model with several periodic parameters.

Nonlinear Analysis of a Functionally Graded Beam Resting on the Elastic Nonlinear Foundation

Abstract This paper evaluates the nonlinear responses of a function- ally graded (FG) beam resting on a nonlinear foundation. After derivation of fundamental nonlinear differential equation using the Euler-Bernouli beam theory, a semi analytical method has been used to study the response of the problem. The responses can be evaluated for both linear and nonlinear isotropic and FG beams individually. Adomians Decomposition and successive approximation methods have been used for solution of nonlinear differential equation. As numerical investigation, the beams with simply supported ends and linear and nonlinear foundations have been analyzed using this method.

Pyragas stabilizability via delayed feedback with periodic control gain

In this paper for the unstable periodic solution of nonlinear differential equation, Pyragas stabilizability problem is studied. An algorithm of stabilization via periodical delayed feedback is obtained. The consideration is based on the theory of nonstationary stabilization.

Wavelet-Galerkin approach for power spectrum determination of nonlinear oscillators

A wavelet-Galerkin method based solution for nonlinear differential equation of motion is presented. Specifically, first, theory background of Periodic Generalized Harmonic Wavelet (PGHW) and its connection coefficients are briefly introduced. Next, wavelet coefficients of response are solved from a set of nonlinear algebra equations obtained via the wavelet-Galerkin approach. In this regard, Newton׳s method is employed to solve the nonlinear algebra equation. Further, stochastic response is determined by evoking a relationship between the wavelet coefficients and the corresponding response power spectrum density. Finally, pertinent numerical simulations demonstrate the reliability of the proposed approach.

Existence of Bounded Solutions of Nonlinear Difference Equations in Banach Spaces

We establish sufficient conditions for the existence of bounded solutions of the nonlinear difference equations $$ x\left( {n+1} \right)=F\left( {n,x(n)} \right)x(n)+f(n),\;n\in \mathbb{Z} $$ , in a Banach space.

On the asymptotic behaviour of solutions of certain differential equations of the third order

In this article, Lyapunov second method is used to obtain criteria for uniform ultimate boundedness and asymptotic behaviour of solutions of nonlinear differential equations of the third order. The results obtained in this investigation include and extend some well known results on third order nonlinear differential equations in the literature.

Conditions of Almost Periodicity for Bounded Solutions of Nonlinear Difference Equations with Continuous Argument

We establish conditions for the existence of almost periodic solutions of nonlinear almost periodic difference equations with continuous argument in a Banach space without using the H-classes of these equations. Translated from Neliniini Kolyvannya, Vol. 16, No. 1, pp. 118–124, January–March, 2013.