On Exact Solutions of Second Order Nonlinear Ordinary Differential Equations

In this paper, a new approach for solving the second order nonlinear ordinary differential equation y’’ + p(x;y)y’ = G(x;y) is considered. The results obtained by this approach are illustrated by examples and show that this method is powerful for this type of equations.

Numerical Treatment of Initial Value Problems of Nonlinear Ordinary Differential Equations by Duan-Rach-Wazwaz Modified Adomian Decomposition Method

We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.

UNIQUENESS OF SOLUTIONS OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR THIRD QRDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].

ON THE SINGULAR PERTURBATION OF A NONLINEAR ORDINARY DIFFERENTIAL EQUATION WITH TWO PARAMETERS

In this paper,the method of differential inequalities has been applied to study theboundary value problems of nonlinear ordinary differential equation with two parameters.The asymptotic solutions have been found and the remainders have been estimated.

NUMERICAL SOLUTION OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION FOR A TURNING POINT PROBLEM

By using the method in [3], several useful estimations of the derivatives of the solution of the boundary value problem for a nonlinear ordinary differential equation with a turning point are obtained. With the help of the technique in [4], the uniform convergence on the small parameter e for a difference scheme is proved. At the end of this paper, a numerical example is given. The numerical result coincides with theoretical analysis.

INTEGRABLE TYPES OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION SETS OF HIGHER ORDERS

Because of the extensive applications of nonlinear ordinary differential equation in physics,mechanics and cybernetics,there have been many papers on the exact solution to differential equation in some major publications both at home and abroad in recent years Based on these papers and in virtue of Leibniz formula,and transformation set technique,this paper puts forth the solution to nonlinear ordinary differential equation set of higher-orders, moveover,its integrability is proven.The results obtained are the generalization of those in the references.

Superconvergence of Continuous Finite Elements with Interpolated Coeffcients for Initial Value Problems of Nonlinear Ordinary Differential Equation

In this paper, n-degree continuous finite element method with interpolated coefficients for nonlinear initial value problem of ordinary differential equation is introduced and analyzed. An optimal superconvergence u-uh = O(hn+2), n ≥ 2, at (n + 1)-order Lobatto points in each element respectively is proved. Finally the theoretical results are tested by a numerical example.

REDUCTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION AND EXACT SOLUTIONS

Nonlinear partial differetial equation(NLPDE) is converted into ordinary differential equation(ODE) via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.

INSTABILITY OF SOLUTION FOR A CLASS OF THE THIRD ORDER NONLINEAR DIFFERENTIAL EQUATION

In Ref [1] the asymptotic stability of nonlinear slowly changing system has been discussed .In Ref [2] the instability of solution for the order linear differential equaiton with varied coefficient has been discussed .In this paper,we have discussed instability of solution for a class of the third order nonlinear diffeential equation by means of the metod of Refs [1] and [2] .

Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method

In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improved, so it is especially suitable for large scale systems. For Brown’s equations, an existing article reported that when the dimension of the equation N = 40, the subroutines they used could not give a solution, as compared with our method, we can easily solve this equation even when N = 100. Other two large equations have the dimension of N = 1000, all the existing available methods have great difficulties to handle them, however, our method proposed in this paper can deal with those tough equations without any difficulties. The sigularity and choosing initial values problems were also mentioned in this paper.

Stochastic Chebyshev-Picard Iteration Method for Nonlinear Differential Equations with Random Inputs

This work presents a stochastic Chebyshev-Picard iteration method to efficiently solve nonlinear differential equations with random inputs.If the nonlinear problem involves uncertainty,we need to characterize the uncer-tainty by using a few random variables.The nonlinear stochastic problems require solving the nonlinear system for a large number of samples in the stochastic space to quantify the statistics of the system of response and explore the uncertainty quantification.The computational cost is very expensive.To overcome the difficulty,a low rank approximation is introduced to the solution of the corresponding nonlinear problem and admits a variable-separation form in terms of stochastic basis functions and deterministic basis functions.No it-eration is performed at each enrichment step.These basis functions are model-oriented and involve offline computation.To efficiently identify the stochastic basis functions,we utilize the greedy algorithm to select some optimal sam-ples.Then the modified Chebyshev-Picard iteration method is used to solve the nonlinear system at the selected optimal samples,the solutions of which are used to train the deterministic basis functions.With the deterministic basis functions,we can obtain the corresponding stochastic basis functions by solv-ing linear differential systems.The computation of the stochastic Chebyshev-Picard method decomposes into an offline phase and an online phase.This is very desirable for scientific computation.Several examples are presented to illustrate the efficacy of the proposed method for different nonlinear differential equations.

A superlinear numerical scheme for multi-term fractional nonlinear ordinary differential equations

In this paper,we present a superlinear numerical method for multi-term fractional nonlinear ordinary differential equations(MTFNODEs).First,the presented problem is equivalently transformed into its integral form with multi-term Riemann-Liouville integrals.Second,the compound product trapezoidal rule is used to approximate the fractional integrals.Then,the unconditional stability and convergence with the order 1+αN−1−αN−2 of the proposed scheme are strictly established,whereαN−1 andαN−2 are the maximum and the second maximum fractional indexes in the considered MTFNODEs,respectively.Finally,two numerical examples are provided to support the theoretical results.

On the Existence of Periodic Solutions for a Nonlinear System of Ordinary Differential Equations

This paper is concerned with the existence of periodic solutions for a nonlinear system of ordinary differential equations.We obtain a Nagumo-type a priori bound for the periodic solutions and then by using this a priori bound we prove the existence of at least one T-periodic solution under some general conditions

Application of Hyperbola Function Method to the Family of Third Order Korteweg-de Vries Equations

In this work, we apply a hyperbola function method to solve the nonlinear family of third order Korteweg-de Vries equations. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions and trigonometric functions. The method used is a promising method to solve other nonlinear evaluation equations.

NEW PROGRESS IN THEORY OF SURFACES DEFINED BY ORDINARY DIFFERENTIAL EQUATIONS~*

Here singular points of the system (E_n~*) includes both finite and infinite singular points defined by H. Poincare, but extended to complex domain.

EXISTENCE AND UNIQUENESS OF THE SOLUTION OF A CLASS OF NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

The unicity of the solution, if any, of a class of nonlinear functional differential equations (fde) is established with the help of a transformation. The transformation reduces the fde to an ordinary differential equation. Existence of the solution is established by means of a fixed point theorem.

A novel (G'/G)-expansion method and its application to the Boussinesq equation

In this article, a novel （G＇/G）-expansion method is proposed to search for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the Boussinesq equation by means of the suggested method. The performance of the method is reliable and useful, and gives more general exact solutions than the existing methods. The new （G＇/G）-expansion method provides not only more general forms of solutions but also cuspon, peakon, soliton, and periodic waves.

Self-adaptive one-dimensional nonlinear finite element method based on element energy projection method

The element energy projection （EEP） method for computation of super- convergent resulting in a one-dimensional finite element method （FEM） is successfully used to self-adaptive FEM analysis of various linear problems, based on which this paper presents a substantial extension of the whole set of technology to nonlinear problems. The main idea behind the technology transfer from linear analysis to nonlinear analysis is to use Newton＇s method to linearize nonlinear problems into a series of linear problems so that the EEP formulation and the corresponding adaptive strategy can be directly used without the need for specific super-convergence formulation for nonlinear FEM. As a re- sult, a unified and general self-adaptive algorithm for nonlinear FEM analysis is formed. The proposed algorithm is found to be able to produce satisfactory finite element results with accuracy satisfying the user-preset error tolerances by maximum norm anywhere on the mesh. Taking the nonlinear ordinary differential equation （ODE） of second-order as the model problem, this paper describes the related fundamental idea, the imple- mentation strategy, and the computational algorithm. Representative numerical exam- ples are given to show the efficiency, stability, versatility, and reliability of the proposed approach.

Exact Solutions of the Cubic Duffing Equation by Leaf Functions under Free Vibration

Exact solutions of the cubic Duffing equation with the initial conditions are presented.These exact solutions are expressed in terms of leaf functions and trigonometric functions.The leaf function r=sleafn(t)or r=cleafn(t)satisfies the ordinary differential equation dx2/dt2=-nr2n-1.The second-order differential of the leaf function is equal to-n times the function raised to the(2n-1)power of the leaf function.By using the leaf functions,the exact solutions of the cubic Duffing equation can be derived under several conditions.These solutions are constructed using the integral functions of leaf functions sleaf2(t)and cleaf2(t)for the phase of a trigonometric function.Since the leaf function and the trigonometric function are used in combination,a highly accurate solution of the Duffing equation can be easily obtained based on the data of leaf functions.In this study,seven types of the exact solutions are derived from leaf functions;the derivation of the seven exact solutions is detailed in the paper.Finally,waves obtained by the exact solutions are graphically visualized with the numerical results.

Unsupervised neural network model optimized with evolutionary computations for solving variants of nonlinear MHD Jeffery-Hamel problem

A heuristic technique is developed for a nonlinear magnetohydrodynamics （MHD） Jeffery-Hamel problem with the help of the feed-forward artificial neural net- work （ANN） optimized with the genetic algorithm （GA） and the sequential quadratic programming （SQP） method. The twodimensional （2D） MHD Jeffery-Hamel problem is transformed into a higher order boundary value problem （BVP） of ordinary differential equations （ODEs）. The mathematical model of the transformed BVP is formulated with the ANN in an unsupervised manner. The training of the weights of the ANN is carried out with the evolutionary calculation based on the GA hybridized with the SQP method for the rapid local convergence. The proposed scheme is evaluated on the variants of the Jeffery-Hamel flow by varying the Reynold number, the Hartmann number, and the an- gles of the walls. A large number of simulations are performed with an extensive analysis to validate the accuracy, convergence, and effectiveness of the scheme. The comparison of the standard numerical solution and the analytic solution establishes the correctness of the proposed designed methodologies.