Solving Fractional Differential Equations via Fixed Points of Chatterjea Maps

In this paper,we present the existence and uniqueness of fixed points and common fixed points for Reich and Chatterjea pairs of self-maps in complete metric spaces.Furthermore,we study fixed point theorems for Reich and Chatterjea nonexpansive mappings in a Banach space using the Krasnoselskii-Ishikawa iteration method associated withSλand consider some applications of our results to prove the existence of solutions for nonlinear integral and nonlinear fractional differential equations.We also establish certain interesting examples to illustrate the usability of our results.

Exact Solution to Nonlinear Differential Equations of Fractional Order via (<i>G’</i>/<i>G</i>)-Expansion Method

In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards, the (G'/G)-expansion method has been implemented, to celebrate the exact solutions of these Equations, in the sense of modified Riemann-Liouville derivative. As application, the exact solutions of time-space fractional Burgers’ Equation have been discussed.

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.

Existence of Positive Solutions of Three-point Boundary Value Problem for Higher Order Nonlinear Fractional Differential Equations

In this paper, we consider the positive solutions of fractional three-point boundary value problem of the form Dο^α＋u（t）＋f（t,u（t）,u＇（t）,…,u^（n-3）（5）,u^（n-2）（t））=0,u^（i）（0）=0,0≤i≤n-2,u^（n-2）（1）-βu^（n-2）（ξ）=0,where 0〈t〈1,n-1〈α≤n,n≥2,ξ Е（0,1）,βξ^a-n〈1. We first transform it into another equivalent boundary value problem. Then, we derive the Green＇s function for the equivalent boundary value problem and show that it satisfies certain properties. At last, by using some fixed-point theorems, we obtain the existence of positive solution for this problem. Example is given to illustrate the effectiveness of our result.

Finite-Time Stability for Nonlinear Fractional Differential Equations with Time Delay

The finite-time stability and the finite-time contractive stability of solutions for nonlinear fractional differential equations with bounded delay are investigated. The derivative of Lyapunov function along solutions of the considered system is defined in terms of the Caputo fractional Dini derivative. Based on the Lyapunov-Razumikhin method, several sufficient criteria are established to guarantee the finite-time stability and the finite-time contractive stability of solutions for the related systems. An example is provided to illustrate the effectiveness of the obtained results.

The Multi-scale Method for Solving Nonlinear Time Space Fractional Partial Differential Equations

In this paper, we present a new algorithm to solve a kind of nonlinear time space-fractional partial differential equations on a finite domain. The method is based on B-spline wavelets approximations, some of these functions are reshaped to satisfy on boundary conditions exactly. The Adams fractional method is used to reduce the problem to a system of equations. By multiscale method this system is divided into some smaller systems which have less computations. We get an approximated solution which is more accurate on some subdomains by combining the solutions of these systems. Illustrative examples are included to demonstrate the validity and applicability of our proposed technique, also the stability of the method is discussed.

The Solution to Impulse Boundary Value Problem for a Class of Nonlinear Fractional Functional Differential Equations

In this paper, we investigate the existence of solution for a class of impulse boundary value problem of nonlinear fractional functional differential equation of mixed type. We obtain the existence results of solution by applying some well-known fixed point theorems. An example is given to illustrate the effectiveness of our result.

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].

EXISTENCE AND STABILITY RESULTS FOR GENERALIZED FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper,a sufficient conditions to guarantee the existence and stability of solutions for generalized nonlinear fractional differential equations of orderα(1<α<2)are given.The main results are obtained by using Krasnoselskii's fixed point theorem in a weighted Banach space.Two examples are given to demonstrate the validity of the proposed results.

FIXED POINTS OF α-TYPE F-CONTRACTIVE MAPPINGS WITH AN APPLICATION TO NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

In this paper, we introduce new concepts of a-type F-contractive mappings which are essentially weaker than the class of F-contractive mappings given in [21, 22] and different from a-GF-contractions given in [8]. Then, sufficient conditions for the existence and uniqueness of fixed point are established for these new types of contractive mappings, in the setting of complete metric space. Consequently, the obtained results encompass various generalizations of the Banach contraction principle. Moreover, some examples and an application to nonlinear fractional differential equation are given to illustrate the usability of the new theory.

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.

A FRACTIONAL NONLINEAR EVOLUTIONARY DELAY SYSTEM DRIVEN BY A HEMI-VARIATIONAL INEQUALITY IN BANACH SPACES

This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space.Utilizing the KKM theorem,a result concerned with the upper semicontinuity and measurability of the solution set of a hemivariational inequality is established.By using a fixed point theorem for a condensing setvalued map,the nonemptiness and compactness of the set of mild solutions are also obtained for such a system under mild conditions.Finally,an example is presented to illustrate our main results.

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.

Fixed Point Theorem and Fractional Differential Equations with Multiple Delays Related with Chaos Neuron Models

In this paper, we show a fixed point theorem which deduces to both of Lou’s fixed point theorem and de Pascale and de Pascale’s fixed point theorem. Moreover, our result can be applied to show the existence and uniqueness of solutions for fractional differential equations with multiple delays. Using the theorem, we discuss the fractional chaos neuron model.

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.

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.

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.

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.