Multiple solutions for fractional differential equations with nonlinear boundary conditions

In this paper, we study certain fractional differential equations with nonlinear boundary conditions. By means of the Amann theorem and the method of upper and lower solutions, some new results on the multiple solutions are obtained.     

Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions

This paper investigates the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order α∈(1,2]. Our results are based on some standard fixed point theorems. Some examples are presented to illustrate the main results.     

Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions

In the this paper, we establish sufficient conditions for the existence and nonexistence of positive solutions to a general class of integral boundary value problems for a coupled system of fractional differential equations. The differential operator is taken in the Riemann-Liouville sense. Our analysis rely on Banach fixed point theorem, nonlinear differentiation of Leray-Schauder type and the fixed point theorems of cone expansion and compression of norm type. As applications, some examples are also provided to illustrate our main results.     

Direct methods of solution of partial differential equations with periodic boundary conditions

In this paper, suitable discrete approximations for the general elliptic and parabolic partial differential equation with periodic boundary conditions are derived and appropriate direct and fast solution methods of the resulting linear systems proposed.     

Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions

In this paper we discuss the blow-up for classical solutions to the following class of parabolic equations with Robin boundary condition: $\{\begin{array}{cc}{\left(b\left(u\right)\right)}_t=??\left(g\left(u\right)?u\right)+f\left(u\right)\hfill & \text{in }\Omega \times (0,T)\text{,}\hfill \\ \frac{?u}{?n}+\gamma u=0\hfill & \text{on }?\Omega \times (0,T)\text{,}\hfill \\ u(x,0)=h\left(x\right)\ge 0\hfill & \text{in }\overline{\Omega }\text{,}\hfill \end{array}$ where $\Omega$ is a bounded domain of ${\mathbb{R}}^N(N\ge 2)$ with smooth boundary $?\Omega$. By constructing some appropriate auxiliary functions and using a first-order differential inequality technique, we derive conditions on the data which guarantee the blow-up or the global existence of the solution. For the blow-up solution, a lower bound on blow-up time is also obtained. Moreover, some examples are presented to illustrate the applications.     

Symbolic computation of analytic approximate solutions for nonlinear differential equations with initial conditions

The Adomian decomposition method (ADM) is one of the most effective methods for constructing analytic approximate solutions of nonlinear differential equations. In this paper, based on the new definition of the Adomian polynomials, and the two-step Adomian decomposition method (TSADM) combined with the Padé technique, a new algorithm is proposed to construct accurate analytic approximations of nonlinear differential equations with initial conditions. Furthermore, a MAPLE package is developed, which is user-friendly and efficient. One only needs to input a system, initial conditions and several necessary parameters, then our package will automatically deliver analytic approximate solutions within a few seconds. Several different types of examples are given to illustrate the validity of the package. Our program provides a helpful and easy-to-use tool in science and engineering to deal with initial value problems.Program summaryProgram title: NAPACatalogue identifier: AEJZ_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEJZ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 4060No. of bytes in distributed program, including test data, etc.: 113 498Distribution format: tar.gzProgramming language: MAPLE R13Computer: PCOperating system: Windows XP/7RAM: 2 GbytesClassification: 4.3Nature of problem: Solve nonlinear differential equations with initial conditions.Solution method: Adomian decomposition method and Padé technique.Running time: Seconds at most in routine uses of the program. Special tasks may take up to some minutes.     

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.     

Multiplicity of positive solutions of a class of nonlinear fractional differential equations

This paper is concerned with the nonlinear fractional differential equation L(D)u=f(x,u), u(0)=0, 0<x<1,where L(D) = Dsn − an−1Dsn−1 − … − a1Ds11 < s2 < … < sn < 1, and aj > 0, j = 1,2,…, n − 1. Some results are obtained for the existence, nonexistence, and multiplicity of positive solutions of the above equation by using Krasnoselskii's fixed-point theorem in a cone. In particular, it is proved that the above equation has N positive solutions under suitable conditions, where N is an arbitrary positive integer.     

Nonhomogeneous boundary value problem for first-order impulsive differential equations with delay

The authors employ the method of upper and lower solution coupled with the monotone iterative technique to obtain results of existence and uniqueness for a nonhomogeneous boundary value problem of impulsive differential equations with delay.     

Existence and multiplicity of positive periodic solutions for a class of higher-dimension functional differential equations with impulses

This paper deals with the existence of multiple periodic solutions for n-dimensional functional differential equations with impulses. By employing the Krasnoselskii fixed point theorem, we obtain some easily verifiable sufficient criteria which extend previous results.     

Antiperiodic boundary value problem for first-order impulsive ordinary differential equations

This paper discusses the antiperiodic boundary value problem for first-order impulsive ordinary differential equations. We establish several existence results by using the Leray-Schauder alternative, the lower and upper solution method and the monotone iterative technique.     

The global solution for a class of systems of fractional nonlinear Schrödinger equations with periodic boundary condition

In this paper, we consider a class of systems of fractional nonlinear Schrödinger equations. We prove the existence and uniqueness of the global solution to the periodic boundary value problem by using the Faedo-Galërkin method.     

Positive solutions for a fourth order equation with nonlinear boundary conditions

Existence of positive solutions for a fourth order equation with nonlinear boundary conditions, which models deformations of beams on elastic supports, is considered using fixed points theorems in cones of ordered Banach spaces. Iterative and numerical solutions are also considered.     

A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations

This paper studies the existence and uniqueness of solutions for a four-point nonlocal boundary value problem of nonlinear impulsive differential equations of fractional order q∈(1,2]. Our results are based on some standard fixed point theorems. Some illustrative examples are also discussed.     

Robust stability of systems of linear differential equations with periodic impulsive influence

We give sufficient robust stability conditions for matrix polytopes of linear systems with impulsive influence. The methods of our study are based on logarithmic matrix measure theory and linear operator theory in Banach spaces. Our results reduce the robust stability problem to the feasibility problem for a system of linear matrix inequalities in the class of positive definite matrices.