Eigenvalue problems of second order impulsive differential equations

This paper is concerned with an eigenvalue problem for second order differential equations with impulse. The existence of a countably infinite set of eigenvalues and eigenfunctions is proved.     

Anti-periodic fractional boundary value problems

We study a class of anti-periodic boundary value problems of fractional differential equations. Some existence and uniqueness results are obtained by applying some standard fixed point principles. Several examples are given to illustrate the results.     

Integral boundary value problems for first order integro-differential equations with impulsive integral conditions

This paper is concerned with the integral boundary value problems for first order integro-differential equations with impulsive integral conditions. Sufficient conditions are obtained for the existence of extremal solutions.     

Multiple solutions to semipositone Dirichlet boundary value problems with singular dependent nonlinearities for second order three-point differential equations

In this paper we present some new existence results for a singular semipositone Dirichlet boundary value problem for second order three-point differential equations by using the upper and lower solutions method and the fixed point theorem in cones.     

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.     

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.     

A fourth order multiderivative method for linear second order boundary value problems

Fourth order methods are developed and analysed for the numerical solution of linear second order boundary value problems.

The methods are developed by replacing the exponential terms in a three-point recurrence relation by Padé approximants.

The derivations of second order and sixth order methods from the recurrence relation are outlined briefly.

One method is tested on two problems from the literature, one of which is mildly nonlinear.     

Some boundary value problems of fractional differential equations and inclusions

In this paper, we study the existence of solutions for nonlinear fractional differential equations and inclusions of order q∈(1,2] with families of mixed and closed boundary conditions. In case of inclusion problems, the existence results are established for convex as well as nonconvex multivalued maps. Our results are based on Leray-Schauder degree theory, nonlinear alternative of Leray-Schauder type, and some fixed point theorems for multivalued maps. Some interesting special cases are also discussed.     

Impulsive boundary value problem for nonlinear differential equations of fractional order

In this paper, we prove the existence and uniqueness of solutions for the boundary value problem of nonlinear impulsive differential equations of fractional order q∈(1,2]. Our results are based on Altman’s fixed point theorem and Leray-Schauder’s fixed point theorem.     

Numerical techniques for solving systems of second order boundary value problems

In this paper, we compare some collocation, finite difference and spline methods for solving a system of second order boundary value problems associated with obstacle problems. An example is given to illustrate the efficiency of these results.     

Existence results for the three-point impulsive boundary value problem involving fractional differential equations

In this paper, we consider the existence of solutions for a class of three-point boundary value problems involving nonlinear impulsive fractional differential equations. By use of Banach’s fixed point theorem and Schauder’s fixed point theorem, some existence results are obtained.     

Discretization and error estimates for boundary value problems of the second order

Summary Boundary value problems for the equationLy≡−[p(x)y′]′+q(x)y=f(x) are considered. The method of finite differences is applied in a usual way andO (h ²) estimates are given for the discretization error as well as for its first and second difference quotients.     

Remarks on the periodic boundary value problems for first-order differential equations

This paper is concerned with the existence of solutions and the monotone method of first-order periodic boundary value problems when the lower and upper solutions α and β violate the boundary conditions α(0) ≤ α(2π) and β(0) ≥ β(2π). Using the topological degree theory, two existence theorems are established under weaker conditions than the one-side Lipschitz conditions. An example is given, which illustrates that PBVP may not have solutions between α and β without further restrictions to f(t, u). The monotone method is also discussed with some new results.     

Double solutions of boundary value problems for 2mth-order differential equations and difference equations

A double fixed-point theorem is applied to obtain the existence of at least two positive solutions for the boundary value problem, (−1)^my^(2m)(t) = f(y(t)), t ϵ [0, 1], y⁽²ⁱ⁾(0) = y⁽²ⁱ⁺¹⁾(1) = 0, 0 ≤ i ≤ m−1. It is later applied to obtain the existence of at least two positive solutions for the analogous discrete boundary value problem, (−1)^mΔ^2mu(k) = g(u(k)), k ϵ {0, …, N}, Δ²ⁱu(0) = Δ²ⁱ⁺¹u(N + 1) = 0, 0 ⩽ m − 1.     

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.     

Computer-assisted reduction of second order linear differential equations

So as to explore the possibilities of representing its solutions in terms of special functions, and using factorization techniques, a process is defined to decide whether a second order linear differential equation with polynomial coefficients can be brought to the hypergeometric or the confluent hypergeometric equation by a rational change of variable. In the first case, an upper bound has to be provided for the degree of the numerator of the rational function which defines the change of variable.     

A new method for solving boundary value problems for partial differential equations

This paper proposes a symmetry–iteration hybrid algorithm for solving boundary value problems for partial differential equations. First, the multi-parameter symmetry is used to reduce the problem studied to a simpler initial value problem for ordinary differential equations. Then the variational iteration method is employed to obtain its solution. The results reveal that the proposed method is very effective and can be applied for other nonlinear problems.