Problems For Second-order Differential Equations Research Articles

Neumann Problem for Second-Order Differential Equation with Fractional Derivative in the Analysis and Modeling of Structures Made of Viscoelastic Elements

This article addresses a second-order differential equation containing a Gerasimov-Kaputo fractional differentiation operator of order less than two. The Neumann problem is formulated for this equation. A system of eigenfunctions and eigenvalues for the considered homogeneous boundary problem of the second kind is found. A conjugate boundary problem for the Gerasimov-Kaputo fractional derivative is introduced. A biorthogonal system is obtained that is orthogonal to the found system of eigenfunctions. Visualizations of the eigenfunction system, biorthogonal system, and an example of eigenvalue distribution on the real axis are provided.

Lower and Upper Solution Method for Semilinear, Quasi-Linear and Quadratic Singularly Perturbed Neumann Boundary Value Problems

In this paper, by using the method of lower and upper solutions and notion of (Iq)-stability, we established sufficient conditions for the uniform convergence of the solutions of singularly perturbed Neumann boundary value problems for second-order differential equations to the solution of their reduced problems.

Multiple Solutions for a Class of BVPs of Second-Order Discontinuous Differential Equations with Impulse Effects

This paper deals with a class of boundary value problems of second-order differential equations with impulses and discontinuity. The existence of single or multiple positive solutions to discontinuous differential equations with impulse effects is established by using the nonlinear alternative of Krasnoselskii’s fixed point theorem for discontinuous operators on cones. Finally, an example is given to illustrate the main results.

Periodic boundary value problem for second-order differential equations from geophysical fluid flows

In this paper, we study a new model for a jet component of the Antarctic Circumpolar Current. In the case of vorticity with perturbation, we present the existence results for positive solutions to periodic boundary value problems for general nonlinear, weak nonlinearity and semilinear terms. A computational method is established and an approximate scheme of solution is also given.

Reconstruction of Singular Second-order Differential Equations From Spectral Characteristics

In this work, we study boundary value problems for second-order differential equations with singularities of second kind inside an interval. First, we investigate properties of the spectrum, then prove a theorem on completeness of eigenfunctions, and study the inverse problem of reconstructing the differential equation from spectral characteristics.

Energy Levels of a Physical System and Eigenvalues of an Operator with a Singular Potential

In this paper, we formulate the boundary value problems for second-order differential equations having singular potentials of the form Ax−2 + f (x)x−1 + q0(x), where q0(x) ∈ Lloc(ω) and ω = (0, b). First, the existence and uniqueness theorems are proved, and the characteristic function is constructed. Then, the method of providing the energy levels of singular systems is described with some examples.

Existence of solutions for second-order integral boundary value problems

In this paper, using a new comparison result and monotone iterative method, we consider the existence of solution of integral boundary value problem for second-order differential equation. To obtain corresponding results, we also discuss second order differential inequalities. The interesting point is that the one-sided Lipschitz constant is related to the first eigenvalues corresponding to the relevant operators.

An iterative method for solving boundary value problems for second order differential equations

The purpose of this paper is to investigate the application of the Adomian decomposition method (ADM) for solving boundary value problems for second-order differential equations with Robin boundary conditions. We first reformulate the boundary value problems for linear equations as a fixed point problems for a linear Fredholm integral operator, and then apply the ADM. We also extend our approach to include second-order nonlinear differential equations subject Robin boundary conditions.

Positive solutions of periodic boundary value problems for second-order differential equations with the nonlinearity dependent on the derivative

In this article, we study the existence of positive solutions of periodic boundary value problems for the second-order differential equation with the nonlinearity depends on the derivative $$\begin{aligned} \left\{ {\begin{array}{l}\left( { Lu} \right) \left( t \right) = h\left( t \right) f\left( {u\left( t \right) ,{u}'\left( t \right) } \right) ,\quad 0 \le t \le \omega ,\\ R_1 \left( \! u \right) \equiv u\left( 0 \right) - u\left( \omega \right) = 0,\\ R_2 \left( \! u \right) \equiv p\left( 0 \right) {u}'\left( 0 \right) - p\left( \omega \right) {u}'\left( \omega \right) = 0, \end{array}} \right. \end{aligned}$$ where \(\left( {Lu} \right) \left( t \right) = - \left( {p\left( t \right) {u}'} \right) ^\prime + q\left( t \right) u\). By applying coincidence degree theorem, some conditions guaranteeing the existence of at least one positive solution are given in terms of the relative behaviors of the quotient \(\displaystyle \frac{f\left( {u,v} \right) }{\left| u \right| + \left| v \right| }\) for \(|u|+ |v|\) near 0 and \( + \infty \). The result discussed in the paper is a generalization of recent one and the difference is that a nonlinear term depends on the derivative of unknown function.

Positive solutions to Sturm–Liouville problems with non-local boundary conditions

In this paper, the existence of at least three non-negative solutions to non-local boundary-value problems for second-order differential equations with deviating arguments α and ϛ is investigated. Sufficient conditions, which guarantee the existence of positive solutions, are obtained using the Avery–Peterson theorem. We discuss our problem for both advanced and delayed arguments. An example is added to illustrate the results.

A Singular Initial-Value Problem for Second-Order Differential Equations

We are interested in the existence of solutions to initial-value problems for second-order nonlinear singular differential equations. We show that the existence of a solution can be explained in terms of a more simple initial-value problem. Local existence and uniqueness of solutions are proven under conditions which are considerably weaker than previously known conditions.

Nonlocal boundary value problems with resonant or non-resonant conditions

We study solvability of nonlocal boundary value problems for second-order differential equations with resonance or non-resonance. The method of proof relies on Schauder’s fixed point theorem. Some examples are presented to illustrate the main results. MSC: 34B05

Global structure of positive solutions for three-point boundary value problems

Abstract In this paper, we are concerned with the three-point boundary value problem for second-order differential equations { u ″ ( t ) + w ( t ) f ( u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = β u ′ ( 0 ) , u ( 1 ) = α u ( η ) , where β ≥ 0 , 0 < η < 1 , 0 < α η < 1 and 1 + β − α η − α β > 0 ; w ∈ C ( [ 0 , 1 ] , ( 0 , + ∞ ) ) and f ∈ C ( R + , R + ) , R + = [ 0 , ∞ ) satisfies f ( u ) > 0 for u > 0 . The existence of the continuum of a positive solution is established by utilizing the Leray-Schauder global continuation principle. Furthermore, the interval of α about the nonexistence of a positive solution is also given. MSC:34B10, 34B18, 34G20.

Galoisian approach for a Sturm-Liouville problem on the infinite interval

We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval. Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue the differential equations are integrable in the setting of differential Galois theory under general assumptions. Our result is illustrated with two examples for a stationary Schroedinger equation having a generalized Hulthen potential and an eigenvalue problem for a traveling front in the Allen-Cahn equation.

Existence of positive solutions for nonlocal second-order boundary value problem with variable parameter in Banach spaces

By obtaining intervals of the parameter λ, this article investigates the existence of a positive solution for a class of nonlinear boundary value problems of second-order differential equations with integral boundary conditions in abstract spaces. The arguments are based upon a specially constructed cone and the fixed point theory in cone for a strict set contraction operator.

On Differential Properties Rν-Generalized Solution of the Dirichlet Problem with Coordinated Degeneration of the Input Data

We consider the first boundary value problem for second-order differential equation with strong singularity caused by coordinated degeneration of the input data. For this problem, we study the differential properties of the Rν-generalized solution, that is, the fact that it belongs to the space H2,ν+β/2k+2(Ω).

Fast multilevel augmentation methods for nonlinear boundary value problems

We develop a multilevel augmentation method for solving nonlinear boundary value problems. We first describe the multilevel augmentation method for solving nonlinear operator equations of the second kind which has been proposed in our recent paper Chen et al. (2011) [16] , and then apply it to solving the nonlinear two-point boundary value problems of second-order differential equations. The theoretical analysis of convergence order and computational complexity are proposed. Finally numerical experiments are presented to confirm the theoretical estimates and illustrate the efficiency of the method.

Existence and Uniqueness Results for Perturbed Neumann Boundary Value Problems

Using a fixed point theorem of general -concave operators, we present in this paper criteria which guarantee the existence and uniqueness of positive solutions for two classes of nonlinear perturbed Neumann boundary value problems for second-order differential equations. The theorems for Neumann boundary value problems obtained are very general.

On the exact solutions for initial value problems of second-order differential equations

In this paper, the solutions of initial value problems for a class of second-order linear differential equations are obtained in the exact form by writing the equations in the general operator form and finding an inverse differential operator for this general operator form.

Multiple positive solutions of the singular boundary value problems for second-order differential equations on the half-line

In this paper, the existence of multiple positive solutions for singular Sturm–Liouville boundary value problems on the half-line is investigated. By applying the fixed point index theorem of cone map, some existence and multiplicity results of positive solutions are derived. Our results improve and generalize many well-known results. Two examples are presented to demonstrate the application of our main results.