Second-order Boundary Value Problems Research Articles

Solvability and positive solution of a system of second-order boundary value problem with integral boundary conditions

The main purpose of this paper is to establish the existence, uniqueness and positive solution of a system of second-order boundary value problem with integral conditions. Using Banach’s fixed point theorem and the Leray-Schauder nonlinear alternative, we discuss the existence and uniqueness solution of this problem, and we apply Guo-Krasnoselskii’s fixed point theorem in cone to study the existence of positive solution. We also give some examples to illustrate our results.

Existence of Positive Solutions for Nonlinear Second-Order Impulsive Boundary Value Problems on Time Scales

This paper is concerned with the existence of positive solutions of second-order impulsive boundary value problem with integral boundary conditions on time scales. Existence results of at least three positive solutions are established via a new fixed point theorem in a cone. Also, an example is given to illustrate the effectiveness of our result.

A Necessary and Sufficient Condition for Pseudo-symmetric Positive Solutions of Boundary Value Problems

We apply the monotone iterative technique to the second-order boundary value problems. We obtain a necessary and sufficient condition and discuss the uniqueness, a iterative sequence and an error estimation for pseudo-symmetric positive solutions. Moreover, an example is given to illustrate the applicability of our results.

Harmonic analysis for the characterization and correction of geometric distortion in MRI.

Magnetic resonance imaging (MRI) is gaining widespread use in radiation therapy planning, patient setup verification, and real-time guidance of radiation delivery. Successful implementation of these technologies relies on the development of simple and efficient methods to characterize and monitor the geometric distortions arising due to system imperfections and gradient nonlinearities. To this end, the authors present the theory and validation of a novel harmonic approach to the quantification of system-related distortions in MRI. The theory of spatial encoding in MRI is applied to demonstrate that the 3D distortion vector field (DVF) is given by the solution of a second-order boundary value problem (BVP). This BVP is comprised of Laplace's equation and a limited measurement of the distortion on the boundary of a specified region of interest (ROI). An analytical series expansion solving this BVP within a spherical ROI is obtained, and a statistical uncertainty analysis is performed to determine how random errors in the boundary measurements propagate to the ROI interior. This series expansion is then evaluated to obtain volumetric DVF mappings that are compared to reference data obtained on a 3 T full-body scanner. This validation is performed within two spheres of 20 cm diameter (one centered at the scanner origin and the other offset +3 cm along each of the transverse directions). Initially, a high-order mapping requiring measurements at 5810 boundary points is used. Then, after exploring the impact of the boundary sampling density and the effect of series truncation, a reduced-order mapping requiring measurements at 302 boundary points is evaluated. The volumetric DVF mappings obtained from the harmonic analysis are in good agreement with the reference data. Following distortion correction using the high-order mapping, the authors estimate a reduction in the mean distortion magnitude from 0.86 to 0.42 mm and from 0.93 to 0.39 mm within the central and offset ROIs, respectively. In addition, the fraction of points with a distortion magnitude greater than 1 mm is reduced from 35.6% to 2.8% and from 40.4% to 1.5%, respectively. Similarly, following correction using the reduced-order mapping, the mean distortion magnitude reduces to 0.45-0.42 mm within the central and offset ROIs, and the fraction of points with a distortion magnitude greater than 1 mm is reduced to 2.8% and 1.5%, respectively. A novel harmonic approach to the characterization of system-related distortions in MRI is presented. This method permits a complete and accurate mapping of the DVF within a specified ROI using a limited measurement of the distortion on the ROI boundary. This technique eliminates the requirement to exhaustively sample the DVF at a dense 3D array of points, thereby permitting the design of simple, inexpensive phantoms that may incorporate additional modules for auxiliary quality assurance objectives.

FINITE ELEMENT MODELING OF NONLINEAR WAVE TRANSFORMATION USING ELLIPTIC MILD SLOPE EQUATION

Accurate modeling of nonlinear wave transformation is important for studies related to harbor and nearshore design. While sophisticated finite-element models based on the elliptic mild-slope equation are often used for wave prediction, they do not include wave-wave interactions. These interactions, in general, involve transfer of energy and wave phase coupling among spectral components and are known to be quite significant especially in shoaling regions and inside harbors. To overcome this limitation, and to provide a basis for the eective modeling of nonlinear wave transformation in complex coastal and harbor environments, the development of a finite-element model based on the second-order nonlinear mild-slope equation of Kaihatu and Kirby (1995) is considered. The model uses an iterative procedure for solving the second-order boundary value problem. To ensure eective boundary treatment, a combination of frequently-used boundary conditions and the method of internal wave generation is used. Two cases involving nonlinear shoaling and harbor resonance are considered for model validation. Modeled results are compared with experimental data, and good agreement is observed in most cases. The methodology described in this paper can improve the applicability of existing finite-element based mild-slope models.

NONLINEAR WAVE EVOLUTION ABOVE RECTANGULAR SUBMERGED STRUCTURES

In this study, an analytic approach for the complete second-order solution proposed by Sulisz and Hudspeth [13] was applied to solve a problem of waves propagating over a rectangular submerged structure. In addition, nonlinear wave evolutions above the submerged structure were studied. The nonlinear problem was expressed up to the second-order boundary value problems by using a Taylor series expansion and the perturbation method. In solving the problem, the nonhomogeneous problem was divided into Stokes wave and free wave counterparts. The solutions of neighboring regions were combined and solved by applying kinematic and dynamic matching conditions. Convergence of the presented theory is examined. The experimental results with and without evanescent modes were compared with previous solutions and effects of evanescent modes can be identified. Further comparisons of the presented theory with previous experimental results also indicated favorable consistency. Using the presented theory, the second-order effects of structural submergence, relative water depth, and wave steepness on wave evolutions were investigated. Parametric studies have indicated that shallow water depths above the structure and shallow relative water depth induce high-shoaling second-order waves. In addition, the second-order wave evolution above the structure increased with the wave steepness.

Solvability for nonlocal boundary value problems on a half line with "Equation missing"

Abstract The existence of at least one solution to the second-order nonlocal boundary value problems on a half line is investigated by using Mawhin’s continuation theorem. MSC: 34B10, 34B40, 34B15.

Positive solutions of second-order non-local boundary value problem with singularities in space variables

We discuss a non-local boundary value problem of second-order, where the involved nonlinearity depends on the derivative and may be singular. The boundary conditions are given by Riemann-Stieltjes integrals. We establish sufficient conditions for the existence of positive solutions of the considered problem. Our approach is based on the Krasnoselskii-Guo fixed point theorem on cone expansion and compression.

Approximate solution of three-point boundary value problems for second-order ordinary differential equations with variable coefficients

Some three-point boundary value problems for a second-order ordinary differential equation with variable coefficients are investigated in the present paper. By using the integration method, the second-order three-point boundary value problems are transformed into a Fredholm integral equation of the second kind. The solutions and Green’s functions for some special cases of the second-order three-point boundary value problems can be determined easily. The existence and uniqueness of the solutions of the given Fredholm integral equations are considered by using the fixed point theorem in Banach spaces. A new numerical method is further proposed to solve the second-kind Fredholm integral equation and an approximate solution is made. The convergence and error estimate of the obtained approximate solution are further analyzed. Numerical results are carried out to verify the feasibility and novelty of the proposed solution procedures.

Modeling nonlinear wave–wave interactions with the elliptic mild slope equation

An approach is developed to simulate wave–wave interactions using nonlinear elliptic mild-slope equation in domains where wave reflection, refraction, diffraction and breaking effects must also be considered. This involves the construction of an efficient solution procedure including effective boundary treatment, modification of the nonlinear equation to resolve convergence issues, and validation of the overall approach. For solving the second-order boundary-value problem, the Alternating Direction Implicit (ADI) scheme is employed, and the use of approximate boundary conditions is supplemented, for improved accuracy, with internal wave generation method and dissipative sponge layers. The performance of the nonlinear model is investigated for a range of practical wave conditions involving reflection, diffraction and shoaling in the presence of nonlinear wave–wave interactions. In addition, the transformation of a wave spectrum due to nonlinear shoaling and breaking, and nonlinear resonance inside a rectangular harbor are simulated. Numerical calculations are compared with the results from other relevant nonlinear models and experimental data available in literature. Results show that the approach developed here performs reasonably well, and has thus improved the applicability of this class of wave transformation models.

The existence of symmetric positive solutions for a seconder-order difference equation with sum form boundary conditions

Abstract In this paper, we consider the existence of positive solutions for a second-order discrete boundary value problem Δ ( g ( k − 1 ) Δ u ( k − 1 ) ) + w ( k ) f ( k , u ( k ) ) = 0 subject to the boundary conditions: a u ( 0 ) − b g ( 0 ) Δ u ( 0 ) = ∑ i = 1 n − 1 h ( i ) u ( i ) , a u ( n ) + b g ( n − 1 ) Δ u ( n − 1 ) = ∑ i = 1 n − 1 h ( i ) u ( i ) , where a , b > 0 , Δ u ( k ) = u ( k + 1 ) − u ( k ) for k ∈ { 0 , 1 , … , n − 1 } , g ( k ) > 0 is symmetric on { 0 , 1 , … , n − 1 } , w ( k ) is symmetric on { 0 , 1 , … , n } , f : { 0 , 1 , … , n } × [ 0 , + ∞ ) is continuous, f ( k , u ) = f ( n − k , u ) for all ( k , u ) ∈ { 0 , 1 , … , n } × [ 0 , + ∞ ) , and h ( i ) is nonnegative and symmetric on { 0 , 1 , … , n } . By the fixed point theorem and the Hölder inequality, we study the existence of symmetric positive solutions for the above difference equation with sum form boundary conditions.

EXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR SECOND-ORDER BOUNDARY VALUE PROBLEMS WITH IMPULSE ACTIONS

In this paper we consider the existence of positive solutions for singular second-order boundary value problems with impulse actions. By con- structing a cone K1 × K2, which is the Cartesian product of two cones in the space C(0,1) and computing the fixed point index in the K1 × K2, we establish the existence of positive solutions for the system.

Existence of solution for a three-point boundary value problem for a second-order impulsive differential equation

In this paper, we study the existence of nontrivial solutions for the second-order three-point boundary value problem with impulses. Under certain growth conditions on the nonlinearity function and by using Leray-Schauder nonlinear alternative, sufficient conditions for the existence of a nontrivial solution are obtained. Furthermore, we illustrate our results with some examples.

Numerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equations

In this study, the numerical solution of Fredholm–Volterra integro-differential equations for two-point, second-order periodic boundary value problems is discussed in a reproducing kernel Hilbert space. A reproducing kernel Hilbert space is constructed, in which the periodic boundary conditions of the problem are satisfied. The exact solution u(x) is represented in the form of series in the space W23. In the mean time, the n-term approximate solution un(x) is obtained and is proved to converge to the exact solution u(x). Furthermore, we present an iterative method for obtaining the solution in the space W23. Some examples are displayed to demonstrate the validity and applicability of the proposed method. The numerical result indicates that the proposed method is straightforward to implement, efficient, and accurate for solving linear and nonlinear equations.

Sign-Changing and Multiple Solutions of Impulsive Boundary Value Problems Via Critical Point Methods

In this paper, we study the second-order impulsive boundary value problem ? Lu = f ( x , u ) , x ? [ 0 , 1 ] ? { x 1 , x 2 , ? , x l } , ? Δ ( p ( x i ) u ? ( x i ) ) = I i ( u ( x i ) ) , i = 1 , 2 , ? , l , R 1 ( u ) = 0 , R 2 ( u ) = 0 , $$\left\{\begin{array}{ll} -Lu=f(x, u), \, \, x\in [0, 1]\backslash\{x_{1}, x_{2}, \cdots, x_{l}\}, \\ -{\Delta} (p(x_{i}) u'(x_{i}))=I_{i}(u(x_{i})), \quad i=1, 2, \cdots, l, \\ R_{1}(u)=0, \, \, \, R_{2}(u)=0, \end{array}\right.$$ where Lu = (p(x)u?)? ? q(x)u is a Sturm-Liouville operator, R 1(u) = ?u?(0) ? βu(0) and R 2(u) = ?u?(1) + ?u(1). The existence of sign-changing and multiple solutions is obtained. The technical approach is based on minimax methods and invariant sets of descending flow.

A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods

This paper presents a comparative study on the newly introduced weak Galerkin finite element methods (WGFEMs) with the widely accepted discontinuous Galerkin finite element methods (DGFEMs) and the classical mixed finite element methods (MFEMs) for solving second-order elliptic boundary value problems. We examine the differences, similarities, and connection among these methods in scheme formulations, implementation strategies, accuracy, and computational cost. The comparison and numerical experiments demonstrate that WGFEMs are viable alternatives to MFEMs and hold some advantages over DGFEMs, due to their properties of local conservation, normal flux continuity, no need for penalty factor, and definiteness of discrete linear systems.

Existence of solutions for second-order periodic-integrable boundary value problems

The existence of solutions for a second-order periodic-integrable boundary value problem is discussed by a variational method, and the application to the forced oscillation equation is given.

Existence of positive solutions for p-Laplacian impulsive boundary value problems on time scales

In this paper, by using fixed point index theory and a double fixed point theorem, we study the existence of many positive solutions for a class of second-order p-Laplacian boundary value problems with impulse on time scales. An example which supports our theoretical results is also indicated.MSC:34B18, 34B37, 34K10.

A new efficient technique for solving two-point boundary value problems for integro-differential equations

In this paper, we propose a new efficient method based on a combination of Adomian decomposition method (ADM) and Green’s function for solving second-order boundary value problems (BVPs) for integro-differential equations (IDEs). The proposed method depends on constructing Green’s function before establishing the recursive scheme for the solution components. Unlike the ADM or modified ADM , the proposed method avoids solving a sequence of difficult nonlinear equations (transcendental equations) for the unknown parameters. The proposed method provides a direct recursive scheme for obtaining the series solution with easily calculable components. We also provide a sufficient condition that guarantees a unique solution to the second-order BVPs for IDEs. Convergence and error analysis of the proposed method are also discussed. Convergence analysis is reliable enough to estimate the error bound of the series solution. Some numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed approach. The numerical results reveal that the proposed method is very effective and simple.

Periodic boundary value problems for second-order functional differential equations with impulse

In this paper, we study the existence of multiple positive solutions of the second-order periodic boundary value problems for functional differential equations with impulse. The proof of our main results is based upon the fixed point index theorem in cones.