Second-order Boundary Value Problems Research Articles

Multiple positive solutions for a second-order boundary value problem on the half-line

Multiple positive solutions for a second-order boundary value problem on the half-line

Trigonometric symmetric boundary value method for oscillating solutions including the sine-Gordon and Poisson equations

We construct a continuous linear multistep method with trigonometric coefficients from which a symmetric main method as well as additional methods are reproduced. The main and additional methods whose coefficients depend on the frequency and step length are then applied as a trigonometric symmetric boundary value method (SBVM) to solve systems of second-order initial and boundary value problems of the form y″=f(x,y) without first reducing the ordinary differential equation into an equivalent first-order system. Moreover, the method is successfully applied to solve hyperbolic and elliptic partial differential equations, such as the sine-Gordon and the Poisson equations. The stability property of the SBVM is discussed and numerical experiments are performed to show the accuracy of the method.

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.

A Fixed-Point Iteration Approach for Solving a BVP Arising in Chemical Reactor Theory

A numerical scheme aimed at solving a broad class of second-order nonlinear boundary value problems (BVPs) is presented, described, and then applied to a problem that appears in chemical reactor theory. In particular, the steady-state solution of the equation governing the model of the adiabatic tubular chemical reactor is investigated. The proposed method is based on expressing the particular solution of the governing equation in terms of an integral involving Green’s function. Then, the Krasnoselskii–Mann’s fixed-point scheme is implemented to an amended version of the resulting integral operator. The convergence of the iterative scheme is proved and its efficiency and applicability are demonstrated by solving the equation for selected values of the parameters that appear in the model. Residual error computation is adopted to confirm the accuracy of the results. In addition, the numerical outcomes of the proposed method are compared with those obtained by other existing numerical approaches.

The extended generalized Störmer–Cowell methods for second-order delay boundary value problems

This paper deals with the numerical solutions of second-order delay boundary value problems (DBVPs). The generalized Störmer–Cowell methods (GSCMs) for second-order initial value problems, proposed by Aceto et al. (2012), are extended to solve the second-order DBVPs. The existence and uniqueness criterion of the methods is derived. It is proved under the suitable conditions that an extended GSCM is stable, and convergent of order p whenever this method has the consistent order p. The numerical examples illustrate efficiency and accuracy of the methods. Moreover, a comparison between the extended GSCMs and the boundary value methods of first-order BVPs is given. The numerical result shows that the extended GSCMs are comparable.

Existence and approximation of solution for a nonlinear second-order three-point boundary value problem

A nonlinear second-order ordinary differential equation with four cases of three-point boundary value conditions is studied by investigating the existence and approximation of solutions. First, the integration method is proposed to transform the considered boundary value problems into Hammerstein integral equations. Second, the existence of solutions for the obtained Hammerstein integral equations is analyzed by using the Schauder fixed point theorem. The contraction mapping theorem in Banach spaces is further used to address the uniqueness of solutions. Third, the approximate solution of Hammerstein integral equations is constructed by using a new numerical method, and its convergence and error estimate are analyzed. Finally, some numerical examples are addressed to verify the given theorems and methods.

Iterative Galerkin discretizations for strongly monotone problems

In this article we investigate the use of fixed point iterations to solve the Galerkin approximation of strictly monotone problems. As opposed to Newton’s method, which requires information from the previous iteration in order to linearize the iteration matrix (and thereby to recompute it) in each step, the alternative method used in this article exploits the monotonicity properties of the problem, and only needs the iteration matrix calculated once for all iterations of the fixed point method. We outline the abstract a priori and a posteriori analyses for the iteratively obtained solutions, and apply this to a finite element approximation of a second-order elliptic quasilinear boundary value problem. We show both theoretically, as well as numerically, how the number of iterations of the fixed point method can be restricted in dependence of the mesh size, or of the polynomial degree, to obtain optimal convergence. Using the a posteriori error analysis we also devise an adaptive algorithm for the generation of a sequence of Galerkin spaces (adaptively refined finite element meshes in the concrete example) to minimize the number of iterations on each space.

A New Collocation Scheme Using Non-polynomial Basis Functions

In this paper, we construct a set of non-polynomial basis functions from a generalised Birkhoff interpolation problem involving the operator: \({\mathscr {L}}_\lambda ={d^2}/{dx^2}-\lambda ^2 \) with constant \(\lambda .\) With a direct inverting the operator, the basis can be pre-computed in a fast and stable manner. This leads to new collocation schemes for general second-order boundary value problems with (i) the matrix corresponding to the operator \({\mathscr {L}}_\lambda \) being identity; (ii) well-conditioned linear systems and (iii) exact imposition of various boundary conditions. This also provides efficient solvers for time-dependent nonlinear problems. Moreover, we can show that the new basis has the approximability to general functions in Sobolev spaces as good as orthogonal polynomials.

Extended cubic B-spline method for solving a linear system of second-order boundary value problems.

A method based on extended cubic B-spline is proposed to solve a linear system of second-order boundary value problems. In this method, two free parameters, lambda _{1} and lambda _{2}, play an important role in producing accurate results. Optimization of these parameters are carried out and the truncation error is calculated. This method is tested on three examples. The examples suggest that this method produces comparable or more accurate results than cubic B-spline and some other methods.

Optimal Spectral Schemes Based on Generalized Prolate Spheroidal Wave Functions of Order $$-1$$

We introduce a family of generalized prolate spheroidal wave functions (PSWFs) of order $$-1,$$ and develop new spectral schemes for second-order boundary value problems. Our technique differs from the differentiation approach based on PSWFs of order zero in Kong and Rokhlin (Appl Comput Harmon Anal 33(2):226–260, 2012); in particular, our orthogonal basis can naturally include homogeneous boundary conditions without the re-orthogonalization of Kong and Rokhlin (2012). More notably, it leads to diagonal systems or direct “explicit” solutions to 1D Helmholtz problems in various situations. Using a rule optimally pairing the bandwidth parameter and the number of basis functions as in Kong and Rokhlin (2012), we demonstrate that the new method significantly outperforms the Legendre spectral method in approximating highly oscillatory solutions. We also conduct a rigorous error analysis of this new scheme. The idea and analysis can be extended to generalized PSWFs of negative integer order for higher-order boundary value and eigenvalue problems.

Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method

The reproducing kernel method is a numerical as well as analytical technique for solving a large variety of ordinary and partial differential equations associated to different kind of boundary conditions, and usually provides the solutions in term of rapidly convergent series in the appropriate Hilbert spaces with components that can be elegantly computed. The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions for systems of second-order differential equations with periodic boundary conditions. A reproducing kernel space is constructed in which the periodic conditions of the systems are satisfied, whilst, the smooth kernel functions are used throughout the evolution of the method to obtain the required grid points. An efficient construction is given to obtain the approximate solutions for the systems together with an existence proof of the exact solutions is proposed based upon the reproducing kernel theory. Convergence analysis and error behavior of the presented method are also discussed. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed.

SPECIAL SPLINES OF EXPONENTIAL TYPE FOR THE SOLUTIONS OF MASS TRANSFER PROBLEMS IN MULTILAYER DOMAINS

We consider averaging methods for solving the 3-D boundary-value problem of second order in multilayer domain. The special hyperbolic and exponential type splines, with middle integral values of piece-wise smooth function interpolation are considered. With the help of these splines the problems of mathematical physics in 3-D with piece-wise coefficients are reduced with respect to one coordinate to 2-D problems. This procedure also allows to reduce the 2-D problems to 1-D problems and the solution of the approximated problemsa can be obtained analytically. In the case of constant piece-wise coefficients we obtain the exact discrete approximation of a steady-state 1-D boundary-value problem. The solution of corresponding averaged 3-D initial-boundary value problem is also obtained numerically, using the discretization in space with the central diferences. The approximation of the 3-D nonstationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerical solution is compared with the analytical solution.

A Novel Numerical Method for Fuzzy Boundary Value Problems

In the present paper, a new numerical method is proposed for solving fuzzy differential equations which are utilized for the modeling problems in science and engineering. Fuzzy approach is selected due to its important applications on processing uncertainty or subjective information for mathematical models of physical problems. A second-order fuzzy linear boundary value problem is considered in particular due to its important applications in physics. Moreover, numerical experiments are presented to show the effectiveness of the proposed numerical method on specific physical problems such as heat conduction in an infinite plate and a fin.

A Schauder-type theorem for discontinuous operators with applications to second-order BVPs

We prove a new fixed point theorem of Schauder-type which applies to discontinuous operators in non-compact domains. In order to do so, we present a modification of a recent Schauder-type theorem due to Pouso. We apply our result to second-order boundary value problems with discontinuous nonlinearities. We include an example to illustrate our theory.

Positive Green’s function and triple positive solutions of a second-order impulsive differential equation with integral boundary conditions and a delayed argument

In this paper, we first establish the expression of positive Green’s function for a second-order impulsive differential equation with integral boundary conditions and a delayed argument. Furthermore, applying Legget-William’s fixed point theorem and Holder’s inequality, we obtain the existence results of at least three positive solutions under three cases: $p=1$ , $1< p<+\infty$ , and $p=+\infty$ . We discuss our problem with impulsive effects and a delayed argument. In this case, our results cover second-order boundary value problems without impulsive effects and delayed arguments and are compared with some recent results. Finally, we give an example to illustrate our main results.

A Convection-Siffusion Type Boundary Value Problem to Determine the Expected Time for the Generation of Action Potentials in Nerve Cells by Random Synaptic Inputs in the Dendrites

According to Lange and Miura [1], the determination of the expected time for the generation of action potentials in nerve cells by random synaptic inputs in the dendrites can be modeled as general boundary value problem for the linear second order differential difference equation with prescribed boundary condition. This biological problem motivates the study of boundary value problems for singularly perturbed differential difference equation with delay. In this paper, a boundary value problem for a second-order singularly perturbed delay differential equation which models the above biological problem is considered on [0,2]. The solution of this problem exhibits an initial layer at 0 and an interior layer at 1. A numerical method composed of a classical finite difference scheme applied on a piecewise-uniform Shishkin mesh is suggested to solve the problem. The biological implications are studied. [1] C.G.Lange and R.M.Miura, \textit{ Singular perturbation analysis of oundary-value problems for differential-difference equations.V. small shifts with layer behavior }, SIAM J.Appl.Math, Vol.54 1 249--272 (1994).

A Priori Error Estimates of C rank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem

In this article, a Crank–Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O(h2 + k2) is proved for the numerical solution of the control, state and adjoint-state in a discrete L2-norm. To derive this result, we also get the error estimate (convergent order O(h2 + k2)) of Crank–Nicolson finite volume element approximation for the second-order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1331–1356, 2016

Multiple positive solutions for a second-order boundary value problem with integral boundary conditions

In view of the Avery-Peterson fixed point theorem, this paper investigates the existence of three positive solutions for the second-order boundary value problem with integral boundary conditions $$\left \{ \textstyle\begin{array}{@{}l} u''(t)+h(t)f(t,u(t),u'(t))=0,\quad 0< t< 1, u(0)-\alpha u'(0)=\int_{0}^{1}g_{1}(s)u(s)\,ds, u(1)+\beta u'(1)=\int_{0}^{1}g_{2}(s)u(s)\,ds. \end{array}\displaystyle \right . $$ The interesting point is that the nonlinear term involves the first-order derivative explicitly.

A numerical solution of a singular boundary value problem arising in boundary layer theory.

In this paper, a second-order nonlinear singular boundary value problem is presented, which is equivalent to the well-known Falkner–Skan equation. And the one-dimensional third-order boundary value problem on interval [0,infty ) is equivalently transformed into a second-order boundary value problem on finite interval [beta , 1]. The finite difference method is utilized to solve the singular boundary value problem, in which the amount of computational effort is significantly less than the other numerical methods. The numerical solutions obtained by the finite difference method are in agreement with those obtained by previous authors.

Fractional differential equations and inclusions with semiperiodic and three-point boundary conditions

In this article, we investigate the existence of solutions for boundary value problems of fractional differential equations and inclusions with semiperiodic and three-point boundary conditions. The existence results for equations are obtained by applying Banach’s contraction mapping principle, Schaefer-type fixed point theorem, Leray-Schauder degree theory, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative, whereas the existence of solutions for convex and nonconvex set-valued maps (inclusion case) is shown via nonlinear alternative of Leray-Schauder type for multivalued maps and Wegrzyk’s fixed point theorem for generalized contractions, respectively. We emphasize that a variety of fixed point theorems are used to obtain different existence criteria for the problems at hand. Several examples are discussed for illustration of the obtained results. Moreover, an interesting observation related to symmetric second-order three-point boundary value problems is presented.