Second-order Boundary Value Problems Research Articles

A High-Order Numerical Method for a Nonlinear System of Second-Order Boundary Value Problems

This paper is concerned with a high-order numerical scheme for nonlinear systems of second-order boundary value problems (BVPs). First, by utilizing quasi-Newton’s method (QNM), the nonlinear system can be transformed into linear ones. Based on the standard Lobatto orthogonal polynomials, we introduce a high-order Lobatto reproducing kernel method (LRKM) to solve these linear equations. Numerical experiments are performed to investigate the reliability and efficiency of the presented method.

Positive solutions for a system of second-order quasilinear boundary value problems

In this paper we study the existence and multiplicity of positive solutions for the system of second-order quasilinear boundary value problems −((u′)p−1)′=f(t,u,v),−((v′)q−1)′=g(t,u,v),u(0)=u′(1)=0,v(0)=v′(1)=0,where p,q>1 and f,g∈C([0,1]×R+2,R+)(R+≔[0,∞)). Based on a priori estimates achieved by utilizing Jensen’s inequality for nonnegative concave functions and homogeneous operators, we use fixed point index theory to establish the main results.

Numerical Study on Single and Multi-Dimensional Boundary Value Problems by the Method of Weighted Residual

The Galerkin method of weighted residual (MWR) for computing natural frequencies of some physical problems such as the Helmholtz equation and some second-order boundary value problems has been revised in this article. The use of one and two-dimensional characteristic Bernstein polynomials as the basis functions have been presented by the Galerkin MWR. The vibration of non-homogeneous membranes with Dirichlet type boundary conditions is also studied here. The useful properties of Bernstein polynomials, its derivatives and function approximations have also been illustrated. Besides, the efficiency and applicability of the proposed technique have been demonstrated through some numerical experiments.

Reversed <i>S</i> -shaped connected component for second-order periodic boundary value problem with sign-changing weight

In this paper, we consider the existence of an reversed S -shaped connected component in the set of positive solutions for second order periodic boundary value problem with a sign-changing weight function. By bifurcation technique, we identify the interval of bifurcation parameter in which the periodic boundary value problem has one or two or three positive solutions according to the asymptotic behavior of f at 0 and ∞.

Solving system of second-order BVPs using a new algorithm based on reproducing kernel Hilbert space

In this study, a new and reliable technique based on Reproducing Kernel Method (RKM) is introduced to solve the linear and nonlinear system of second-order boundary value problem. We propose that RKM without Gram-Schmidt orthogonalization process significantly reduces the CPU time and increases the accuracy of the approximate solutions. The convergence theorem and error analysis are also verified. An algorithm is presented to indicate the steps of implementing the proposed scheme. Since the system of second-order boundary value problem is solvable with the numerical commands in Mathematica software, we examined these commands for numerical examples, compared the results with the present method, and demonstrated the higher precision of the present algorithm.

Smooth solutions of the surface semi-geostrophic equations

The semi-geostrophic equations have attracted the attention of the physical and mathematical communities since the work of Hoskins in the 1970s owing to their ability to model the formation of fronts in rotation-dominated flows, and also to their connection with optimal transport theory. In this paper, we study an active scalar equation, whose activity is determined by way of a Neumann-to-Dirichlet map associated to a fully nonlinear second-order Neumann boundary value problem on the infinite strip $${\mathbb {R}}^2\times (0,1)$$, that models a semi-geostrophic flow in regime of constant potential vorticity. This system is an expression of an Eulerian semi-geostrophic flow in a coordinate system originally due to Hoskins, to which we shall refer as Hoskins’ coordinates. We obtain results on the local-in-time existence and uniqueness of classical solutions of this active scalar equation in Holder spaces.

A Multiple-Input Deep Neural Network Architecture for Solution of One-Dimensional Poisson Equation

In this letter, we demonstrate that a multiple-input deep neural network architecture can be used for the solution of a one-dimensional (1-D) second-order boundary value problem. We investigate the solution of the 1-D Poisson equation, while using sinc- and cosine-type functions to emulate typically found electromagnetic field distributions. Network architecture, modeling of the derivative, and boundary condition criteria are implemented, and test cases are used for validation. For the considered second-order boundary value problems, we obtain $-$ 80 dB error convergence in 8.2 s, showing a successful demonstration of the method. We further investigate the effect of the number of nodes, number of layers, and learning rate on the convergence of the method.

Applications of maximum modulus method and Phragm\xe9n\u2013Lindel\xf6f method for second-order boundary value problems with respect to the Schr\xf6dinger operator

In this paper, we present a reliable combination of the maximum modulus method with respect to the Schrödinger operator (Meng in J. Syst. Sci. Complex. 16:446–452, 2003) and Phragmén–Lindelöf method (Shehu in Matematiche 64:57–66, 2015) to investigate the solution of a second-order boundary value problem with respect to the Schrödinger operator. We establish the uniqueness of the solution for this problem. The results reveal that this method is effective and simple.

Positive solutions to singular second-order boundary value problems for dynamic equations

We study singular second-order boundary value problems with mixed boundary conditions on an infinitely discrete time scale. We prove the existence of a positive solution by means of a lower and upper solutions method and the Brouwer fixed-point theorem, in conjunction with perturbation methods used to approximate regular problems.

English

Consider a second-order elliptic boundary value problem in three dimensions with locally smooth coefficients and solution. Discuss local superconvergence estimates for the tensor-product finite element approximation on a regular family of rectangular meshes. It will be shown that, by the estimates for the discrete Green’s function and discrete derivative Green’s function, and the relationship of norms in the finite element space such as L2 -norms, W1,∞ -norms, and negative-norms in locally smooth subsets of the domain Ω, locally pointwise superconvergence occurs in function values and derivatives.

Solutions to nonlinear second-order three-point boundary value problems of dynamic equations on time scales

In this paper, we consider existence criteria of three positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales. To show our main results, we apply the well-known Leggett-Williams fixed point theorem. Moreover, we present some results for the existence of single and multiple positive solutions for boundary value problems on time scales, by applying fixed point theorems in cones. The conditions we used in the paper are different from those in [Dogan A. On the existence of positive solutions for the one-dimensional $ p $-Laplacian boundary value problems on time scales. Dynam Syst Appl 2015; 24: 295-304].

Non-optimal fourth-order and optimal sixth-order B-spline collocation methods for Lane-Emden boundary value problems

In the present work, the authors describe two B-spline collocation methods for a class of nonlinear singular Lane-Emden type equations which describe several phenomena in theoretical physics and astrophysics. The first method is based on non-optimal quintic B-spline collocation approach, while the second method is based on optimal quintic B-spline collocation approach. We note that the former method yields fourth-order convergent approximation to the solution of the second-order boundary value problems. To obtain an optimal convergence of order six to the solution of the same problem, the later method is designed by perturbing the original problem. Convergence results for both methods are established via Green's function approach. Four nonlinear examples are provided to illustrate the suggested methods and to verify their theoretical rates of convergence. It is observed that the first method yields non-optimal fourth order accuracy and the second method exhibits optimal sixth order accuracy for the solution of the problem. Moreover, numerical results appear to be higher accurate when compared to other methods.

Fuzzy reproducing kernel space method for solving fuzzy boundary value problems

In the beginning, we describe the fuzzy inner product space and the fuzzy Hilbert space. Our goal is to use the fuzzy reproducing kernel method to solve the second-order fuzzy boundary value problem. The fuzzy convergence analysis of introduced method is discussed in detail. We present some examples in the end.

Numerical approximation of fractional powers of elliptic operators

Abstract In this paper, we develop and study algorithms for approximately solving linear algebraic systems: ${{\mathcal{A}}}_h^\alpha u_h = f_h$, $ 0< \alpha <1$, for $u_h, f_h \in V_h$ with $V_h$ a finite element approximation space. Such problems arise in finite element or finite difference approximations of the problem $ {{\mathcal{A}}}^\alpha u=f$ with ${{\mathcal{A}}}$, for example, coming from a second-order elliptic operator with homogeneous boundary conditions. The algorithms are motivated by the method of Vabishchevich (2015, Numerically solving an equation for fractional powers of elliptic operators. J. Comput. Phys., 282, 289–302) that relates the algebraic problem to a solution of a time-dependent initial value problem on the interval $[0,1]$. Here we develop and study two time-stepping schemes based on diagonal Padé approximation to $(1+x)^{-\alpha }$. The first one uses geometrically graded meshes in order to compensate for the singular behaviour of the solution for $t$ close to $0$. The second algorithm uses uniform time stepping, but requires smoothness of the data $f_h$ in discrete norms. For both methods, we estimate the error in terms of the number of time steps, with the regularity of $f_h$ playing a major role for the second method. Finally, we present numerical experiments for ${{\mathcal{A}}}_h$ coming from the finite element approximations of second-order elliptic boundary value problems in one and two spatial dimensions.

On positive solutions for some second-order three-point boundary value problems with convection term

In this paper, a fixed point theorem in a cone and some inequalities of the associated Green’s function are applied to obtain the existence of positive solutions of second-order three-point boundary value problem with dependence on the first-order derivative \t\t\tx″(t)+f(t,x(t),x′(t))=0,0<t<1,x(0)=0,x(1)=μx(η),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& x''(t) + f\\bigl(t, x(t), x'(t)\\bigr) =0, \\quad 0< t< 1, \\\\& x(0) =0, \\qquad x(1) =\\mu x(\\eta ), \\end{aligned}$$ \\end{document} where f: [0, 1] times [0, infty ) times R rightarrow [0, infty ) is a continuous function, mu >0, eta in (0, 1), mu eta <1. The interesting point is that the nonlinear term is dependent on the convection term.

A third-derivative two-step block Falkner-type method for solving general second-order boundary-value systems

In this article, a third derivative continuous 2-step block Falkner-type method for the general solution of second order boundary value problems of ordinary differential equations (ODEs) with different types of boundary conditions is developed. The approaches of collocation and interpolation are adopted to derive the new Falkner-type method, which is then implemented in a block mode to get approximations at all the grid points simultaneously. This method is said to be a global method since it simultaneously produces a solution over the entire interval, although it may also be categorized as a boundary value method (see Brugnano and Trigiante (1998)). The order and the convergence analysis of the proposed method are studied. The new Falkner-type scheme is applied to solve linear and non-linear systems of second-order boundary value problems of ODEs considering different types of boundary conditions. Numerical results obtained through the implementation of the scheme are very much close to the theoretical solution and found favourably compared with various existing methods in the literature.

Exponential spline method for approximation solution of Fredholm integro-differential equation

ABSTRACT In this paper, we approximate the solution of Fredholm integro-differential equations of the second kind by using exponential spline function. The proposed method reduces to the system of algebraic equations. The convergence analysis of the method has been discussed. The numerical examples are presented to illustrate the applications of the method and to compare the computed results with the method in Chen et al. [A multiscale Galerkin method for second-order boundary value problems of Fredholm integro-differential equation, J. Comput. Appl. Math. 290 (2015), pp. 633–640].

Fast computation of the spectral differentiation by the fast multipole method

This paper is concerned with the application of the fast multipole method (FMM) to the pseudospectral methods. The spectral differentiation in pseudospectral methods involves (dense) matrix–vector products, which, if evaluated directly, entail O(N2) operations. The core of this work lies in the fact that the FMM leads to a significant speed-up in CPU time with a major reduction in the amount of computer memory needed when performing matrix–vector products. The FMM is exploited to estimate the barycentric weights and the pseudospectral differentiation, which reduces the complexity to O(N). Furthermore, we show the convergence analysis and present the applications to the second-order boundary value problems. Numerical examples confirm the advantages of the method.

On multipoint resonant problems on the half-line

In this work we obtain sufficient conditions for the existence of bounded solutions of a resonant multipoint second-order boundary value problem, with a fully differential equation.The noninvertibility of the linear part is overcome by a new perturbation technique, which allows obtaining an existence result and a localization theorem. Our hypotheses are clearly much less restrictive than those existent in the literature and, moreover, they can be applied to higher-order, resonant or nonresonant, boundary value problems defined on the half-line or even on the real line.

Analytic Solution of a Class of Singular Second-Order Boundary Value Problems with Applications

Recently, it was observed that the concentration/heat transfer of pure/nano fluids are finally governed by singular second-order boundary value problems with exponential coefficients. These coefficients were transformed into polynomials and therefore the governing equations become singular in a new independent variable. Unfortunately, the published approximate solutions in the literature suffer from some weaknesses as showed by one of the present coauthors. Hence, the exact solution for such types of problems becomes a challenge. In this paper, a straightforward approach is presented to obtaining the exact solution for such class of singular second-order boundary value problems. The results are also applied to some selected problems within the literature. Accordingly, the published solutions are recovered as special cases of the present ones.