Second-order Two-point Boundary Value Problems Research Articles

A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems

A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems

A superconvergent ultra-weak discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems

A superconvergent ultra-weak discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems

Stiffness Mitigation in Stochastic Particle Flow Filters

The linear convex log-homotopy has been used in the derivation of particle flow filters. One natural question is whether it is beneficial to consider other forms of homotopy. We revisit this question by considering a general linear form of log-homotopy for which we derive particle flow filters, validate the distribution of flows, and obtain conditions for the stability of particle flows. We then formulate the problem of stiffness mitigation as an optimal control problem by minimizing the condition number of the Hessian matrix of the posterior density function. The optimal homotopy can be efficiently obtained by solving a 1-D second-order two-point boundary value problem. Compared with traditional matrix analysis based approaches to improving condition numbers, this approach explicitly exploits the special structure of the stochastic differential equations in particle flow filters. The effectiveness of the proposed approach is demonstrated by numerical examples.

Existence of solutions for boundary value problems and nonlinear matrix equations via F-contraction mappings in b-metric spaces

The purpose of this paper is to present some sufficient conditions for the existence of solutions for nonlinear matrix equations and the second-order two-point boundary value problems for nonlinear ordinary differential equations via fixed point theorem. To do this, we introduce two classes of mappings called the Suzuki–Geraghty-[Formula: see text]-[Formula: see text]-contraction mappings and generalize Suzuki-type [Formula: see text]-F-contraction mappings and establish some fixed point results for these classes of mappings. The results obtained in this work complements, extends, generalizes, and improves some known results in the literature.

A Multiple Variational Iteration Method for Nonlinear Two-Point Boundary Value Problems with Nonlinear Conditions

A multiple variational iteration method (VIM) is proposed to effectively solve the second-order nonlinear two-point boundary value problems. For problems where convergence speed of the original VIM is slow or the original method is divergent, the multiple VIM method (MVIM) presented in this paper can readily improve the rate of convergence.

Analysis of optimal superconvergence of a local discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems

In this paper, we investigate the convergence and superconvergence properties of a local discontinuous Galerkin (LDG) method for nonlinear second-order two-point boundary-value problems (BVPs) of the form u″=f(x,u,u′), x∈[a,b] subject to some suitable boundary conditions at the endpoints x=a and x=b. We prove optimal L2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the derivatives of the LDG solutions are superconvergent with order p+1 toward the derivatives of Gauss-Radau projections of the exact solutions. Moreover, we prove that the LDG solutions are superconvergent with order p+2 toward Gauss-Radau projections of the exact solutions. Finally, we prove, for any polynomial degree p, the (2p+1)th superconvergence rate of the LDG approximations at the upwind or downwind points and for the domain average under quasi-uniform meshes. Our numerical experiments demonstrate optimal rates of convergence and superconvergence. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p≥1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.

Septic B-spline method for solving nonlinear singular boundary value problems arising in physiological models

In this paper, we present a numerical method based on septic B-spline function for nonlinear singular second-order two-point boundary value problems, which depend on different physiological processes as thermal explosions problem and the steady state oxygen diffusion in a spherical cell with Michaelis–Menten uptake kinetics and distribution of heat sources in the human head. Septic B-spline method has a truncation error of O(h^8) and converges to the exact solution with O(h^6). The numerical problems show that our method is very effective. The resulting sets of differential equations are modified at the singular point and are treated by using septic B-spline for finding the numerical solution. The maximum absolute errors and the absolute residual errors are acceptable.

Numerical study of self-adjoint singularly perturbed two-point boundary value problems using collocation method with error estimation

A numerical treatment for self-adjoint singularly perturbed second-order two-point boundary value problems using trigonometric quintic B-splines is presented, which depend on different engineering applications. The method is found to have a truncation error of O(h6) and converges to the exact solution at O(h4). The numerical examples show that our method is very effective and the maximum absolute error is acceptable.

Stability and Convergence of Two-Step Obrechkoff Scheme For Second-Order Two-Point Boundary Value Problem

Mathematical modeling of scientific and engineering processes often yield Boundary Value Problems (BVPs). One of the broad categories of numerical methods for solving BVPs is the finite difference methods, in which the differential equation is replaced by a set of difference equations which are solved by direct or iterative methods. In this paper, we use some properties of matrices to analyze the stability and convergence of the prominent finite difference methods - two-step Obrechkoff method - for solving the boundary value problem $u^{\prime \prime} = f(t,u)$, $a < x < b$, $u(a) = \eta_1$, $u(b) = \eta_2$. Conditions for the stability and convergence of the two-step Obrechkoff method method were established.

A series solution of the Falkner–Skan equation using the crocco–wang transformation

A direct series solution for the Falkner–Skan equation is obtained by first transforming the problem using the Crocco–Wang transformation. The transformation converts the third-order problem to a second-order two-point boundary value problem. The method first constructs a series involving the unknown skin-friction coefficient [Formula: see text]. Then, [Formula: see text] is determined by using the secant method or Newton’s method. The derivative needed for Newton’s method is also computed using a series derived from the transformed differential equation. The method is validated by solving the Falkner–Skan equation for several cases reported previously in the literature.

Biologically inspired computing framework for solving two-point boundary value problems using differential evolution

In the present study, a design of biologically inspired computing framework is presented for solving second-order two-point boundary value problems (BVPs) by differential evolution (DE) algorithm employing finite difference-based cost function. The DE has been implemented to minimize the combined residue from all nodes in a least square sense. The proposed methodology has been evaluated using five numerical examples in linear and nonlinear regime of BVPs in order to demonstrate the process and check the efficacy of the implementation. The assessment and validation of the DE algorithm have been carried out by comparing the DE-computed results with exact solution as well as with the corresponding data obtained using continuous genetic algorithms. These benchmark comparisons clearly establish DE as a competitive solver in this domain in terms of computational competence and precision.

A simple finite element method for boundary value problems with a Riemann–Liouville derivative

We consider a boundary value problem involving a Riemann–Liouville fractional derivative of order α∈(3/2,2) on the unit interval (0,1). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term xα−1 in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value problem with nonlocal low order terms, whose solution can reconstruct directly the solution to the original problem. The stability of the variational formulation, and the optimal regularity pickup of the solution are analyzed. A novel Galerkin finite element method with piecewise linear or quadratic finite elements is developed, and L2(D) error estimates are provided. The approach is then applied to the corresponding fractional Sturm–Liouville problem, and error estimates of the eigenvalue approximations are given. Extensive numerical results fully confirm our theoretical study.

Three-Step Block Method for Solving Nonlinear Boundary Value Problems

We propose a three-step block method of Adam’s type to solve nonlinear second-order two-point boundary value problems of Dirichlet type and Neumann type directly. We also extend this method to solve the system of second-order boundary value problems which have the same or different two boundary conditions. The method will be implemented in predictor corrector mode and obtain the approximate solutions at three points simultaneously using variable step size strategy. The proposed block method will be adapted with multiple shooting techniques via the three-step iterative method. The boundary value problem will be solved without reducing to first-order equations. The numerical results are presented to demonstrate the effectiveness of the proposed method.

Self-adjoint singularly perturbed second-order two-point boundary value problems: A patching approach

The basic aim of this study is to introduce and describe a patching approach based on a novel combination of the variational iterative method (VIM) and adaptive cubic spline collocation scheme for the solution of a class of self-adjoint singularly perturbed second-order two-point boundary value problems that model various engineering problems. The domain of the problem is decomposed into two subintervals: the VIM is implemented in the vicinity of the boundary layer while in the outer region the resulting problem is tackled by applying an adaptive cubic spline collocation scheme (ASS), which comprises the use of mapping/transformation redistribution functions or constructed grading functions.Numerical results, computational comparisons, appropriate error measures and illustrations are provided to testify the convergence, efficiency and applicability of the method. Performance of this method is examined through test examples that reveal that the current approach converges to the exact solution rapidly and with O(h4) accuracy and that the convergence is uniform across the domain. The proposed technique yields numerical solutions in very good agreement with and/or superior to existing exact and approximate solutions.

Sinc-Galerkin and Sinc-Collocation methods in the solution of nonlinear two-point boundary value problems

A comparative study of the Sinc-Galerkin and Sinc-Collocation methods for solving singular and nonsingular nonlinear second-order two-point boundary value problems (BVPs) with nonhomogeneous boundary conditions is given. We developed the Sinc-Galerkin and Sinc-Collocation methods to approximate the nonlinear two-point BVPs. These method are tested on the test examples and compared with several existing methods. The demonstrated results show that both of the presented methods more or less have the same accuracy, but in comparison with the other existing methods are considerable accurate.

Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach

Using an elementary approach, we prove the existence of three positive and concave solutions of the second-order two-point boundary-value problem $$ \begin{array}{lll}&& x^{\prime\prime}(t)=\alpha(t)f(t,x(t),x^{\prime}(t)),\qquad 0 < t < 1,\\&& \qquad \qquad \qquad x(0)=x(1)=0.\end{array}$$ We rely on the analysis of the corresponding vector field on the phase space, Kneser-type properties of the solution funnel, and the Schauder fixed-point theorem. The obtained results demonstrate the simplicity and efficiency (one could study a problem with more general boundary conditions) of our new approach as compared with the commonly used ones, such as the Leggett–Williams fixed-point theorem and its generalizations.

New Spectral Second Kind Chebyshev Wavelets Algorithm for Solving Linear and Nonlinear Second-Order Differential Equations Involving Singular and Bratu Type Equations

A new spectral algorithm based on shifted second kind Chebyshev wavelets operational matrices of derivatives is introduced and used for solving linear and nonlinear second-order two-point boundary value problems. The main idea for obtaining spectral numerical solutions for these equations is essentially developed by reducing the linear or nonlinear equations with their initial and/or boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. Convergence analysis and some efficient specific illustrative examples including singular and Bratu type equations are considered to demonstrate the validity and the applicability of the method. Numerical results obtained are compared favorably with the analytical known solutions.

Cubic Spline Solutions for Two-Point Boundary Value Problems Using Quarter-Sweep SOR Method

In this paper, iterative methods particularly a family of Successive Over Relaxation (SOR) methods are used to solve system of linear equations generated from discretization of second-order two-point boundary value problems through cubic spline approaches. For the proposed problems, family of SOR methods namely Full-Sweep SOR (FSSOR), Half-Sweep SOR (HSSOR) and Quarter-Sweep SOR (QSSOR) has been considered to be the generated linear solver. In addition, the formulation and implementation of these three proposed methods were also presented. Comparison among all tested SOR methods were carried out to show their performance.

Convergence analysis for least-squares finite element approximations of second-order two-point boundary value problems

In this paper, a C0 least-squares finite element method for second-order two-point boundary value problems is considered. The problem is recast as a first-order system. Standard and improved optimal error estimates in maximum-norms are established. Superconvergence estimates at interelement, Lobatto, and Gauss points are developed. Numerical experiments are given to illustrate theoretical results.

Nodal solutions of second-order two-point boundary value problems

Abstract We shall study the existence and multiplicity of nodal solutions of the nonlinear second-order two-point boundary value problems, u ″ + f ( t , u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 . The proof of our main results is based upon bifurcation techniques. Mathematics Subject Classifications: 34B07; 34C10; 34C23.