Second-order Boundary Value Problems Research Articles

Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm

In this paper, continuous genetic algorithm is introduced as an efficient solver for systems of second-order boundary value problems where smooth solution curves are used throughout the evolution of the algorithm to obtain the required nodal values of the unknown variables. The solution methodology is based on representing each derivative in the system of differential equations by its finite difference approximation. After that, the overall residue for all nodes in the given system of differential equations is formulated. The solution to the system of differential equations is then converted into the problem of minimizing the overall residue or maximizing the fitness function based on the nodal values generated from the genetic operators. Three numerical test problems including linear and nonlinear systems were analyzed to illustrate the procedure and confirm the performance of the proposed method. In addition to that, a convergence and sensitivity analysis to genetic operators and control parameters of the algorithm has been carried out. The numerical results show that the proposed algorithm is a robust and accurate procedure for solving systems of second-order boundary value problems. Furthermore, the obtained accuracy for the solutions using CGA is much better than the results obtained using some modern methods.

Existence of positive pseudo-symmetric solution for second-order three-point boundary value problems

The existence results of at least one positive pseudo-symmetric solution for a nonlinear second-order three-point boundary value problem are established by means of the fixed point theorem due to Avery, Anderson and Krueger.

Existence of three symmetric positive solutions for a second-order multi-point boundary value problem on time scales

AbstractIn this article, we investigate the existence of at least three symmetric positive solutions of a second-order multi-point boundary value problem on time scales. The ideas involve Bai and Ge’s fixed point theorem. As an application, we give an example to demonstrate our main result.MSC:34B10, 34B18, 39A10.

On the linear and nonlinear discrete second-order Neumann boundary value problems

By construct the spectrum of the linear eigenvalue problem, we are concerned with the global structure of the set of sign-changing solutions to the discrete second-order Neumann boundary value problem-Δ[p(t-1)Δu(t-1)]+q(t)u(t)=ra(t)f(u(t)),t∈[1,T]Z,Δu(0)=Δu(T)=0.Furthermore, we obtain the existence of the sign-changing solutions of the above nonlinear problem.

Exact multiplicity and stability of solutions of second-order Neumann boundary value problem

The purpose of this paper is to establish the exact multiplicity and stability of solutions of the equation u″+g(x,u)=f(x) with the Neumann boundary value conditions u′(0)=u′(1)=0. Exactly three ordered solutions are obtained by taking advantage of the anti-maximum principle combined with the methods of upper and lower solutions. Moreover, we obtain that one of three solutions is negative, while the other two are positive, the middle solution is unstable, and the remaining two are stable.

Superconvergence patch recovery for the gradient of the tensor-product linear triangular prism element

In this article, we study superconvergence of the finite element approximation to the solution of a general second-order elliptic boundary value problem in three dimensions over a fully uniform mesh of piecewise tensor-product linear triangular prism elements. First, we give the superclose property of the gradient between the finite element solution and the interpolant Πu. Second, we introduce a superconvergence recovery scheme for the gradient of the finite element solution. Finally, superconvergence of the recovered gradient is derived.

Green’s Function and Positive Solutions for a Second-Order Singular Boundary Value Problem with Integral Boundary Conditions and a Delayed Argument

This paper investigates the expression and properties of Green’s function for a second-order singular boundary value problem with integral boundary conditions and delayed argument-x′′t-atx′t+btxt=ωtft, xαt, t∈0, 1; x′0=0, x1-∫01htxtdt=0, wherea∈0, 1, 0, +∞, b∈C0, 1, 0, +∞and,ωmay be singular att=0or/and att=1. Furthermore, several new and more general results are obtained for the existence of positive solutions for the above problem by using Krasnosel’skii’s fixed point theorem. We discuss our problems with a delayed argument, which may concern optimization issues of some technical problems. Moreover, the approach to express the integral equation of the above problem is different from earlier approaches. Our results cover a second-order boundary value problem without deviating arguments and are compared with some recent results.

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.

Multiplicity of Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem with a Parameter

This paper is concerned with the following second-order three-point boundary value problemu″t+β2ut+λqtft,ut=0,t∈0 , 1,u0=0,u(1)=δu(η), whereβ∈(0,π/2),δ>0,η∈(0,1), andλis a positive parameter. First, Green’s function for the associated linear boundary value problem is constructed, and then some useful properties of Green’s function are obtained. Finally, existence, multiplicity, and nonexistence results for positive solutions are derived in terms of different values ofλby means of the fixed point index theory.

带积分边界条件的二阶非线性边值问题的正解

带积分边界条件的二阶非线性边值问题的正解

Existence of positive solutions for second-order impulsive time scale boundary value problems on infinite intervals

In this paper, we consider nonlinear second order m-point impulsive time scale boundary value problems on infinite intervals. By using Leray-Schauder fixed point theorem, Avery-Henderson fixed point theorem and the five functional fixed point theorem, respectively, we establish the criteria for the existence of at least one, two and three positive solutions to the nonlinear impulsive time scale boundary value problems on infinite intervals.

A Variational Principle for Three-Point Boundary Value Problems with Impulse

We construct a variational functional of a class of three-point boundary value problems with impulse. Using the critical points theory, we study the existence of solutions to second-order three-point boundary value problems with impulse.

Nodal Solutions for Some Second-Order Semipositone Integral Boundary Value Problems

Using bifurcation techniques, we first prove a global bifurcation theorem for nonlinear second-order semipositone integral boundary value problems. Then the existence and multiplicity of nodal solutions of the above problems are obtained. Finally, an example is worked out to illustrate our main results.

A new cubic B-spline method for approximating the solution of a class of nonlinear second-order boundary value problem with two dependent variables

In this paper, we will apply cubic B-splines on a uniform mesh to explore the numerical solutions and numerical derivatives of a class of nonlinear second-order boundary value problems with two dependent variables. Our new method is based on the cubic spline interpolation. The analytical solutions and any-order derivatives can be well approximated with 4th order accuracy. Furthermore, our new method is also able to solve general nonlinear 4th-order two-point boundary value problems. Numerical results show that our method is very practical and effective. a0(x)u 00 +a1(x)u 0 +a2(x)u +a3(x)v 00 +a4(x)v 0 +a5(x)v +G1(x;u;v) = f1(x); b0(x)v 00 +b1(x)v 0 +b2(x)v +b3(x)u 00 +b4(x)u 0 +b5(x)u +G2(x;u;v) = f2(x);

Application of Continuous Genetic Algorithm for Nonlinear System of Second-Order Boundary Value Problems

In this paper, a numerical algorithm, based on the use of genetic algorithm technique, is presented for solving a class of nonlinear systems of second-order boundary value problems. In this technique, the system is formulated as an optimization problem by the direct minimization of the overall individual residual error subject to the given constraints boundary condtions, and is then solved using continuous genetic algorithm. In general, the proposed technique uses smooth operators and avoids sharp jumps in the parameter values. The applicability, efficiency, and accuracy of the proposed alg orithm for the solution of different problems is investigated. Meanwhile, the convergence analysis based on the resulting statistical data is also discussed. emphasize that the continuous nature of the optimization problem and the continuity of the resulting solution curves) for the solution of the following nonlinear system of second-order BVPs (4): a1,0 (x)u 00(x) +a1,1 (x)u 0 (x) + a1,2 (x)u1(x) +a1,3 (x)u 00 (x) + a1,4 (x)u 0 (x) +a1,5 (x)u2(x) + G1 (x,u1(x),u2(x)) = f1 (x), a2,0 (x)u 00(x) +a2,1 (x)u 0(x) + a2,2 (x)u2(x) +a2,3 (x)u 00(x) + a2,4 (x)u 0(x) +a2,5 (x)u1(x) + G2 (x,u1(x),u2(x)) = f2 (x),

Some Existence Results of Positive Solution to Second-Order Boundary Value Problems

We study the existence of positive and monotone solution to the boundary value problemu′′(t)+f(t,u(t))=0,0⩽t⩽1,u(0)=ξu(1),u'(1)=ηu'(0), whereξ,η∈(0,1)∪(1,∞). The main tool is the fixed point theorem of cone expansion and compression of functional type by Avery, Henderson, and O’Regan. Finally, four examples are provided to demonstrate the availability of our main results.

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.

Existence results for p-Laplacian boundary value problems of impulsive dynamic equations on time scales

Abstract In this paper, Bai-Ge’s fixed point theorem is used to investigate the existence of positive solutions for second-order boundary value problems of p-Laplacian impulsive dynamic equations on time scales. As an application, we give an example to demonstrate our results. MSC:34B18, 34N05, 39B37.

Existence of positive solutions for a system of second-order multi-point discrete boundary value problems

We investigate the existence of positive solutions for a system of nonlinear second-order difference equations subject to multi-point boundary conditions.

First-order system least squares with inhomogeneous boundary conditions

We derive well-posed first-order system least-squares formulations of second-order elliptic boundary value problems, in which homogeneous essential boundary conditions are appended by means of additional equations, rather than by their incorporation in the trial space. This approach has the advantage that it applies equally well to inhomogeneous boundary conditions.