Nonlinear Second-order Boundary Value Problems Research Articles

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);

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

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.

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.

Triple positive periodic solutions of nonlinear singular second-order boundary value problems

This paper studies the positive solutions of the nonlinear second-order periodic boundary value problem $$u''\left( t \right) + \lambda \left( t \right)u\left( t \right) = f\left( {t,u\left( t \right)} \right), a.e. t \in \left[ {0,2\pi } \right], u\left( 0 \right) = u\left( {2\pi } \right),u'\left( 0 \right) = u'\left( {2\pi } \right),$$ where f(t, u) is a local Caratheodory function. This shows that the problem is singular with respect to both the time variable t and space variable u. By applying the Leggett-Williams and Krasnosel’skii fixed point theorems on cones, an existence theorem of triple positive solutions is established. In order to use these theorems, the exact a priori estimations for the bound of solution are given, and some proper height functions are introduced by the estimations.

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.

Positive Solutions to System of Nonlinear Second-Order Three-Point Boundary Value Problem

In this paper,we investigate the following system of nonlinear second-order three-point boundary value problems — u~" = f(t,v),t ∈(0,1),— v″ = g(t,u),t ∈(0,1),u(0) = αu(η),u(1) =βu(η),u(0) = av(η),v(1) = βv(η),where f,g ∈C([0,1]x R~+,R~+),g{t,0)≡ 0,η∈(0,1) and0 β≤α 1.First,Green's function for associated liner boundary value problem is constructed;next,several useful properties of the Green's function are obtained;in the last place,some existence and multiplicity criteria of positive solutions are established by using the fixed point theorems of cone expansion and compression.

Computation of Positive Solutions for Nonlinear Impulsive Integral Boundary Value Problems withp-Laplacian on Infinite Intervals

This paper deals with the existence and iteration of positive solutions for nonlinear second-order impulsive integral boundary value problems withp-Laplacian on infinite intervals. Our approach is based on the monotone iterative technique.

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.

A variational approach to a BVP arising in the modelling of electrically conducting solids

Abstract The purpose of this paper is to apply an amended variational scheme, based on an adapted domain decomposition, for the solution of a second-order nonlinear boundary value problem that arises in applications involving the diffusion of heat generated by positive temperature dependent sources. The underlying idea of this approach is to decompose the original interval prior to the implementation of the iterative variational approach in order to improve the accuracy and acquire uniform convergence to the exact solution over the entire domain. Convergence analysis is given and then numerical experiments are carried out to make obvious the convergence, accuracy and efficient applicability of the method.

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.

Nonlinear second-order equations with functional implicit impulses and nonlinear functional boundary conditions

In this paper we present existence results for solutions of nonlinear second-order boundary value problems with impulses. Our impulses are applied at p points in the interval and given implicitly by nonlinear functions of the solution. Moreover we allow functional dependence on the solution. Our existence results follow from the existence of a pair of well ordered lower and upper solutions. We generalize earlier results of Cabada and Tomec˘ek, allowing more general compatible boundary conditions, impulses and φ -Laplacian equations.

Existence of Two Solutions for a Second-Order Discrete Boundary Value Problem

Abstract The existence of two nontrivial solutions for a class of nonlinear second-order discrete boundary value problems is established. The approach adopted is based on variational methods.

Second-order two-point boundary value problems using the variational iteration algorithm-II

In this paper, we introduce an improved variational iteration method (VIM) for nonlinear second-order boundary value problems. The main advantage of this modification is that it can avoid additional computation in determining the unknown parameters in initial approximation when solving boundary value problems using the conventional VIM. Also, iterative sequences obtained using the improved VIM do satisfy the boundary conditions while iterative sequences obtained using conventional VIM may not, in general, satisfy the boundary conditions. Numerical results reveal that the improved method is accurate and efficient.

The Comparative Boubaker Polynomials Expansion Scheme (BPES) and Homotopy Perturbation Method (HPM) for solving a standard nonlinear second-order boundary value problem

In this paper, we propose an analytical solution to a nonlinear second-order boundary value problem using the Homotopy Perturbation Method ( HPM) and the 4q-Boubaker Polynomials Expansion Scheme ( BPES). Both methods have been applied to a particular differential problem. The results are plotted, and compared with exact solutions proposed elsewhere, in order to evaluate accuracy.

Quadratic approximation of solutions for boundary value problems with nonlocal boundary conditions

In this paper, using the quasilinearization method coupled with the method of upper and lower solutions, we study a class of second-order nonlinear boundary value problems with nonlocal boundary conditions. We establish some sufficient conditions under which corresponding monotone sequences converge uniformly and quadratically to the unique solution of the problem. An example is also included to illustrate the main result.

Existence of Positive Solutions of Nonlinear Second-Order M-Point Boundary Value Problem

Under suitable conditions on, the nonlinear second-order m-point boundary value problem has at least one positive solution. In this paper, the authors examine the positive solutions of nonlinear second-order m-point boundary value problem.

Upper and lower solutions for BVPs on the half-line with variable coefficient and derivative depending nonlinearity

This paper is concerned with a second-order nonlinear boundary value problem with a derivative depending nonlinearity and posed on the positive half-line. The derivative operator is time dependent. Upon a priori estimates and under a Nagumo growth condition, the Schauder’s fixed point theorem combined with the method of upper and lower solutions on unbounded domains are used to prove existence of solutions. A uniqueness theorem is also obtained and some examples of application illustrate the obtained results.

New Fixed Point Theorems of Mixed Monotone Operators and Applications to Singular Boundary Value Problems on Time Scales

Some new existence and uniqueness theorems of fixed points of mixed monotone operators are obtained, and then they are applied to a nonlinear singular second-order three-point boundary value problem on time scales. We prove the existence and uniqueness of a positive solution for the above problem which cannot be solved by using previously available methods.

Modified variation of parameters method for solving system of second-order nonlinear boundary value problem

In this paper, we use the modified variation of parameters method, which is an elegant coupling of variation of parameters method and Adomian’s decomposition method, for solving the solution system of nonlinear boundary value problems associated with obstacle, contact and unilateral problems. The results are calculated in terms of series with easily computable components. The suggested method is applied without any discretization, transformation and restrictive assumptions. To illustrate the implementation and efficiency of the proposed method, an example is given. Key words: Variation of parameters method, Adomian’s polynomials, system of nonlinear boundary value problems.