Nonlinear Two-point Boundary Value Problems Research Articles

Homotopy perturbation technique for solving two-point boundary value problems – comparison with other methods

The two-point boundary value problems occur in a wide variety of problems in engineering and science. In this paper, we implement the homotopy perturbation method for solving the linear and nonlinear two-point boundary value problems. The main aim of this paper is to compare the performance of the homotopy perturbation method with extended Adomian decomposition method and shooting method. As a result, for the same number of terms, the homotopy perturbation method yields relatively more accurate results with rapid convergence than other methods. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations.

An O(h 6 ) numerical solution of general nonlinear fifth-order two point boundary value problems

A sixth-order numerical scheme is developed for general nonlinear fifth-order two point boundary-value problems. The standard sextic spline for the solution of fifth order two point boundary-value problems gives only O(h 2) accuracy and leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. O(h 6) global error estimates obtained for these problems. The convergence properties of the method is studied. This scheme has been applied to the system of nonlinear fifth order two-point boundary value problem too. Numerical results are given to illustrate the efficiency of the proposed method computationally. Results from the numerical experiments, verify the theoretical behavior of the orders of convergence.

On the nonlinear vibration of heated bimetallic shallow shells of revolution

The axisymmetrically nonlinear free vibration of a bimetallic shallow shell of revolution under uniformly distributed static temperature changes is investigated. Based on the nonlinear bending theory of thin shallow shells, the governing equations are established in forms similar to those of classical single-layered shells theory by redetermination of reference surface of coordinate. These partial differential equations are reduced to corresponding ordinary ones by elimination of the time variable with Kantorovich averaging method following an assumed harmonic time mode. The resulting equations, which form a nonlinear two-point boundary value problem, are then solved numerically by shooting method, and the temperature-dependent characteristic relations of frequency vs. amplitude are obtained successfully. A detailed parametric study is conducted involving shell geometry and temperature parameters. The effects of these variables on the frequency–amplitude characteristics are plotted and discussed.

Existence result for a class of nonlinear third-order two-point boundary-value problems

AbstractThe upper and lower solutions method and Leray–Schauder degree theory are employed to establish the existence result for a class of nonlinear third-order two-point boundary-value problems with a sign-type Nagumo condition.

Third order accurate variable mesh discretization and application of TAGE iterative method for the non-linear two-point boundary value problems with homogeneous functions in integral form

In this paper, we discuss the application of two-parameter alternating group explicit (TAGE) iterative method to an efficient third order variable mesh method for the solution of non-linear differential equations with integral homogeneous functions subject to natural boundary conditions. The proposed method is applicable only when the internal grid points of the solution space are odd in number. The proposed iterative method is also applicable to the integro-differential equations with singular coefficients. Comparative numerical results are given to demonstrate the usefulness of the proposed method.

Constructive proof for existence of nonlinear two-point boundary value problems

Owing to the importance of differential equations in physics, the existence of solutions for differential equations has been paid much attention. In this paper, the existence of solution are obtained for the nonlinear second order two-point boundary value problem in the reproducing kernel space. Under certain assumptions on right-hand side, we propose constructive proof for the existence result, and a method is presented to obtain the exact solution expressed by the form of series. This paper is a extension of previous paper [Wei Jiang, Minggen Cui, The exact solution and stability analysis for integral equation of third or first kind with singular kernel, Appl. Math. Comput. 202 (2) (2008) 666–674], which extends a method of solving linear problems to present method for solving nonlinear problems.

The steady temperature distributions with different types of nonlinearities

The nonlinear two-point boundary value problems are solved and the steady temperature distributions in a rod are found by considering different types of the nonlinear parts of the problems, particularly in the polynomial, exponential and trigonometric forms. In this paper, with the aid of some transformations the variational iteration method’s scheme is reproduced for the nonlinear problems including two-point boundary value problems. The illustrative related problems are solved by means of the method scheme.

A variational approach to nonlinear two-point boundary value problems

A variational formulation is established for a nonlinear two-point boundary value problem, an analytical solution is obtained using the Ritz method, and the obtained solution is valid for the whole solution domain.

On nonlinear two-point boundary value problems for impulsive differential-algebraic problems

In this paper, we investigate boundary value problems for first order impulsive differential-algebraic problems with causal operators. Note that a corresponding boundary condition is given by a nonlinear function. Using a monotone iterative method we formulate general sufficient conditions under which such problems have solutions (extremal or a unique). An example shows that corresponding assumptions are satisfied. The results are new.

Determination of Nonlinear Optimal Feedback Law for Satellite Injection Problem Using Neighboring Optimal Control

An optimal trajectory design of a nonlinear satellite injection problem for transfer to a final target orbit by minimizing the time was investigated. Indeed, this design was an exact solution to the nonlinear two-point boundary value problem which determined optimal control history as well as optimal state trajectories in the open-loop form. Furthermore, the obtained optimal guidance strategy was exerted in the closed-loop form against the environment disturbances using neighboring optimal control method in the exact solution. Neighboring Optimal Control (NOC) law could produce time-variant feedback gains minimizing the performance measure to second order for perturbations from a nominal optimal path. Generally, this law was a function of perturbations appeared in the states and constraints and could be computed utilizing the backward sweep method. The simulation results indicated that the presented methodology was successfully utilized in the real world applications with good robustness to each noise or disturbance produced in each state variable.

Application of optimal control theory in marketing: What is the optimal number of choices on a shopping website?

There have been some recent interests in making websites responsive to the dynamic behaviour of users. We present a method for applying the steps of optimal control theory to internet shopping platforms. The information rate is a state variable to maximise and the rate of change in number of options available to the consumer is our control variable to minimise. We apply optimal control theory to identify the optimal number of options. We specify and solve the necessary and sufficient conditions for the optimal control for a class of objective functions, which gives a nonlinear two-point boundary value problem.

Multiple positive solutions for nonlinear third order general two-point boundary value problems

Multiple positive solutions for nonlinear third order general two-point boundary value problems

A spectral collocation algorithm for two-point boundary value problem in fiber Raman amplifier equations

A novel algorithm implementing Chebyshev spectral collocation (pseudospectral) method in combination with Newton’s method is proposed for the nonlinear two-point boundary value problem (BVP) arising in solving propagation equations in fiber Raman amplifier. Moreover, an algorithm to train the known linear solution for use as a starting solution for the Newton iteration is proposed and successfully implemented. The exponential accuracy obtained by the proposed Chebyshev pseudospectral method is demonstrated on a case of the Raman propagation equations with strong nonlinearities. This is in contrast to algebraic accuracy obtained by typical solvers used in the literature. The resolving power and the efficiency of the underlying Chebyshev grid are demonstrated in comparison to a known BVP solver.

A variational approach to nonlinear two-point boundary value problems

A variational approach is applied to nonlinear two-point boundary value problems. In this method, an initial solution is chosen with some unknown parameters, which can be identified by making J(y) stationary with respect to these parameters. Comparison with the exact solution shows that the method is very effective and convenient.

Existence and iteration of monotone positive solutions for a third-order two-point boundary value problem

The existence of nondecreasing positive solutions for the nonlinear third-order two-point boundary value problem u‴(t) + q(t)f(t, u(t), u′(t)) = 0, 0 < t < 1, u(0) = u″(0) = u′ (1) = 0 is studied. The iterative schemes for approximating the solutions are obtained by applying a monotone iterative method.

Large amplitude vibration of heated corrugated circular plates with shallow sinusoidal corrugations

This paper presents a nonlinear free vibration analysis of corrugated circular plates with shallow sinusoidal corrugations under uniformly static ambient temperature. Based on the nonlinear bending theory of thin shallow shells, the governing equations for corrugated plates are established from Hamilton’s principle. These partial differential equations are reduced to corresponding ordinary ones by elimination of the time variable with Kantorovich method following an assumed harmonic time mode. The resulting equations, which form a nonlinear two-point boundary value problem in spatial variable, are then solved numerically by shooting method, and the temperature-dependent characteristic relations of frequency vs. amplitude for nonlinear vibration of heated corrugated plates are obtained successfully. The comparison with available published results shows that the numerical analysis here is of good reliability. A detailed parametric study is conducted involving the dependency of nonlinear frequency on the depth and density of corrugations along with the temperature change. Effects of these variables on the trend of nonlinearity are plotted and discussed.

Singular non-linear two-point boundary value problems: Existence and uniqueness

A general approach is presented for proving existence and uniqueness of solutions to the singular boundary value problem y ″ ( x ) + m x y ′ ( x ) = f ( x , y ( x ) ) , x ∈ ( 0 , 1 ] , y ′ ( 0 ) = 0 , A y ( 1 ) + B y ′ ( 1 ) = C , A > 0 , B , C ⩾ 0 . The proof is constructive in nature, and could be used for numerical generation of the solution. The only restriction placed on f ( x , y ) is that it not be a singular function of the independent variable x ; singularities in y are easily avoided. Solutions are found in finite regions where ∂ f / ∂ y ⩾ 0 , using an integral equation whose Green’s function contains an adjustable parameter that secures convergence of the Picard iterative sequence. Methods based on the theory are developed and applied to a set of problems that have appeared previously in published works.

Multiple solutions of two-point nonlinear boundary value problems

A quasi-linear boundary value problem has a solution with properties induced by oscillatory properties of the linear part of an equation. This result is proved for two dimensional systems. Consequences for Φ -Laplacian equations and problems with resonant linear parts are discussed.

SOLVING SINGULAR NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS IN THE REPRODUCING KERNEL SPACE

In this paper, we present a new method for solving a nonlinear two-point boundary value problem with finitely many singularities. Its exact solution is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximation <TEX>$u_n(x)$</TEX> to the exact solution u(x) is obtained and is proved to converge to the exact solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.

Accurate numerical solutions to stationary free surface problems from capillarity

The paper considers two static problems from capillarity. The first one consists in the determination of the surface of a liquid in a capillary tube and the second in the computation of the shape of a sessile or pendent drop of liquid of a given volume, both configurations being considered in the gravitational field. From a mathematical point of view, both problems are nonlinear two-point boundary value problems which include in their formulations additional geometrical unknowns, i.e., the so-called free boundary value problems. Computational, they are laborious problems owing to the fact that they involve nonnormal Jacobian matrices. The resulting numerical difficulties are considerably reduced by treating these problems by a collocation method in conjunction with the continuation with respect to the parameters. The method is implemented by the MATLAB code bvp4c. The numerical evaluations of the free surfaces are in reasonable accordance with asymptotic estimations.