A parametrization method for solving nonlinear two-point boundary value problems

A sharper version of the local Hadamard theorem on the solvability of nonlinear equations is proved. Additional parameters are introduced, and a two-parameter family of algorithms for solving nonlinear two-point boundary value problems is proposed. Conditions for the convergence of these algorithms are given in terms of the initial data. Using the right-hand side of the system of differential equations and the boundary conditions, equations are constructed from which initial approximations to the unknown parameters can be found. A criterion is established for the existence of an isolated solution to a nonlinear two-point boundary value problem. This solution is shown to be a continuous function of the data specifying the problem.     

A series method for solving nonlinear two-point boundary value problems

Summary A new method for solving nonlinear boundary value problems based on Taylor-type expansions generated by the use of Lie series is derived and applied to a set of test examples. A detailed discussion is given of the comparative performance of this method under various conditions. The method is of theoretical interest but is not applicable, in its present form, to real life problems; in particular, because of the algebraic complexity of the expressions involved, only scalar second order equations have been discussed, though in principle systems of equations could be similarly treated. A continuation procedure based on this method is suggested for future investigation.     

A new numerical approach for solving a class of singular two-point boundary value problems

In this paper, we consider the following class of singular two-point boundary value problem posed on the interval x ?? (0, 1]

$$\begin{array}{@{}rcl@{}} (g(x)y^{\prime})^{\prime}=g(x)f(x,y),\\ y^{\prime}(0)=0,\mu y(1)+\sigma y^{\prime}(1)=B. \end{array} $$

A recursive scheme is developed, and its convergence properties are studied. Further, the error estimation of the method is discussed. The proposed scheme is based on the integral equation formalism and optimal homotopy analysis method in which a recursive scheme is established without any undetermined coefficients. The original differential equation is transformed into an equivalent integral equation to remove the singularity. The integral equation is then made free of undetermined coefficients by imposing the boundary conditions on it. Finally, the integral equation without any undetermined coefficients is efficiently treated by using optimal homotopy analysis method for finding the numerical solution. The optimal control-convergence parameter involved in the components of the series solution is obtained by minimizing the squared residual error equation. The present method is applied to obtain numerical solution of singular boundary value problems arising in various physical models, and numerical results show the advantages of our method over the existing methods.     

A wavelet method for solving a class of nonlinear boundary value problems

We develop an approximation scheme for a function defined on a bounded interval by combining techniques of boundary extension and Coiflet-type wavelet expansion. Such a modified wavelet approximation allows each expansion coefficient being explicitly expressed by a single-point sampling of the function, and allows boundary values and derivatives of the bounded function to be embedded in the modified wavelet basis. By incorporating this approximation scheme into the conventional Galerkin method, the interpolating property makes the solution of boundary value problems with strong nonlinearity to be very effective and accurate. As an example, we have applied the proposed method to the solution of the Bratu-type equations. Results demonstrate a much better accuracy than most methods developed so far. Interestingly, unlike most existing methods, numerical errors of the present solutions are not sensitive to the nonlinear intensity of the equations.     

Cubic spline for a class of singular two-point boundary value problems

In this paper we have presented a method based on cubic splines for solving a class of singular two-point boundary value problems. The original differential equation is modified at the singular point then the boundary value problem is treated by using cubic spline approximation. The tridiagonal system resulting from the spline approximation is efficiently solved by Thomas algorithm. Some model problems are solved, and the numerical results are compared with exact solution.     

Sinc collocation method with boundary treatment for two-point boundary value problems

Sinc collocation method is proven to provide an exponential convergence rate in solving linear differential equations, even in the presence of singularities. But in order to treat the derivatives on boundaries, people often relied on the finite difference method, which would be expected to limit the accuracy. The present paper develops a Sinc collocation method with boundary treatment for two-point boundary value problems. Numerical results show that the method can directly and efficiently handle the boundary derivatives.     

On Numerov's method for a class of strongly nonlinear two-point boundary value problems

The purpose of this paper is to give a numerical treatment for a class of strongly nonlinear two-point boundary value problems. The problems are discretized by fourth-order Numerov's method, and a linear monotone iterative algorithm is presented to compute the solutions of the resulting discrete problems. All processes avoid constructing explicitly an inverse function as is often needed in the known treatments. Consequently, the full potential of Numerov's method for strongly nonlinear two-point boundary value problems is realized. Some applications and numerical results are given to demonstrate the high efficiency of the approach.     

A numerical method for solving singular boundary value problems

Summary A numerical method is treated for solving singular boundary value problems with solutions that can be represented as series expansions on a subinterval near the singularity. A regular boundary value problem is derived on the remaining interval, for which a difference method is used. Convergence theorems are given for general schemes and for schemes of positive type for second order equations.     

Variational iteration method for solving two-point boundary value problems

Variational iteration method is introduced to solve two-point boundary value problems. Numerical results demonstrate that the method is promising and may overcome the difficulty arising in Adomian decomposition method.     

A new method for solving a class of mixed boundary value problems with singular coefficient

In [1], [2], [3], [4], [5], [6] and [7], it is very difficult to deal with initial boundary value conditions. In this paper, we give a new method to deal with boundary value conditions, the main contribution of this paper is to put mixed boundary value conditions into reproducing kernel Hilbert space. The numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.     

Wavelet analysis method for solving linear and nonlinear singular boundary value problems

In this paper, a robust and accurate algorithm for solving both linear and nonlinear singular boundary value problems is proposed. We introduce the Chebyshev wavelets operational matrix of derivative and product operation matrix. Chebyshev wavelets expansions together with operational matrix of derivative are employed to solve ordinary differential equations in which, at least, one of the coefficient functions or solution function is not analytic. Several examples are included to illustrate the efficiency and accuracy of the proposed method.     

A rational spectral collocation method for solving a class of parameterized singular perturbation problems

A new kind of numerical method based on rational spectral collocation with the sinh transformation is presented for solving parameterized singularly perturbed two-point boundary value problems with one boundary layer. By means of the sinh transformation, the original Chebyshev points are mapped onto the transformed ones clustered near the singular points of the problem. The results from asymptotic analysis as regards the singularity of the solution are employed to determine the parameters in the transformation. Numerical experiments including several nonlinear cases illustrate the high accuracy and efficiency of our method.     

A fourth-order accurate method for fourth-order two-point nonlinear boundary value problems

A fourth-order accurate finite difference method is developed for a class of fourth order nonlinear two-point boundary value problems. The method leads to a pentadiagonal scheme in the linear cases, which often arise in the beam deflection theory. The convergence of the method is tested numerically on examples from the literature.     

B-spline collocation for solution of two-point boundary value problems

A numerical method based on B-spline is developed to solve the general nonlinear two-point boundary value problems up to order 6. The standard formulation of sextic spline for the solution of boundary value problems 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. The error analysis and convergence properties of the method are studied via Green’s function approach. O(h⁶) global error estimates are obtained for numerical solution of these classes of problems. Numerical results are given to illustrate the efficiency of the proposed method. Results of numerical experiments verify the theoretical behavior of the orders of convergence.     

A modification of the Adomian decomposition method for a class of nonlinear singular boundary value problems

In this paper, the Adomian decomposition method is modified to solve a class of nonlinear singular boundary value problems which arise as nonlinear normal modal equations in nonlinear conservative vibratory systems. The effectiveness of the modified method is verified by three examples.