High accuracy variable mesh method for nonlinear two-point boundary value problems in divergence form☆
Introduction
Consider the class of two-point boundary value problem subjected to the boundary conditions: where A and B are two constants.
We assume that for
- (i)
f(x, y) is continuous,
- (ii)
exists and is continuous,
- (iii)
.
- (iv)
p(x) is continuous,
- (v)
p′(x)exists and is continuous,
- (vi)
p(x) > 0.
These conditions ensure the existence of a unique solution of the boundary value problem (1.1) and (1.2) (see [6]) and we assumed to be satisfied in the problem under consideration.
The above nonlinear differential equation is an essential tool for modeling many physical situations: chemical reactions, pendulums, bending of beams & so forth. Further, nonlinear problems of type (1.1) arise in various fields of science and engineering, like quantum mechanics, fluid dynamics, turbulent interaction of waves & current, etc. In most of the cases it is extremely difficult to find analytical solution for such boundary value problems. Thus, the solution of boundary value problem is extremely important to conclude the study of physical situation. The numerical methods are therefore found to be an attractive alternate.
Some numerical and analytical methods such as shooting method [22], finite-element method [1], the Adomain decomposition method [2], homotopy perturbation method [3], homotopy analysis method [8], have been studied for obtaining approximate solutions of boundary value problems.
There is a considerable literature on numerical methods for such boundary value problems e.g. [7], [10], [13], [14], [15], [18], [21]. Keller [6] and Marchuk [4], [5] have developed the second order three-point finite difference schemes for the solution of two-point boundary value problem (1.1) whereas Chawla and Katti [16] gave fourth order method. TAGE iteration methods based on cubic spline approximations on non-uniform mesh for the numerical solution of nonlinear two-point boundary value problems in non-divergence form have been studied by Mohanty et al. [19] and Mohanty and Khosla [20].
In this paper, we have discussed the variable mesh method of order three for the class of two-point boundary value problem (1.1). In Section 2, we first establish a certain identity from which general three-point finite difference approximation of various orders can be obtained. Subsequently, the method is discussed in detail. We obtain a family of third order discretizations for the differential Eq. (1.1). We select the free parameters available in these discretizations which lead to a simplest third order method. In Section 3, the third order accuracy of the method is shown. To illustrate the applicability of the proposed method several numerical examples have been solved in Section 4 and the results are presented along with their comparison with schemes which have been developed for nonlinear two-point boundary value problems in non-divergence form. Finally, the discussion on these numerical results presented in Section 5. The choice of the mesh is depend on the given problem.
Section snippets
Variable mesh discretization
We first establish certain identities from which general three-point finite difference methods of various orders can be obtained for the two-point boundary value problem (1.1).
Let N be a positive integer, N ≥ 2, We discretize the solution region [0, 1] such that Let be the mesh sizes and the mesh ratio We set and .
In (1.1) we also set and Integrating (1.1)
Convergence of the method
To show the convergence of the method, we consider (a constant). Let denote the tridiagonal matrix with Let and
Numerical results
The given interval [0, 1] is divided into (N+1) parts with where and
We can write Thus, This determines the starting value of the first step length and the subsequent step-lengths are given by etc.
For simplicity, we consider (a positive constant), , then h1 reduces to By prescribing
Final discussion
In this paper, we have presented variable mesh methods of accuracy for two-point boundary value problems in divergence form and tested for convergence. The method has been successful when applied to both linear and nonlinear problems. Several examples have been solved to illustrate the applicability of proposed simplest third order method. We have observed that our method has produced accurate and oscillation free solution. We have compared our results with the schemes [18], [19], [20]
Acknowledgments
The authors thank the reviewers for their valuable suggestions, which substantially improved the standard of the paper. This work is supported by “Council of Scientific and Industrial Research (CSIR), New Delhi, India” under the grant no. 09/045(1161)/2012-EMR-I.
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This work is supported by CSIR-JRF, grant no. 09/045(1161)/2012-EMR-I.
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Present address: 4076, C/4, Vasant Kunj, New Delhi 110070, India.