A three POINT variable mesh finite difference method of third order, have been derived to solve the singular TWO-POINT BOUNDARY VALUE PROBLEMs. The method reduces to a method of order four for the uniform mesh case and may be considered as a modification of the well known Numerov"s method. The methods are self starting and are exact for y=1/x. The convergence of the fourth order method has also been discussed. The various measures of error for solution of TWO PROBLEMs, using the methods and the fourth order method in [1] are listed. The numerical results are also compared with five well known classical methods to show the self starting and accuracy of the present methods. Subject Classifications: AMS 64L10, CR: G1.7

The present study addresses an exponentially fitted finite difference method to obtain the solution of singularly perturbed TWO-POINT BOUNDARY VALUE PROBLEMs (BVPs) having a BOUNDARY layer at one end (left or right) POINT on uniform mesh. A fitting factor is introduced in the derived scheme using the theory of singular perturbations. Thomas algorithm is employed to solve the resulting tri-diagonal system of equations. The convergence of the presented method is investigated. Several model example PROBLEMs are solved using the proposed method. The results are presented with terms of maximum absolute errors, which demonstrate the accuracy and efficiency of the method. It is observed that the proposed method is capable of producing highly accurate results with minimal computational effort for a fixed VALUE of step size h, when the perturbation parameter tends to zero. From the graphs, we also observed that a numerical solution approximates the exact solution very well in the BOUNDARY layers for smaller VALUE of ε.

In this paper, we suggest a simple iterative method to find the geodesic path on a surface parameterized by orthogonal curvilinear system between TWO given POINTs based on solving BOUNDARY VALUE PROBLEM. In this supposed method, an iterative algorithm is used for finding the sufficient initial VALUEs as the destination POINT agree with the BOUNDARY conditions. Geodesic determination between TWO given POINTs is formulated for a general surface, and specially tested for reference ellipsoid which has many applications in geosciences and geodesy. Accuracy of the method is independent on the distant between TWO POINTs on the surface. Moreover, it can be used in aviation and sailings for finding the shortest path between start and destination POINTs.

We use cubic spline functions to develop a numerical method for the solution of second-order linear TWO-POINT BOUNDARY VALUE PROBLEMs. The resulting linear system of equations has been solved using a tri-diagonal solver. Convergence of the method is shown through standard convergence analysis. Numerical examples are given to show the applicability and efficiency of our method. Also we compared our method with finite difference, finite element, B-spline and finite volume methods which applied to the TWO-POINT BOUNDARY VALUE PROBLEMs.

In this work the collocation method based on quartic B-spline is developed and applied to TWO-POINT BOUNDARY VALUE PROBLEM in ordinary differential equations. The error analysis and convergence of presented method is discussed. The method illustrated by TWO test examples which verify that the presented method is applicable and considerable accurate.

The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude, long waves in dispersive media. This evolution equation has been used in both laboratory and field studies of water waves. The principal new result is an exact theory of convergence of the TWO-POINT BOUNDARY-VALUE PROBLEM to the initial-VALUE PROBLEM posed on an infinite stretch of the medium of propagation. In addition to their intrinsic interest, our results provide justification for the use of the TWO-POINT BOUNDARY-VALUE PROBLEM in numerical studies of the initial-VALUE PROBLEM posed on the entire line.

Variable mesh finite difference method of fifth order have been derived for solution of singular TWO POINT BOUNDARY VALUE PROBLEM, which reduces to the method of order six, for uniform mesh case. These methods are self starting and are exact for u=r-1. The convergence of the sixth order method is discussed. The absolute error, for solution of four PROBLEMs, TWO linear and TWO nonlinear are listed to show the accurate and effectiveness of the present method.

We consider a class of singularly perturbed semilinear three-POINT BOUNDARY VALUE PROBLEMs. An accelerated uniformly convergent numerical method is constructed via the exponential fitted operator method using Richardson extrapolation techniques to solve the PROBLEM. To treat the semilinear term, we use quasi-linearization techniques. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and ε-uniformly convergent for h ≥ ε, where the classical numerical methods fail to give a good result. It also improves the results of the methods existing in the literature. The method is shown to be second-order convergent independent of perturbation parameter ε.