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Differential transform method is adopted, for the first time, for solving linear singularly perturbed two-point boundary value problems. Four numerical examples are given to demonstrate the effectiveness of the present method. Results show that the numerical scheme is very effective and convenient for solving a large number of linear singularly perturbed two-point boundary value problems with high accuracy.

Singularly perturbed second-order two-point boundary value problems, which received a significant amount of attention in past and recent years, arise very frequently in fluid mechanics, quantum mechanics, optimal control, chemical-reactor theory, aerodynamics, reaction-diffusion process, geophysics, and so forth. In these problems a small parameter multiplies to a highest derivative. A well-known fact is that the solution of such problems display sharp boundary or interior layers when the singular perturbation parameter

The aim of our study is to introduce the differential transform method [

The rest of the paper is organized as follows. In Section

In this section, the concept of the differential transformation method (DTM) is briefly introduced. The concept of differential transform was first introduced by Pukhov [

The differential transform of the

If

If

If

If

If

If

In order to evaluate the accuracy of DTM for solving singularly perturbed two-point boundary value problems, we will consider the following examples. These examples have been chosen because they have been widely discussed in the literature and also approximate solutions are available for a concrete comparison.

We first consider the following problem [

The approximate solution (dotted curve) versus the analytic solution (solid curve) for

Secondly, we consider the following problem:

The constant

Then, by using the inverse transform rule in (

In Figure

The approximate solution (dotted curve) versus the analytic solution (solid curve) for

Thirdly, we consider the following problem [

By taking

From (

Comparison of the approximate solution with the exact solution (

The approximate solution (dotted curve) versus the analytic solution (solid curve) for

Finally, we consider the following problem [

Graphical result for

The approximate solution (dotted curve) versus the analytic solution (solid curve) for

In this study, the differential transformation method (DTM) has been employed, for the first time, successfully for solving linear singularly perturbed two-point boundary value problems. Four examples with boundary layers have been treated. This new method accelerated the convergence to the solutions. As it can be seen, this method leads to tremendously accurate results. It provides the solutions in terms of convergent series with easily computable components in a direct way without using linearization, discretization, or restrictive assumptions. The Mathematica software system has been used for all the symbolic and numerical computations in this paper.