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Following a six-step flow chart, exponentially-fitted variant of the 2-step Simpson’s method suitable for solving ordinary differential equations with periodic/oscillatory behaviour is constructed. The qualitative properties of the constructed methods are also investigated. Numerical experiments on standard problems confirming the theoretical expectations regarding the constructed methods compared with other existing standard methods are also presented. Our results unify and improve the existing classical 2-step Simpson’s method.

In this paper, we consider the first-order initial value problem of the form

with oscillatory/periodic solution.

Several classical methods ( [

be expressed as linear combinations of functions of the form

Here, we analyze the construction and implementation of the exponentially-fitted variants of the 2-step Simpson method for solving problems of the form (1) which possess oscillatory/periodic solution, taking into account the six-step flow chart described by Ixaru and Vanden Berghe in [

The main interest of this work is to modify the classical 2-step Simpson method for adaptation to oscillatory/ periodic problems.

The classical 2-step Simpson method for solving (1) is given by

To begin the construction of the exponentially-fitted variants of (2), we rewrite (2) in a more general way as

Following the six-step flow chart, the corresponding linear difference operator

where

where

which are the coefficients of the classical method (2).

Applying step III, we find that

where

To implement step IV, consider the reference set of M functions:

with

•

•

•

The coefficients of the method for each case are obtained by the implementation of step V as follows:

S1:

S2:

S3:

As expected, the exponentially fitted variants reduce to the the classical method as

The general expression of the leading term of the local truncation error (lte) for an exponentially fitted method with respect to the basis functions

takes the form (see [

with K, P and M satisfying the condition

For the three methods constructed above, one finds the following results:

• S1:

• S2:

• S3:

The following theorem states conditions of

Theorem 1. Let

then if

The requirement (12) is known as the Lipschitz Condition, and the constant

This condition may be thought of as being intermediate between differentiability and continuity, in the sense that

•

• Þ

• Þ

In particular, if

where

is chosen.

In the sequel, we shall apply the following Contraction Mapping Theorem:

Theorem 2. (Contraction Mapping Theorem). Consider a set

• D is closed (i.e., it contains all limit points of sequences in D)

•

• The mapping g is a contraction on D: There exists

Then

• there exists a unique

• for any

•

and the a-posteriori error estimate

If h is sufficiently small, implicit LMM methods also have unique solutions given h and

where

That is, we are looking for a fixed point of

If

So by the Contraction Mapping Fixed Point Theorem,

Theorem 3 (Dahlquist Theorem) The necessary and sufficient conditions for a linear multistep method to be convergent are that it be consistent and zero-stable

Dahlquist theorem (3) holds also true for EF-based algorithms but, because their coefficients are no longer constants the concepts of consistency and stability have to be adapted.

Definition 4. An exponentially fitted method associated with the fitting space (9) is said to be of order

Since

Definition 5. A linear s-step method is said to be weakly stable if there is more than one simple root of the polynomial equation

To investigate the stability of (3), one applies the method to the test problems

From the above, one finds that

where

setting

Definition 6. A region of stability is a region of the q--z plane, throughout which

For each of the constructed methods, the region of stability is presented in

Numerical experiments confirming the theoretical expectations regarding the constructed methods are now performed. The constructed methods are applied to two test problems and the result obtained compared with the classical fourth-order Taylor method, explicit four stage fourth-order Runge-Kutta method and the classical 2-step Simpson method.

Consider the IVP:

Consider the IVP:

The exponentially-fitted versions of the classical 2-step Simpson method have been constructed and imple- mented in this paper. The stability and convergence properties of the constructed methods were also analysed. The results obtained from the numerical examples show that the theoretical expectations are meet (i.e. the expo- nentially-fitted variants of the classical 2-step Simpson method are suitable for solving periodic/oscillatory problems).

We thank the Editor and the referee for their comments.

Ashiribo SenaponWusu,Bosede AlfredOlufemi,Akanbi MosesAdebowale, (2016) Exponentially-Fitted 2-Step Simpson’s Method for Oscillatory/Periodic Problems. Journal of Applied Mathematics and Physics,04,368-375. doi: 10.4236/jamp.2016.42043