Exponentially-Fitted Methods

In each case a Table with numerical properties of the methods is given. 

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The P-stable Exponentially Fitted Method developed by Kalogiratou and Simos"" (which is indicated as Method III)... 

B. Stabilization of a multistep exponentially-fitted methods and their application to the Schrodinger equation... 

In this part of the paper we present the study of the stabilization of the multistep exponentially-fitted methods. More specifically we present a family of singularly P-stable exponentially-fitted four-step methods and a family of six-step exponentially-fitted methods for the numerical solution of the radial Schrodinger equation. 

T. E. Simos and P. S. Williams, A family of Numerov-type exponentially-fitted methods for the numerical integration of the Schrodinger equation, Comput. Chem., 1997, 21,403-417. 

J. R. Cash, A. D. Raptis and T. E. Simos, A sixth order exponentially-fitted method for the numerical solution of the radial Schrodinger equation, J. Comput. Phys., 1990, 91, 413-423. 

T. E. Simos, Stabilization of a Four-Step Exponentially-Fitted Method and its Application to the Schrodinger Equation, International Journal of Modern Physics C, 2007,18(3), 315-328. 

T. E. Simos, P-stable Four-Step Exponentially-Fitted Method for the Numerical Integration of the Schrodinger Equation, Computing Letters, 2005, 1(1), 37-45. 

In Section 5 trigonometrically fitted Fifth algebraic order Runge-Kutta methods are presented. For these methods we present the construction and the error analysis from which one can see that the classical method is dependent on the third power of energy, the first exponentially-fitted methods is depended on a second power of the energy and finally the second exponentially-fitted methods is depended on first power of the energy. 

Figure 1 Behavior of the coefficient b0 of the exponentially-fitted method of second exponential order for the interval [3.0,33]...

The new developed exponentially-fitted methods are the most efficient ones... 

The exponentially-fitted method developed in Section 2.3 is much more efficient than the exponentially-fitted method developed in Section 2.2. 

Exponentially-Fitted Method developed by Raptis 20 (indicated as Method... 

Finally, the Explicit Four-Step Method developed, the Four-step method developed in 19 and the Exponentially-Fitted Method developed by Raptis and Allison 21 are less efficient than the others. 

Remark 2. (See 24.) Every exponentially fitted method corresponds in a unique way to an algebraic method (by setting o, = 0 for all i). 

In 50 the author presents a further investigation of the frequency evaluation techniques which are recently proposed by Ixaru et al. for exponentially fitted multistep algorithms for the solution of first-order ordinary differential equations (ODEs). These studies have a scope which is to maximize the benefits of the exponentially-fitted methods via the evaluation of the frequency of the problem. The proposed by Ixaru and co-workers method for frequency... 

The Derivation of Exponentially-Fitted Methods for General Problems. -Consider the construction of an exponentially-fitted multistep method (2) which exactly integrates the set of functions exp( v oc). =0. We will use this for the numerical solution of the general problem (1). 

Two-step Methods. - The exponentially-fitted methods of this category are of the form... 

Using (12) a group of exponentially-fitted methods has been produced. These are found in ... 

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