Bessel and Neumann Fitted Methods

2 Bessel and Neumann Fitted Methods. - Raptis and Cash13 have constructed a method which integrates exactly the spherical Bessel and Neumann functions. They considered the following second algebraic order symmetric two-step method, 

Solving the above system of equations (91), the coefficients of the method (90) are produced. 

Simos and Raptis41 have considered the following symmetric fourth algebraic order symmetric four-step method 

If we require that the method (92) integrates the functions krji(kr)and r/i/(fcr)exactly, the following system of equations is produced  

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T. E. Simos, Simple and Accurate Explicit Bessel and Neumann Fitted Methods for the Numerical Solution of the Schrodinger Equation, International Journal of Modern Physics C, 2000, 11(1), 79-89. 

T. E. Simos and P. S. Williams, Bessel and Neumann-fitted methods for the numerical solution of the radial Schrodinger equation, Comput. Chem., 1997, 21(3), 175-179. 

Table 3 Properties of the Bessel and Neumann fitted methods. N.o.S = Number of steps. A. O. = Algebraic order. I = Implicit.

Numerical Illustrations for Exponentially-Fitted Methods and Phase Fitted Methods. - In this section we test several finite difference methods with coefficients dependent on the frequency of the problem to the numerical solution of resonance and eigenvalue problems of the Schrodinger equations in order to examine their efficiency. First, we examine the accuracy of exponentially-fitted methods, phase fitted methods and Bessel and Neumann fitted methods. We note here that Bessel and Neumann fitted methods will also be examined as a part of the variable-step procedure. We also note that Bessel and Neumann fitted methods have a large penalty in a constant step procedure (it is known that the coefficients of the Bessel and Neumann fitted methods are position dependent, i.e. they are required to be recalculated at every step). 

Avdelas and Simos,93 (13) the exponentially-fitted variable-step method developed by Simos,8 (14) the variable-step phase-fitted method developed by Simos,51 (15) the variable-step P-stable method developed by Simos,74 (16) the exponentially-fitted variable-step method developed by Thomas and Simos,25 (17) the variable-step Bessel and Neumann fitted method developed by Simos,43 (18) the variable-step Bessel and Neumann fitted method developed by Simos,44 (19) the new exponentially-fitted variable step method based on the new exponentially-fitted tenth algebraic order method developed in Section... 

In this section recent developments on exponentialy fitted methods and Bessel and Neumann-fitted methods are presented. 

The most efficient variable-step methods for the solution of coupled differential equations arising from the Schrodinger equation is the P-stable exponentialy fitted variable-step method developed by Aguiar and Simos. Another very efficient variable-step method is the variable-step Bessel- and Neumann-fitted method of Simos. Efficient variable-step methods for the solution of the above problem are also the variable-step Bessel- and Neumann-fitted method of Simos and the variable-step exponentialy fitted method developed by Konguetsof and Simos. Finally efficient methods for the solution of the above problem are the generator and the optimized generator developed by Avdelas et... 

T. E. Simos, A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrodinger equation, J. Math. Chem., 2000, 27(4), 343-356. 

Bessel-fitted and Neumann-fitted Methods. — Simos has considered the following second algebraic order explicit method ... 

PET = P-stable exponentially fitted and trigonometrically fitted method. BNE = explicit Bessel-fitted and Neumann-fitted method. BNI = implicit Bessel-fitted and Neumann-fitted method. 

In [79], [119] numerical methods fitted to other functions than exponential or trigonometric (for example Bessel and Neumann functions) is presented. 

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