A new high order implicit four-step methods with vanished phase-lag and some of its derivatives for the numerical solution of the radial Schr¨odinger equation

Abstract

A new four-step implicit linear eight algebraic order method with vanished phase-lag and its first, second and third derivatives is constructed in this paper. The purpose of this paper is to develop an efficient algorithm for the approximate solution of the one-dimensional radial Schrödinger equation and related problems. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. An error analysis and a stability analysis is also investigated and a comparison with other methods is also studied. The efficiency of the new methodology is proved via theoretical analysis and numerical applications.

Spectral Rectangular Collocation Formula: An Approach for Solving Oscillatory Initial Value Problems or/and Boundary Value Problems in Ordinary Differential Equations

“…However, these does not imply that the Newton formula have no relevance again. There are many cases where Newton interpolation is preferable, see for example, the construction of the multi-step methods for the numerical solution of ordinary differential equations (ODEs) in [15,16,17,18,19]. In addition, the barycentric formula of the first kind (5) can be further modified into a more elegant method, which is usually used in practice (see [5]).…”

A new efficient implicit four-step method with vanished phase-lag and some of its derivatives for the numerical solution of the radial Schr¨odinger equation

“…Computational methods involving a parameter proposed by Gautschi [12], Jain et al [14] and Steifel and Bettis [30] yield numerical solution of problems of class (1). Chawla and et al [7,8], Anantha krishnaiah [3], Shokri and et al [20,21,22,23,24], Dahlquist [9], Franco [10], Lambert and Watson [15], Simos and et al [25,26,27], Saldanha and Achar [19], Achar [1], and Daele and Vanden Berghe [31] have developed methods to solve problems of class (2). We have organized the paper as follows: In Section 2, we present the preliminary concepts that requisite for theory of the new methodology.…”