Zero Dissipative, Explicit Numerov-Type Methods for Second Order IVPs with Oscillating Solutions

New explicit, zero dissipative, hybrid Numerov type methods are presented in this paper. We derive these methods using an alternative which avoids the use of costly high accuracy interpolatory nodes. We only need the Taylor expansion at some internal points then. The method is of sixth algebraic order at a cost of seven stages per step while their phase lag order is fourteen. The zero dissipation condition is satisfied, so the methods possess an non empty interval of periodicity. Numerical results over some well known problems in physics and mechanics indicate the superiority of the new method.     

A new explicit hybrid four-step method with vanished phase-lag and its derivatives

A hybrid explicit sixth algebraic order four-step method with phase-lag and its first, second and third derivatives vanished is obtained in this paper. We present the development of the new method, its comparative error analysis and its stability analysis. The resonance problem of the Schrödinger equation, is used in order to study the efficiency of the new developed method. After the presentation of the theoretical and the computational results it is easy to see that the new constructed method is more efficient than other well known methods for the approximate solution of the Schrödinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions.     

An explicit four-step method with vanished phase-lag and its first and second derivatives

In this paper we will develop an explicit fourth algebraic order four-step method with phase-lag and its first and second derivatives vanished. The comparative error and the stability analysis of the above mentioned paper is also presented. The new obtained method is applied on the resonance problem of the Schrödinger equationIn order in order to examine its efficiency. The theoretical and the computational results shown that the new obtained method is more efficient than other well known methods for the numerical solution of the Schrödinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions.     

A family of explicit linear six-step methods with vanished phase-lag and its first derivative

A family of explicit linear sixth algebraic order six-step methods with vanished phase-lag and its first derivative is obtained in this paper. The investigation of the above family of methods contains:

theoretical study of the new family of methods and

computational study of the new family of methods.

The theoretical study of the above mentioned family of methods contains:

1. the development of the method,
2. the computation of the local truncation error,
3. the comparative local truncation error analysis. The comparison is taken place between the new family of methods with the corresponding method with constant coefficients and
4. the stability analysis of the new family of methods. The stability analysis is taken place using test equation with different frequency than the frequency of the test equation used for the phase-lag analysis of the methods.

The application of the new family of linear six-step sixth algebraic order methods to the resonance problem of the one-dimensional time independent Schrödinger equation is used for the computational study of the new family of methods. The result of the above mentioned theoretical and computational investigation is that the new proposed family of linear explicit schemes are computationally and theoretically more effective than other well known methods for the approximate solution of the radial Schrödinger equation and related initial or boundary value problems with periodic and/or oscillating solutions.     

A family of ten-step methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation

A family of high algebraic order ten-step methods is obtained in this paper. The new developed methods have vanished phase-lag (the first one) and phase-lag and its first derivative (the second one). We apply the new developed methods to the resonance problem of the radial Schrödinger equation. The efficiency of the new proposed methods is shown via error analysis and numerical applications.