Numerical integration and error analysis

1999 ◽  
pp. 59-67
Author(s):  
P. A. Leffelaar
Keyword(s):  
Error Analysis ◽  
Numerical Integration

Modified THDRK methods for the numerical integration of the Schrödinger equation

2020 ◽  
Vol 31 (10) ◽  
pp. 2050149
Author(s):  
Yonglei Fang ◽  
Yanping Yang ◽  
Xiong You ◽  
Lei Ma
Keyword(s):  
Schrödinger Equation ◽  
Error Analysis ◽  
Numerical Solution ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Resonant State ◽  
Order Conditions ◽  
Large Energy ◽  
New Methods ◽  
Fifth Order

Modified three-derivative Runge–Kutta (MTHDRK) methods for the numerical solution of the resonant state for the Schrödinger equation are investigated. Order conditions are presented and oscillation-fitting conditions are derived. Two practical fifth-order explicit MTHDRK methods are constructed and the error analysis is carried out for large energy. The numerical results are presented for the numerical solution of the Schrödinger equation to show the robustness of our new methods.

Wavelet packets: Uniform approximation and numerical integration

2019 ◽  
Vol 18 (02) ◽  
pp. 2050004
Author(s):  
Nikhil Khanna ◽  
S. K. Kaushik ◽  
A. M. Jarrah
Keyword(s):  
Error Analysis ◽  
Numerical Integration ◽  
Uniform Approximation ◽  
Integrable Function ◽  
Wavelet Packet ◽  
Sufficient Conditions ◽  
Wavelet Packets ◽  
Definite Integral ◽  
Approach Zero ◽  
Square Integrable Function

In this paper, it is proved that under some conditions, wavelet packet basis of [Formula: see text] can be used as a tool for the uniform approximation of an [Formula: see text]-times ([Formula: see text]) continuously differentiable and square integrable function [Formula: see text]. Sufficient conditions which establish that the approximations of wavelet packet sequences of square integrable function [Formula: see text] at lower levels are uniformly reliable and they uniformly approach zero as [Formula: see text] are given. Finally, a method based on wavelet packet expansion to find the definite integral of a function in [Formula: see text] is given and its error analysis has been discussed.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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