A remark on Moore's new method of obtaining approximate solutions of the Dirac equation

1979 ◽  
Vol 57 (12) ◽  
pp. 2114-2119 ◽  
Author(s):  
Yasuo Tomishima
Keyword(s):  
Exact Solution ◽  
Wave Function ◽  
Approximate Solution ◽  
Dirac Equation ◽  
Approximate Solutions ◽  
Zero Order ◽  
New Method ◽  
Hydrogenic Atom ◽  
Approximate Wave Function ◽  
Approximate Wave

The new method for obtaining an approximate solution of the Dirac equation proposed by Moore is applied rigorously in zero order to the hydrogenic atom, and the results are compared with the exact solution and with those obtained by Pauli's approximation. It is concluded that up to the order of α2 the three methods give the same energy values, but the higher order contribution included in Moore's method makes the energy worse than Pauli's. Moore's zeroth approximate wave function is also examined.

Electronic wave functions II. A calculation for the ground state of the beryllium atom

1950 ◽  
Vol 201 (1064) ◽  
pp. 125-137 ◽  
Keyword(s):  
Wave Function ◽  
Variational Method ◽  
Ground State ◽  
Accurate Result ◽  
Energy Criterion ◽  
Wave Functions ◽  
Exponential Functions ◽  
Approximate Wave Function ◽  
Electronic Wave Functions ◽  
Approximate Wave

An approximate wave function expressed in terms of exponential functions, spherical harmonics, etc., with numerical coefficients has been calculated for the ground state of the beryllium atom . Judged by the energy criterion this gives a more accurate result than the Hartree result which was the best previously known. This has been calculated as a trial of a fresh method of calculating atomic wave functions. A linear combination of Slater determinants is treated by the variational method. The results suggest that this will provide a more powerful and convenient method than has previously been available for atoms with more than two electrons.

scholarly journals The Laplace-Adomian-Pade Technique for the ENSO Model

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Yi Zeng
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Decomposition Methods ◽  
High Accuracy ◽  
Padé Method

The Laplace-Adomian-Pade method is used to find approximate solutions of differential equations with initial conditions. The oscillation model of the ENSO is an important nonlinear differential equation which is solved analytically in this study. Compared with the exact solution from other decomposition methods, the approximate solution shows the method’s high accuracy with symbolic computation.

Variational study on the ground state of the spin XY magnet

1978 ◽  
Vol 56 (7) ◽  
pp. 902-912 ◽  
Author(s):  
Masuo Suzuki ◽  
Seiji Miyashita
Keyword(s):  
Wave Function ◽  
Variational Method ◽  
Ground State ◽  
State Energy ◽  
Finite Lattice ◽  
Long Range Order ◽  
Lattice Method ◽  
Approximate Wave Function ◽  
Approximate Wave ◽  
Variational Study

An approximate wave function of the ground state of the spin [Formula: see text] XY magnet is derived using a variational method. This wave function yields estimates of the ground state energy and long-range order which agree very well with the results obtained by Betts and Oitmaa by a finite lattice method.

Bilateral set-valued stochastic integral equations

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3253-3274
Author(s):  
Marek Malinowski ◽  
Donal O'Regan
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Integral Equations ◽  
Existence And Uniqueness ◽  
Stochastic Integral ◽  
Approximate Solutions ◽  
Existence And Uniqueness Theorem ◽  
Initial State ◽  
Stochastic Integral Equations ◽  
Stability Of The Solution

We investigate bilateral set-valued stochastic integral equations and these equations combine widening and narrrowing set-valued stochastic integral equations studied in literature. An existence and uniqueness theorem is established using approximate solutions. In addition stability of the solution with respect to small changes of the initial state and coefficients is established, also we provide a result on boundedness of the solution, and an estimate on a distance between the exact solution and the approximate solution is given. Finally some implications for deterministic set-valued integral equations are presented.

Time-of-flight diffraction method for joint with linear misalignment

2021 ◽  
Vol 63 (11) ◽  
pp. 654-658
Author(s):  
Y Kurokawa ◽  
T Kawaguchi ◽  
H Inoue
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Geometric Model ◽  
Time Of Flight ◽  
Diffraction Method ◽  
Approximate Solutions ◽  
Weld Joints ◽  
Time Of Flight Diffraction ◽  
Flaw Sizing ◽  
Probe Spacing

The time-of-flight diffraction (TOFD) method is known as one of the most accurate flaw sizing methods among the various ultrasonic testing techniques. However, the standard TOFD method cannot be applied to weld joints with linear misalignment because of its basic assumptions. In this study, a geometric model of the TOFD method for weld joints with linear misalignment is introduced and an exact solution for calculating the flaw tip depth is derived. Since the exact solution is extremely complex, a simple approximate solution is also derived assuming that the misalignment is sufficiently small relative to the probe spacing and the flaw tip depth. The error in the approximate solution is confirmed to be negligible if the assumptions are satisfied. Numerical simulations are conducted to assess the flaw sizing accuracy of both the exact and approximate solutions considering the constraint of the probe spacing and the influence of the excess metal shape. Finally, experiments are conducted to prove the applicability of the proposed method. As a result, the proposed method is proven to enable accurate flaw sizing of weld joints with linear misalignment.

Hypervirial theorems and perturbation theory in quantum mechanics

1965 ◽  
Vol 283 (1393) ◽  
pp. 229-237 ◽  
Keyword(s):  
Wave Function ◽  
Perturbation Theory ◽  
Wave Functions ◽  
First Order ◽  
Optimum Energy ◽  
Arbitrary Operator ◽  
Approximate Wave Function ◽  
Hypervirial Theorem ◽  
Variational Wave Functions ◽  
Approximate Wave

Previous ideas about the way in which hypervirial theorems might be used to improve approximate wave functions are discussed. To provide a firmer foundation for these ideas, a link is established between hypervirial theorems and perturbation theory. It is proved that if the first-order perturbation correction to the expectation value of an arbitrary operator vanishes, then the approximate wave function used satisfies a certain hypervirial theorem. Conversely, if an arbitrary hypervirial theorem is satisfied by the wave function, then it is proved that the expectation values of certain operators have vanishing first-order corrections. Some consequences of the theory as applied to variational wave functions with optimum energy are developed. The results are illustrated by the use of a simple approximate wave function for the ground state of the helium atom.

PSEUDODIFFERENTIAL EQUATIONS ON THE SPHERE WITH SPHERICAL SPLINES

2011 ◽  
Vol 21 (09) ◽  
pp. 1933-1959 ◽  
Author(s):  
T. D. PHAM ◽  
T. TRAN ◽  
A. CHERNOV
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Theoretical Result ◽  
Galerkin Method ◽  
Unit Sphere ◽  
Approximate Solutions ◽  
Numerical Results ◽  
Pseudodifferential Equations ◽  
Optimal Convergence ◽  
Sobolev Norms

Spherical splines are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by using Galerkin method. We prove optimal convergence (in Sobolev norms) of the approximate solution by spherical splines to the exact solution. Our numerical results underlie the theoretical result.

The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems

2011 ◽  
Vol 17 (13) ◽  
pp. 2059-2065 ◽  
Author(s):  
SA Yousefi ◽  
A Lotfi ◽  
M Dehghan
Keyword(s):  
Optimal Control ◽  
Exact Solution ◽  
Approximate Solution ◽  
Collocation Method ◽  
Algebraic System ◽  
Optimal Control Problems ◽  
New Method ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

In this article the Legendre multiwavelet basis with the aid of a collocation method has been applied to give the approximate solution for the fractional optimal control problems (FOCPs). The properties of the Legendre multiwavelet are presented. These properties together with the collocation method are then utilized to reduce the problem to the solution of an algebraic system. Numerical results and a comparison with the exact solution in the cases when we have an exact solution are given to demonstrate the applicability and efficiency of the new method.

scholarly journals Application of Jacobi Polynomials to the Approximate Solution of a Singular Integral Equation with Cauchy Kernel on the Real Half-line

2006 ◽  
Vol 6 (3) ◽  
pp. 326-335
Author(s):  
D. Pylak
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Jacobi Polynomials ◽  
Approximate Solutions ◽  
Half Line ◽  
Cauchy Kernel ◽  
The Real

AbstractIn this paper, exact solution of the characteristic equation with Cauchy kernel on the real half-line is presented. Next, Jacobi polynomials are used to derive approximate solutions of this equation. Moreover, estimations of errors of the approximated solutions are presented and proved.

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