Exact solution for the interaction between a harmonic and an angular momentum oscillator by means of a Taylor expansion

1977 ◽  
Vol 60 (3) ◽  
pp. 173-176
Author(s):  
S. Ruschin ◽  
Y. Ben-Aryeh
Keyword(s):  
Exact Solution ◽  
Angular Momentum ◽  
Taylor Expansion

scholarly journals Computational Methods for Solving Linear Fuzzy Volterra Integral Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jihan Hamaydi ◽  
Naji Qatanani
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Numerical Methods ◽  
Volterra Integral Equation ◽  
Taylor Expansion ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Iteration Methods ◽  
Good Agreement

Two numerical schemes, namely, the Taylor expansion and the variational iteration methods, have been implemented to give an approximate solution of the fuzzy linear Volterra integral equation of the second kind. To display the validity and applicability of the numerical methods, one illustrative example with known exact solution is presented. Numerical results show that the convergence and accuracy of these methods were in a good agreement with the exact solution. However, according to comparison of these methods, we conclude that the variational iteration method provides more accurate results.

scholarly journals Free Axial Vibrations of Non-Uniform Rods

2011 ◽  
Vol 2-3 ◽  
pp. 882-889
Author(s):  
Shu Qi Guo ◽  
Shao Pu Yang
Keyword(s):  
Exact Solution ◽  
Material Properties ◽  
Taylor Expansion ◽  
Series Solution ◽  
Wkb Method ◽  
Kummer Functions ◽  
Axial Vibrations ◽  
Special Case

Free axial vibrations of non-uniform rods are investigated by a proposed method, which results in a series solution. In a special case, with the proposed method an exact solution with a concise form can be obtained, which imply four types of profiles with variation in geometry or material properties. However, the WKB (Wentzel-Kramers-Brillouin) method leads to a series solution, which is a Taylor expansion of the results of the pro-posed method. For the arbitrary non-uniform rods, the comparison indicates that the WKB method is simpler, but the convergent speed of the series solution resulting from the proposed method is faster than that of the WKB method, which is also validated numerically using an exact solution of a kind of non-uniform rods with Kummer functions.

scholarly journals Asymmetric hypergeometric laser beams

2019 ◽  
Vol 43 (5) ◽  
pp. 735-740
Author(s):  
V.V. Kotlyar ◽  
A.A. Kovalev ◽  
E.G. Abramochkin
Keyword(s):  
Exact Solution ◽  
Angular Momentum ◽  
Hypergeometric Function ◽  
Orbital Angular Momentum ◽  
Topological Charge ◽  
Optical Axis ◽  
Type Equation ◽  
Laser Beams ◽  
Propagation Equation ◽  
Degenerate Hypergeometric Function

Here we study asymmetric Kummer beams (aK-beams) with their scalar complex amplitude being proportional to the Kummer function (a degenerate hypergeometric function). These beams are an exact solution of the paraxial propagation equation (Schrödinger-type equation) and obtained from the conventional symmetric hypergeometric beams by a complex shift of the transverse coordinates. On propagation, the aK-beams change their intensity weakly and rotate around the optical axis. These beams are an example of vortex laser beams with a fractional orbital angular momentum (OAM), which depends on four parameters: the vortex topological charge, the shift magnitude, the logarithmic axicon parameter and the degree of the radial factor. Changing these parameters, it is possible to control the beam OAM, either continuously increasing or decreasing it.

scholarly journals Exact Solution for Relativistic Trajectories Using Modal Transseries

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1505
Author(s):  
Luis Acedo ◽  
Abraham J. Arenas ◽  
Nicolas De La Espriella
Keyword(s):  
Exact Solution ◽  
Angular Momentum ◽  
High Precision ◽  
Limit Cycles ◽  
Fundamental Problem ◽  
Orbital Plane ◽  
Relativistic Effects ◽  
Geodesic Equation ◽  
Novel Method ◽  
Analytical Expressions

In this article, we design a novel method for finding the exact solution of the geodesic equation in Schwarzschild spacetime, which represents the trajectories of the particles. This is a fundamental problem in astrophysics and astrodynamics if we want to incorporate relativistic effects in high precision calculations. Here, we show that exact analytical expressions can be given, in terms of modal transseries for the spiral orbits as they approach the limit cycles given by the two circular orbits that appear for each angular momentum value. The solution is expressed in terms of transseries generated by transmonomials of the form e−nθ, n=1, 2, …, where θ is the angle measured in the orbital plane. Examples are presented that verify the effect of the solutions.

scholarly journals A Representation of the Exact Solution of Generalized Lane-Emden Equations Using a New Analytical Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Omar Abu Arqub ◽  
Ahmad El-Ajou ◽  
A. Sami Bataineh ◽  
I. Hashim
Keyword(s):  
Exact Solution ◽  
Analytical Method ◽  
Taylor Expansion ◽  
Computational Cost ◽  
Test Problems ◽  
Analytic Method ◽  
Computational Results ◽  
Initial Value ◽  
Practical Usefulness ◽  
And Performance

A new analytic method is applied to singular initial-value Lane-Emden-type problems, and the effectiveness and performance of the method is studied. The proposed method obtains a Taylor expansion of the solution, and when the solution is polynomial, our method reproduces the exact solution. It is observed that the method is easy to implement, valuable for handling singular phenomena, yields excellent results at a minimum computational cost, and requires less time. Computational results of several test problems are presented to demonstrate the viability and practical usefulness of the method. The results reveal that the method is very effective, straightforward, and simple.

scholarly journals A Numerical Scheme to Solve Fuzzy Linear Volterra Integral Equations System

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
A. Jafarian ◽  
S. Measoomy Nia ◽  
S. Tavan
Keyword(s):  
Exact Solution ◽  
Integral Equations ◽  
Fuzzy System ◽  
Numerical Scheme ◽  
Taylor Expansion ◽  
Expansion Method ◽  
Volterra Integral Equations ◽  
Taylor Coefficients ◽  
Linear Volterra Integral Equations ◽  
The Given

The current research attempts to offer a new method for solving fuzzy linear Volterra integral equations system. This method converts the given fuzzy system into a linear system in crisp case by using the Taylor expansion method. Now the solution of this system yields the unknown Taylor coefficients of the solution functions. The proposed method is illustrated by an example and also results are compared with the exact solution by using computer simulations.

scholarly journals Series Solution for Single and System of Non-linear Volterra Integral Equations

2020 ◽  
Vol 4 (2) ◽  
pp. 6-8
Author(s):  
Narmeen N. Nadir
Keyword(s):  
Exact Solution ◽  
Integral Equations ◽  
Taylor Expansion ◽  
Series Solution ◽  
Volterra Integral Equations ◽  
Non Linear ◽  
Linear Volterra Integral Equations

In this paper, Taylor expansion has been used for solving non-linear Volterra integral equations (VIEs) of the second kind. This method allows us to overcome the difficulty caused by integrals and non-linearity; also, it has more precise and rapidly convergent to the exact solution. Two examples are presented for illustrate the performance of this method.

On perturbation problems associated with finite boundaries

1948 ◽  
Vol 44 (2) ◽  
pp. 251-262 ◽  
Author(s):  
G. D. Wassermann
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Conditions ◽  
Taylor Expansion ◽  
Displacement Vector ◽  
Surface Displacement ◽  
Equation Boundary ◽  
Subsequent Separation ◽  
Perturbation Problems ◽  
Good Agreement

The methods of a previous paper (9) are improved and extended. It is assumed that the eigenfunctions and eigenvalues of an eigenvalue problem given by an elliptic differential equation are known subject to given boundary conditions on a finite boundary. It is shown how the corresponding quantities can be obtained for a similar problem in which the original differential equation, boundary and boundary conditions are simultaneously perturbed. The introduction of a surface displacement vector allows of a Taylor expansion of all quantities and a subsequent separation of orders. The problem of finding the perturbed eigenfunctions for each order then reduces to the solution of an inhomogeneous differential equation subject to known boundary conditions. These equations are solved by a variational method. An application of Green's theorem at each stage enables us to find the perturbed eigenvalues. The method is applied to a problem of which an exact solution is known and good agreement is obtained.The author is greatly indebted to Dr H. Fröhlich for many interesting discussions and some valuable suggestions.

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