scholarly journals Bivariate Infinite Series Solution of Kepler’s Equations

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 785
Author(s):  
Daniele Tommasini
Keyword(s):  
Parameter Space ◽  
Infinite Series ◽  
Iterative Procedure ◽  
Series Solution ◽  
Numerical Algorithms ◽  
Base Point ◽  
Series Solutions ◽  
Theoretical Interest ◽  
Analytical Computation ◽  
Derivatives Of

A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This result is based on an iterative procedure for the analytical computation of all the higher-order partial derivatives of the eccentric anomaly with respect to the eccentricity e and mean anomaly M in a given base point (ec,Mc) of the (e,M) plane. Explicit examples of such bivariate infinite series are provided, corresponding to different choices of (ec,Mc), and their convergence is studied numerically. In particular, the polynomials that are obtained by truncating the infinite series up to the fifth degree reach high levels of accuracy in significantly large regions of the parameter space (e,M). Besides their theoretical interest, these series can be used for designing 2-D spline numerical algorithms for efficiently solving Kepler’s equations for all values of the eccentricity and mean anomaly.

Voltammetric current–potential calculations using infinite series solution

2002 ◽  
Vol 4 (10) ◽  
pp. 803-807 ◽  
Author(s):  
Jan Mocak
Keyword(s):  
Infinite Series ◽  
Series Solution ◽  
Current Potential

The complete infinite series solution of systems governed by the wave equation with boundary damping

Wave Motion ◽  
2014 ◽  
Vol 51 (1) ◽  
pp. 114-124 ◽  
Author(s):  
Lea Sirota ◽  
Yoram Halevi
Keyword(s):  
Wave Equation ◽  
Infinite Series ◽  
Series Solution ◽  
Boundary Damping

Sinusoidal Torsional Buckling of Bars of Angle Section Under Bending Loads, as a Problem in Plate Theory

1951 ◽  
Vol 18 (3) ◽  
pp. 285-292
Author(s):  
H. J. Plass
Keyword(s):  
Infinite Series ◽  
Plate Theory ◽  
Ritz Method ◽  
Compressive Load ◽  
Series Solutions ◽  
Torsional Buckling ◽  
Bending Load ◽  
Angle Section ◽  
Bending Loads

Abstract Timoshenko has applied plate theory to each leg of an angle-section bar to determine the critical compressive load needed to cause sinusoidal torsional buckling. In this paper his idea is used to calculate the critical bending load needed to cause sinusoidal torsional buckling of an angle bar. The bending is assumed to be applied so that the extreme fibers of the angle are in compression, the vertex in tension. Approximate results are first obtained by means of the Rayleigh-Ritz method. The approximate deflection functions from which the energy terms are computed are based upon certain infinite-series solutions. After having obtained approximate results, exact values are obtained, using the approximate values as a guide to limit the amount of calculation. The results of this calculation are shown in Fig. 5, where they are compared with those predicted by bar theory. Differences between the two theories become more noticeable as the bar becomes short compared to its flange width. It is found that the critical bending load becomes larger very rapidly as the ratio of length to width of the flanges decreases. Bar theory predicts no such increase. The reason for this difference is explained.

Derivation of the Explicit Solution of the Inverse Involute Function and its Applications

1992 ◽  
Author(s):  
Harry Hui Cheng
Keyword(s):  
Series Solution ◽  
Asymptotic Series ◽  
Explicit Solutions ◽  
Series Solutions ◽  
Perturbation Techniques ◽  
Pocket Calculator ◽  
Practical Applications ◽  
Involute Gears ◽  
Simple Expansion ◽  
Explicit Series Solutions

Abstract The involute function ε = tanϕ – ϕ or ε = invϕ, and the inverse involute function ϕ = inv−1(ε) arise in the tooth geometry calculations of involute gears, involute splines, and involute serrations. In this paper, the explicit series solutions of the inverse involute function are derived by perturbation techniques in the ranges of |ε| < 1.8, 1.8 < |ε| < 5, and |ε| > 5. These explicit solutions are compared with the exact solutions, and the expressions for estimated errors are also developed. Of particular interest in the applications are the simple expansion ϕ = inv−1(ε) = (3ε)1/3 – 2ε/5 which gives the angle ϕ (< 45°) with error less than 1.0% in the range of ε < 0.215, and the economized asymptotic series expansion ϕ = inv−1 (ε) = 1.440859ε1/3 – 0.3660584ε which gives ϕ with error less than 0.17% in the range of ε < 0.215. The four, seven, and nine term series solutions of ϕ = inv−1 (ε) are shown to have error less than 0.0018%, 4.89 * 10−6%, and 2.01 * 10−7% in the range of ε < 0.215, respectively. The computation of the series solution of the inverse involute function can be easily performed by using a pocket calculator, which should lead to its practical applications in the design and analysis of involute gears, splines, and serrations.

Axisymmetric Thermal Stresses in a Spherical Shell of Arbitrary Thickness

1957 ◽  
Vol 24 (3) ◽  
pp. 376-380
Author(s):  
E. L. McDowell ◽  
E. Sternberg
Keyword(s):  
Steady State ◽  
Spherical Shell ◽  
Spherical Cavity ◽  
Series Solution ◽  
Thermal Stresses ◽  
Solid Sphere ◽  
Theory Of Elasticity ◽  
Infinite Medium ◽  
Series Solutions ◽  
Arbitrary Thickness

Abstract This paper contains an explicit series solution, exact within the classical theory of elasticity, for the steady-state thermal stresses and displacements induced in a spherical shell by an arbitrary axisymmetric distribution of surface temperatures. The corresponding solutions for a solid sphere and for a spherical cavity in an infinite medium are obtained as limiting cases. The convergence of the series solutions obtained is discussed. Numerical results are presented appropriate to a solid sphere if two hemispherical caps of its boundary are maintained at distinct uniform temperatures.

Homotopy Analysis Method

2017 ◽  
pp. 173-226
Keyword(s):  
Homotopy Analysis Method ◽  
Series Solution ◽  
Nonlinear Partial Differential Equations ◽  
Analytic Method ◽  
Series Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Homotopy Analysis ◽  
And Control

In this chapter, the analytic solution of nonlinear partial differential equations arising in heat transfer is obtained using the newly developed analytic method, namely the Homotopy Analysis Method (HAM). The homotopy analysis method provides us with a new way to obtain series solutions of such problems. This method contains the auxiliary parameter provides us with a simple way to adjust and control the convergence region of series solution. By suitable choice of the auxiliary parameter, we can obtain reasonable solutions for large modulus.

Infinite Series Solutions

1969 ◽  
pp. 98-131
Author(s):  
ZALMAN RUBINSTEIN
Keyword(s):  
Infinite Series ◽  
Series Solutions

Appearance of resonance conditions in certain continuous systems: Closed versus infinite series solutions

1987 ◽  
Vol 118 (1) ◽  
pp. 186-187 ◽  
Author(s):  
P.A.A. Laura ◽  
V.H. Cortinez ◽  
W. Meyer
Keyword(s):  
Infinite Series ◽  
Continuous Systems ◽  
Series Solutions

CONVERGENCE IN RELAXATION SPECTRUM RECOVERY

2016 ◽  
Vol 95 (1) ◽  
pp. 121-132 ◽  
Author(s):  
R. J. LOY ◽  
F. R. DE HOOG ◽  
R. S. ANDERSSEN
Keyword(s):  
Infinite Series ◽  
Relaxation Spectrum ◽  
Sufficient Conditions ◽  
Convolution Equation ◽  
Series Expansions ◽  
Practical Application ◽  
Rigorous Analysis ◽  
Spectrum Recovery ◽  
Derivatives Of

Because of its practical and theoretical importance in rheology, numerous algorithms have been proposed and utilised to solve the convolution equation $g(x)=(\text{sech}\,\star h)(x)\;(x\in \mathbb{R})$ for $h$, given $g$. There are several approaches involving the use of series expansions of derivatives of $g$, which are then truncated to a small number of terms for practical application. Such truncations can only be expected to be valid if the infinite series converge. In this note, we examine two specific truncations and provide a rigorous analysis to obtain sufficient conditions on $g$ (and equivalently on $h$) for the convergence of the series concerned.

Using Maple to Study the Partial Differential Problems

2013 ◽  
Vol 479-480 ◽  
pp. 800-804 ◽  
Author(s):  
Chii Huei Yu
Keyword(s):  
Infinite Series ◽  
Partial Derivative ◽  
Higher Order ◽  
The Other ◽  
Mathematical Software ◽  
Partial Differential ◽  
Differential Problem ◽  
Partial Derivatives ◽  
Order Partial Derivative ◽  
Derivatives Of

This paper uses the mathematical software Maple for the auxiliary tool to study the partial differential problem of two types of multivariable functions. We can obtain the infinite series forms of any order partial derivatives of these two types of multivariable functions by using differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order partial derivative values. On the other hand, we propose two examples of multivariable functions to evaluate their any order partial derivatives, and some of their higher order partial derivative values practically. At the same time, we employ Maple to calculate the approximations of these higher order partial derivative values and their infinite series forms for verifying our answers.

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