A Successive Approximation Approach to Problems in Radiative Transfer with a Differential Formulation

1980 ◽  
Vol 102 (1) ◽  
pp. 86-91 ◽  
Author(s):  
W. W. Yuen ◽  
C. L. Tien
Keyword(s):  
Radiative Transfer ◽  
Solution Method ◽  
Practical Importance ◽  
Approximate Solutions ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Dimensional Problem ◽  
Weighted Residuals ◽  
First Order ◽  
Differential Formulation

The radiation intensity in a gray participating medium is expressed in a differential form. The energy equation for radiative transfer becomes an infinite-order differential equation. Utilizing the method of weighted residuals and introducing some appropriate formulations for the intensity boundary conditions, a method of successive approximations is developed. The solution method is applied to a one-dimensional problem with linear-anisotropic scattering. This problem is chosen because of its practical importance and the availability of exact solutions. A first-order closed-form result, which has never been derived analytically before, is obtained and shown to have good accuracy. Successive higher-order approximate solutions are also presented. These solutions are easily attainable algebraically and converge quickly to the exact result. To illustrate the possible applicability of the solution method for multidimensional problems, the first-order solution to a simple two-dimensional problem is presented. Results show that based on the present approach, reasonably accurate approximate solutions can be generated with some simple mathematical developments.

scholarly journals A Series-Solution Method for Cometary Orbits

1972 ◽  
Vol 45 ◽  
pp. 43-51
Author(s):  
P. E. Nacozy
Keyword(s):  
Series Solution ◽  
Solution Method ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Reference Solution ◽  
Successive Interval ◽  
First Order ◽  
Order Solution ◽  
The Mean ◽  
The Masses

A series-solution method for highly-eccentric perturbed orbits using a modified form of Hansen's method of partial anomalies is presented. Series in Chebyshev polynomials in the eccentric anomaly of a comet and the mean anomaly at an epoch of a planet provide a theory valid to first order with respect to the masses. The first-order solution becomes a reference solution about which higher-order perturbations are obtained by the method of successive approximations. The first-order solutions are valid approximations for long durations of time, whereas the higher orders are valid only over the interval of time that is selected for the Chebyshev expansions. The method is somewhat similar to Encke's method of special perturbations except that for each successive interval of time perturbations about a first-order solution are calculated instead of perturbations about a conic solution.

scholarly journals Spin-Dependent Coupled Altarelli-Parisi Equations in the NLO and the Method of Successive Approximations

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Ranjit Choudhury ◽  
D. K. Choudhury
Keyword(s):  
Gluon Distribution ◽  
Quark Distribution ◽  
Distribution Functions ◽  
Integrodifferential Equations ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Leading Order ◽  
Order Linear ◽  
First Order ◽  
Analytic Expressions

The coupled Altarelli-Parisi (AP) equations for polarized singlet quark distribution and polarized gluon distribution, when considered in the small x limit of the next to leading order (NLO) splitting functions, reduce to a system of two first order linear nonhomogeneous integrodifferential equations. We have applied the method of successive approximations to obtain the solutions of these equations. We have applied the same method to obtain the approximate analytic expressions for spin-dependent quark distribution functions with individual flavour and polarized structure functions for nucleon.

First Order Partial Functional Differential Equations with Unbounded Delay

2003 ◽  
Vol 10 (3) ◽  
pp. 509-530
Author(s):  
Z. Kamont ◽  
S. Kozieł
Keyword(s):  
Differential Equations ◽  
Integral Functional ◽  
Functional Differential Equations ◽  
Generalized Solutions ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
First Order ◽  
Unbounded Delay ◽  
The Cauchy Problem ◽  
Functional Differential

Abstract The phase space for nonlinear hyperbolic functional differential equations with unbounded delay is constructed. The set of axioms for generalized solutions of initial problems is presented. A theorem on the existence and continuous dependence upon initial data is given. The Cauchy problem is transformed into a system of integral functional equations. The existence of solutions of this system is proved by the method of successive approximations and by using theorems on integral inequalities. Examples of phase spaces are given.

Least-Squares Radial Point Interpolation Collocation Meshless Method for Radiative Heat Transfer

2006 ◽  
Vol 129 (5) ◽  
pp. 669-673 ◽  
Author(s):  
J. Y. Tan ◽  
L. H. Liu ◽  
B. X. Li
Keyword(s):  
Radiative Transfer ◽  
Least Squares ◽  
Meshless Method ◽  
Solution Method ◽  
Approximate Solutions ◽  
Discrete Ordinates ◽  
Scattering Media ◽  
Radial Point Interpolation ◽  
Point Interpolation ◽  
Transfer Problems

A least-squares radial point interpolation collocation meshless method based on the discrete ordinates equation is developed for solving the radiative transfer in absorbing, emitting, and scattering media, in which compact support radial basis functions augmented with polynomial basis are employed to construct the trial functions. In addition to the collocation nodes, a number of auxiliary points are also adopted to form the total residuals of the problem. The least-squares technique is used to obtain the solution of the problem by minimizing the summation of residuals of all collocation and auxiliary points. Three typical examples of radiative transfer in semitransparent media are examined to verify this new solution method. The numerical results are compared with other benchmark approximate solutions in references. By comparison, the results show that the least-squares radial point interpolation collocation meshless method has good accuracy in solving radiative transfer problems within absorbing, emitting, and scattering media.

New classes of integrals inherent in the mathematical structure of extended equations describing superconducting systems

2015 ◽  
Vol 29 (17) ◽  
pp. 1550117 ◽  
Author(s):  
Ryszard Gonczarek ◽  
Mateusz Krzyzosiak ◽  
Adam Gonczarek ◽  
Lucjan Jacak
Keyword(s):  
Mathematical Structure ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Superconducting Gap ◽  
S Wave ◽  
First Order ◽  
Order Expansion ◽  
Quantitative Characteristics ◽  
Fundamental Equations ◽  
New Generation

In this paper, we discuss the mathematical structure of the s-wave superconducting gap and other quantitative characteristics of superconducting systems. In particular, we evaluate and discuss integrals inherent in fundamental equations describing superconducting systems. The results presented here extend the approach formulated by Abrikosov and Maki, which was restricted to the first-order expansion. A few infinite families of integrals are derived and allow us to express the fundamental equations by means of analytic formulas. They can be then exploited in order to find some quantitative characteristics of superconducting systems by the method of successive approximations. We show that the results can be applied to some modern formalisms in order to study high-Tc superconductors and other superconducting materials of the new generation.

Numerical Methods for Solving Physically Nonlinear Problems for Inhomogeneous Thick-Walled Shells

2017 ◽  
Vol 865 ◽  
pp. 325-330 ◽  
Author(s):  
Vladimir I. Andreev ◽  
Lyudmila S. Polyakova
Keyword(s):  
Numerical Method ◽  
Nonlinear Problems ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Deformation Diagram ◽  
Method Of Solution ◽  
Properties Of Materials ◽  
Physical Fields ◽  
Cylinders And Spheres ◽  
Numerical Method Of Solution

The paper proposes the numerical method of solution the problems of calculation the stress state in thick-walled cylinders and spheres from physically nonlinear inhomogeneous material. The urgency of solved problem due to the change of mechanical properties of materials under the influence of different physical fields (temperature, humidity, radiation, etc.). The deformation diagram describes the three-parameter formula. The numerical method used the method of successive approximations. The results of numerical calculation are compared with the test analytical solutions obtaining the authors with some restrictions on diagram parameters. The obtained results can be considered quite satisfactory.

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