On a method of solution of the initial value problem for the linearized Boltzmann equation

1971 ◽  
Vol 4 (2) ◽  
pp. 817-825
Author(s):  
A. D. Khon'kin
Keyword(s):  
Boltzmann Equation ◽  
Initial Value Problem ◽  
Initial Value ◽  
Linearized Boltzmann Equation ◽  
Method Of Solution

On the initial value problem for the Boltzmann equation with a force term

1989 ◽  
Vol 18 (1) ◽  
pp. 87-102 ◽  
Author(s):  
N. Bellomo ◽  
M. Lachowicz ◽  
A. Palczewski ◽  
G. Toscani
Keyword(s):  
Boltzmann Equation ◽  
Initial Value Problem ◽  
Force Term ◽  
Initial Value ◽  
The Boltzmann Equation

scholarly journals Approximate Solution of the Fuzzy Triangular Initial Value Problem with Different Fractional Operator

2020 ◽  
Vol 9 (1) ◽  
pp. 1242-1249
Keyword(s):  
Initial Value Problem ◽  
Reproducing Kernel ◽  
Fractional Operator ◽  
Initial Value ◽  
Derivative Operator ◽  
Method Of Solution ◽  
The Difference ◽  
Definition Of ◽  
Reproducing Kernel Theory ◽  
Solution Behaviour

This study aims to conduct a comparison regarding the process of solving the fuzzy triangular initial value problem (FTIVP). The series solution of this problem is acquired through the reproducing kernel theory (RKT), although there have been past studies on FTIVP, there is no specialist study to compare solutions for the definition of different fractional operator. The comparisons where located through the difference in the use of an operator in the process of solution by using Riemann-Liouville integral operator (RLIO) and then by using Caputo fractional derivative operator (CFDO). Algorithm was presented to validate the method of solution and to view the effect of changing the operators on the solution behaviour in the two cases. During this comparison, the effectiveness of RKT was cleared and the notion of difference between using RLIO and CFDO were fixedly identified. Applications: The results identified in this research pronounced active difference in the behavior of errors, CDFO variations, and the behavior of error in favour of RLIO.

scholarly journals Analytical formulations to the method of variation of parameters in terms of universal Y's functions

2015 ◽  
pp. 33-40 ◽  
Author(s):  
M.A. Sharaf ◽  
A.S. Saad ◽  
H.H. Selim
Keyword(s):  
Initial Value Problem ◽  
Series Solution ◽  
Continued Fraction ◽  
Short Note ◽  
Time Transformation ◽  
Universal Functions ◽  
Initial Value ◽  
Variation Of Parameters ◽  
Method Of Solution ◽  
Analytical Expressions

The method of variation of parameters still has a great interest and wide applications in mathematics, physics and astrodynamics. In this paper, universal functions (the Y's functions) based on Goodyear's time transformation formula were used to establish a variation of parameters method which is useful in slightly perturbed two-body initial value problem. Moreover due to its universality, the method avoids the switching among different conic orbits which are commonly occurring in space missions. The position and velocity vectors are written in terms of f and g series. The method is developed analytically and computationally. For the analytical developments, exact literal formulations for the differential system of variation of the epoch state vector are established. Symbolical series solution of the universal Kepler's equation was also established, and the literal analytical expressions of the coefficients of the series are listed in Horner form for efficient and stable evaluation. For computational developments of the method, an efficient algorithm was given using continued fraction theory. Finally, a short note on the method of solution was given just for the reader guidance.

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