A spectral representation of the linear multivelocity transport problem
-
Mariam Avalishvili
and
Dazmir Shulaia
Abstract
The transformation of the original characteristic equation of the multivelocity linear transport theory was carried out by expanding the scattering function for the problem to be solved as a spectral integral over a complete set of eigenfunctions for the previously solved transport problem. The obtained equation represents a singular integral equation containing a spectral integral over the spectrum of the solved problem, whose kernel depends on the difference between the scattering of the problem to be solved and that of the already solved problem. We consider also the examples illustrating the validity of such a transformation. M. Kanal and J. Davies made a similar transformation of the characteristic equation of the one-velocity transport theory.
A Appendix
Suppose that the kernel
The kernel
Let us denote
and
Using the orthogonality and recursion properties of the Legendre polynomials from equation (2.2) yields
for
From equation (3.8) we get
An important special case of equation (A.3) occurs when
A corollary of the latter is that
for all
and
respectively. From equation (A.3) we obtain
References
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© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Non-trivial solutions for nonlocal elliptic problems of Kirchhoff-type
- Existence of renormalized solutions for strongly nonlinear parabolic problems with measure data
- On the modification of the Szaśz–Durrmeyer operators
- A spectral representation of the linear multivelocity transport problem
- A Tauberian theorem for the product of Abel and Cesàro summability methods
- The spaces of bilinear multipliers of weighted Lorentz type modulation spaces
- Summations of Schlömilch series containing Struve function terms
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- Modified relativistic Laguerre polynomials. Monomiality and Lie algebraic methods
- On the difference between a Vitali–Bernstein selector and a partial Vitali–Bernstein selector
- The Hardy--Littlewood maximal operator and BLO1/log class of exponents
- Bicritical domination and double coalescence of graphs
- The asymptotic behavior of a counting process in the max-scheme. A discrete case
- Functions of bounded fractional differential variation – A new concept
- Sensitivity analysis of the optimal exercise boundary of the American put option
- Estimates for the asymptotic convergence of a non-isothermal linear elasticity with friction
- A coupled system of nonlinear differential equations involving m nonlinear terms
Articles in the same Issue
- Frontmatter
- Non-trivial solutions for nonlocal elliptic problems of Kirchhoff-type
- Existence of renormalized solutions for strongly nonlinear parabolic problems with measure data
- On the modification of the Szaśz–Durrmeyer operators
- A spectral representation of the linear multivelocity transport problem
- A Tauberian theorem for the product of Abel and Cesàro summability methods
- The spaces of bilinear multipliers of weighted Lorentz type modulation spaces
- Summations of Schlömilch series containing Struve function terms
- On nondifferentiable minimax semi-infinite programming problems in complex spaces
- Modified relativistic Laguerre polynomials. Monomiality and Lie algebraic methods
- On the difference between a Vitali–Bernstein selector and a partial Vitali–Bernstein selector
- The Hardy--Littlewood maximal operator and BLO1/log class of exponents
- Bicritical domination and double coalescence of graphs
- The asymptotic behavior of a counting process in the max-scheme. A discrete case
- Functions of bounded fractional differential variation – A new concept
- Sensitivity analysis of the optimal exercise boundary of the American put option
- Estimates for the asymptotic convergence of a non-isothermal linear elasticity with friction
- A coupled system of nonlinear differential equations involving m nonlinear terms
