Double Laplace transformation in mixed boundary-initial value problems and its application to multi-component plasmas
Evans, Kenneth Edward, Jr.
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https://hdl.handle.net/2142/16623
Description
Title
Double Laplace transformation in mixed boundary-initial value problems and its application to multi-component plasmas
Author(s)
Evans, Kenneth Edward, Jr.
Issue Date
1970
Department of Study
Physics
Discipline
Physics
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
double laplace transformation
Language
en
Abstract
"The application of the double Laplace transform (Laplace transformation in both space and time) to the solution of systems of linear, homogenous, hyperbolic, partial differential equations with real, constant
coefficients is treated. The purpose of this treatment is to discuss comprehensively a method whereby the mixed boundary-initial value problem for these equations can be solved. The treatment is limited to one dimensional systems. Certain features of the double Laplace transform method which regularly appear in the solution of equations of the type described are examined in detail. Two of these features are the important role played by the characteristics of the partial differential equations and the restrictions among the boundary and initial conditions which are necessary for a well-defined solution.
The method is applied to the moment equations for a multi-component plasma, and the connection between the general solution and the usual ""normal mode"" solution is discussed. The case of a monoenergetic beam injected into a cold, semi-infinite plasma is treated in detail. The effect of the collisions of the plasma particles with the background is included. A solution for the growth of an initial thermal disturbance in the plasma is obtained in the asymptotic limit t→∞, The dominant feature of this solution is a growth proportional to exp (γx2/ 3 t l / 3 - 1/2νt) where Y is a constant and v is the collision frequency, The relationship between this solution and the spatially inhomogenous stationary state resulting from recurring thermal disturbances is discussed. This treatment yields the first picture of the relationship between the temporal and spatial growth in a finite, unstable plasma."
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