Solution of the One-Speed Transport Eqn. by K M Case method

[Sharfu]

Homework Statement

In the book, Nuclear Reactor Theory, Glasstone, Bell, under section 2.2
SOLUTION OF THE ONE-SPEED TRANSPORT EQUATION BY THE SEPARATION OF VARIABLES, I have difficulty in understanding the derivation. Hope some one can explain the derivation or give a reference where the derivation has been derived with explanation for the different (read each and every) steps. (The book is also available as TID 25606). The problem definition is equation 2.12. Then solved according to separation of variable methods for PDE.

Homework Equations

The ansatz solution thing is a new one (for someone with an engineering course in DE only) but mathematically viable. But why Mr. Case sought solutions with eigenvalues of Ψ? what is mathematical reason for it? As I am thinking the solution for "Source-Free Infinite Medium" as a sloped line going to zero as x increase toward infinity.
RHS of equation 2.14 contain an integral. Can someone show some link or reference where such PDE have been solved.
How integral of equation 2.19 is determined! What is the mathematical name for such calculation or some reference or link?
How can I see two discrete eigenvalues +vo and -vo satisfy equation 2.16?

The Attempt at a Solution

Spoiler: I admit I am a very slow learner, so do not get annoyed if I could not understand some very basic concepts. Secondly it would look too "demoralizing" if I skip a topic within first hundred pages of the book!

 

[member 553083]

The ansatz solution of the one-speed transport equation involves making an assumption of the form Ψ(x,t)=X(x)T(t). This is a mathematical technique that is often used to solve partial differential equations (PDEs). The idea is to assume that the solution can be written as the product of two functions, one of which depends only on x and the other only on t. By doing this, it is possible to separate the variables in the PDE and reduce the problem to two ordinary differential equations (ODEs), one for X(x) and one for T(t). This can then be solved by standard ODE techniques.Mr. Case sought solutions with eigenvalues of Ψ because the ansatz he was using assumed that the solution could be written as a product of two functions - one depending on x and the other on t. He wanted to find solutions which would satisfy this form. To do this, he separated the variables in the PDE and reduced the problem to two ODEs, one for X(x) and one for T(t). He then looked for solutions which would have the form Ψ(x,t)=X(x)T(t), where X(x) and T(t) are the solutions of the two ODEs. These solutions are called eigenvalues of Ψ.The integral in equation 2.14 is an integral of the form ∫φ(x)dx, where φ(x) is a function of x. This type of integral is called an indefinite integral. It is used to calculate the area under a curve and can be evaluated using standard integration techniques.The two discrete eigenvalues +vo and -vo satisfy equation 2.16 because they are solutions of the two ODEs that Mr. Case derived from the one-speed transport equation. The ODEs are X'(x)=voX(x) and T'(t)=voT(t). The solutions of these two ODEs are X(x)=Ae^(+vo*x) and T(t)=Be^(+vo*t) for +vo, and X(x)=Ce^(-vo*x) and T(t)=De^(-vo*t) for -vo. Substituting these solutions into equation 2.12 gives equation 2.16.