Citation
Ferguson, Warren E. (1975) A singularly perturbed linear twopoint boundaryvalue problem. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/3T9FVQ87. https://resolver.caltech.edu/CaltechTHESIS:04112013102123813
Abstract
We consider the following singularly perturbed linear twopoint boundaryvalue problem:
Ly(x) ≡ Ω(ε)D_xy(x)  A(x,ε)y(x) = f(x,ε) 0≤x≤1 (1a)
By ≡ L(ε)y(0) + R(ε)y(1) = g(ε) ε → 0^+ (1b)
Here Ω(ε) is a diagonal matrix whose first m diagonal elements are 1 and last m elements are ε. Aside from reasonable continuity conditions placed on A, L, R, f, g, we assume the lower right mxm principle submatrix of A has no eigenvalues whose real part is zero. Under these assumptions a constructive technique is used to derive sufficient conditions for the existence of a unique solution of (1). These sufficient conditions are used to define when (1) is a regular problem. It is then shown that as ε → 0^+ the solution of a regular problem exists and converges on every closed subinterval of (0,1) to a solution of the reduced problem. The reduced problem consists of the differential equation obtained by formally setting ε equal to zero in (1a) and initial conditions obtained from the boundary conditions (1b). Several examples of regular problems are also considered.
A similar technique is used to derive the properties of the solution of a particular difference scheme used to approximate (1). Under restrictions on the boundary conditions (1b) it is shown that for the stepsize much larger than ε the solution of the difference scheme, when applied to a regular problem, accurately represents the solution of the reduced problem.
Furthermore, the existence of a similarity transformation which block diagonalizes a matrix is presented as well as exponential bounds on certain fundamental solution matrices associated with the problem (1).
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  29 May 1975 
Record Number:  CaltechTHESIS:04112013102123813 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:04112013102123813 
DOI:  10.7907/3T9FVQ87 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  7610 
Collection:  CaltechTHESIS 
Deposited By:  Benjamin Perez 
Deposited On:  11 Apr 2013 18:14 
Last Modified:  21 Dec 2019 04:01 
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