A singularly perturbed linear two-point boundary-value problem

• Main Author: Ferguson, Warren E.
• Format: Others
• Published: 1975
• Online Access: https://thesis.library.caltech.edu/7610/1/Ferguson_we_1975.pdf; Ferguson, Warren E. (1975) A singularly perturbed linear two-point boundary-value problem. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/3T9F-VQ87. https://resolver.caltech.edu/CaltechTHESIS:04112013-102123813 <https://resolver.caltech.edu/CaltechTHESIS:04112013-102123813>

<p>We consider the following singularly perturbed linear two-point boundary-value problem:</p> <p>Ly(x) ≡ Ω(ε)D_xy(x) - A(x,ε)y(x) = f(x,ε) 0≤x≤1 (1a)</p> <p>By ≡ L(ε)y(0) + R(ε)y(1) = g(ε) ε → 0^+ (1b)</p> <p>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.</p> <p>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.</p> <p>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).</p>