We consider the following singularly perturbed linear two-point boundary-value 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 elementsare 1 and last m elements are ε. Aside from reasonablecontinuity conditions placed on A, L, R, f, g, we assume thelower right mxm principle submatrix of A has no eigenvalueswhose real part is zero. Under these assumptions a constructive technique is used to derive sufficient conditions for the existenceof a unique solution of (1). These sufficient conditions are used todefine when (1) is a regular problem. It is then shown that asε → 0^+ the solution of a regular problem exists and converges onevery closed subinterval of (0,1) to a solution of the reduced problem.The reduced problem consists of the differential equationobtained by formally setting ε equal to zero in (1a) and initialconditions obtained from the boundary conditions (1b). Severalexamples of regular problems are also considered.
A similar technique is used to derive the properties of thesolution of a particular difference scheme used to approximate (1).Under restrictions on the boundary conditions (1b) it is shown thatfor the stepsize much larger than ε the solution of the difference scheme, when applied to a regular problem, accurately representsthe solution of the reduced problem.
Furthermore, the existence of a similarity transformationwhich block diagonalizes a matrix is presented as well as exponentialbounds on certain fundamental solution matrices associated with the problem (1).
