A singularly perturbed linear two-point boundary-value problem
<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...
| Summary: | <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> |
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