Using the 5-point finite difference scheme, (16) is reduced to the following algebraic system
The matrix in a linear algebraic system
obtained from a Shishkin mesh discretization of a singularly perturbed convection-diffusion equation is nonsymmetric and often highly nonnormal and ill-conditioned.
We carry out a direct substitution of (4) into (3) and gather the coefficients of the resulting polynomial in x, y, t, and z, to obtain a nonlinear algebraic system
To solve a nonlinear algebraic system
(3), the method of extension by parameter in conjunction with the iterative Newton method is applied.
For converting the system of integral equations to an algebraic system
, at first by means of change of variables in the system and in conditions (15) we reduce all the integration integrals to one interval [-1;1].
This leads to the nonlinear algebraic system
of equations: [mathematical expression not reproducible] (A2)
The new algebraic system
of equations can be written in form given in (10):
So the sparsity of the Jacobian matrix based on these different basis functions indicates how to solve the nonlinear algebraic system
. Next, we give the formulation for two kinds of wavelet bases.
Step 4 collecting all the terms with the same power of [e.sup.[alpha][xi]] yields a set of algebraic system
for [a.sub.i], [b.sub.i](i = 0, 1, ..., 2m), [alpha], where [a.sub.i], [b.sub.i](i = 0, 1, ..., 2m), [alpha] are coefficients to be determined later;
In order to find an algebraic system
of four equations with four unknowns, it is necessary to integrate over the area (r,[theta]), where r = [0,a], and [theta] = [0,27[pi]], by using the orthogonal-relations [[integral].sup.2[pi].sub.0] cos(n[theta]) cos(n'[theta])d[theta] = [pi][[delta].sub.nn]', [[integral].sup.2[pi].sub.0] sin(n[theta]) sin(n'[theta])d[theta] = [pi][[delta].sub.nn]', and [[integral].sup.2[pi].sub.0] sin(n[theta]) cos(n'[theta])d[theta] = 0, where [[delta].sub.nn]' is the Kronecker delta which equals unity for n = n', and zero otherwise .
By solving the algebraic system
(7), we obtain [[phi].sup.I.sub.ki] in n distinct points from the boundary of [L.sub.0].