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 in [[alpha].sub.k].

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].

The

algebraic system was used because, in Brazil, it is the most usual and cannot be ignored.

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 [13].

By solving the

algebraic system (7), we obtain [[phi].sup.I.sub.ki] in n distinct points from the boundary of [L.sub.0].