**1. ****The implement of the first method: kernel.mpl**

Function *KernelMethod*
solve the equation system

*
K(x,y)F(x,y)=A(x,y)G(x)+B(x,y)*

The basic input is

*KernelMethod**(K,A,B,x,y)*

where *K,A* are *n*
by *n* matrices and *B* is an *n* by *1* matrix. The output
is a set of solutions. Each solution is an array of length n, indicating
G1,G2,...Gn.

** **For example, the command

*KernelMethod**(matrix([[z*x^2-x+z]]),matrix([[z]]),matrix([[-x]]),z,x);*

returns

*{[-1/2*(-1+(1-4*z^2)^(1/2))/z^2]}*

** **As mentioned in Remark
3, we may get extra equations by setting *v=1*. This can be done by adding
an extra parameter *[v=1]*.

The *K,A,B* corresponding
to Sections 4.1 and 4.2 can be generated by the functions *K01(n),A01(n),B01(n)*
and *K02(n),A02(n),B02(n)* respectively.** **For the examples in the
paper, see paperex.mws**.**

**2. ****The implement of the second method: kernel-wu.mpl**

We use
the Maple package wsolve to eliminate extra variables. The input is the
same as above and the output are the equations that *Gi* satisfy.** **For examples, see eqns.mws**.**