12. The New Transform Method for Elliptic PDEs in Simple Polygonal Domains
ATHANASSIOS S. FOKAS · Society for Industrial and Applied Mathematics eBooks · 2008
The implementation of the new transform method to linear evolution PDEs makes crucial use of the following facts: (a) There exist transformations in the complex k-plane which leave invariant the transforms of the boundary values. (b) The contribution of the unknown function q^ (k, T) to the integral representation either vanishes or yields a contribution which can be computed explicitly through a residue calculation. It turns out that the implementation of the new transform method to elliptic PDEs uses similar facts. Consider for example the Laplace equation in the quarter plane (see Example 11.1); for the Dirichlet problem, the unknown boundary values are given by −iU1/2 and −iU2/2, whereU1(k)=∫0∞ekyqx(0,y)dy,Rek≤0,andU2(−ik)=∫0∞e−ikxqy(x,0)dx,Imk≤0.It is important to recall that for second order elliptic PDEs there exist two global relations. The second global relation involves the Schwarz conjugate of the integrals of the boundary values, which in this example are U1(k) and U2(ik). Hence, the two global relations involve the following two vectors (U1(k), U1(k)), (U2(ik), U2(−ik)). Regarding fact (a) mentioned above, we note that the components of the second vector remain invariant under the transformation k → −k. Thus, in this case we must supplement the two global relations with the two equations obtained by replacing k with −k. Regarding fact (b) above, we note that using the global relations and the equations obtained under the transformation k → −k, it is possible to express U1(k) and U2(−ik) in terms of U2(ik). This function is analytic in the first quadrant of the complex k-plane, which is the domain involved in the integral representation (see Figure 11.4), and it turns out that its contribution vanishes.