An existence theorem for the reduced wave equation

Some distribution solutions of integral equations and their application to partial differential equations

“…The (This follows from the fact that, as pointed out in [4] and [5] the derivative of the left hand side of (1.12) is precisely the left hand side of (1.10)). For our purposes it is convenient to renormalize the sets {Tn},"=, and (Un},"=, so that they become orthonormal.…”

“…Similarly Eq. (1.2) is related to the Helmholtz equation These equations have been studied by the author in two previous papers [4], [5]. In order to be able to present the results of the previous work and to state the aims of the present work we introduce the following notation: Let p(t) = (1 -t2)-*, -1 < t < 1 and q(t) = (I -t2)*, -1 < t < 1.…”

The Reduced Wave Equation with a Different Radiation Condition

“…Previously Wilcox explicitly used this integral form in [8] with stronger regularity conditions. Using Levine's result and the Fredholm's theorem, P. Wolfe [9] proved the existence of the solution with the radiation condition (4) in the two-dimensional case. The result since Y r IDul z dx o(log R) o-<lxl__<R because u is outgoing.…”

“…We assert that L(Ax) R is dense in R. Otherwise there exists g 0 in R such that (9) fe rZg(Lu) 0 for all u coo(E) f'l A with IIAu + ull finite. In particular, (9) Thus we can expand rZg in terms of spherical harmonics in r >= a for some a" (11) rZg a,.,j(r) Ym,j(O, dp). m=Oj=-m Let h.,(r) equal "'m+4() /2(r)/ and U,j(r, 0, ) equal h(r)Y,j(O, ).…”

The Helmholtz equation in the exterior of slits in a plane with different impedance boundary conditions on opposite sides of the slits

“…Neumann problems for the Laplace and Helmholtz equations in the exterior of several single-sided open arcs in a plane have been studied in [9,21,22]. If we consider each open arc as a slit in a plane, then by singlesided open arcs we mean the case when the same boundary data is specified on both sides of the slits.…”