Adjoining appropriate singular elements to transport theory computations

Convergence of Product Integration Rules Over (0, ∞) for Functions with Weak Singularities at the Origin

“…In particular, to test our bounds (2.12), we have applied the «-point Gauss-Laguerre formula to the following integral: (3.3) h= / e-xEx(\x-y\)xbdx, Jo where Ex(t) denotes the exponential integral (see [1]). Their kernels satisfy condition (2.14) with a = 0 and appear in certain well-known Wiener-Hopf integral equations (see [2,13,22]). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use…”

“…Finally, we consider one further application of the product rule used in the last example. We use it to construct a Nyström interpolant for the following case of the well-known linear transport equation (see [2,13]): (3.21) u(y) -± jH Ex(\x -y\)u(x) dx = \ , whose solution at x = 0 assumes the known value u(0) = ^ .…”

“…Since it is known (see [13]) that lim;c_,00 u(x) = 1 and u(x) -1 = o(e~ßX) as x -> oo, for any 0 < p < p*, p* = 0.957504, we set u(x) = u(x) -1 and rewrite (3.21) in the following form: We recall that ß e Cq°[0, oo) and, furthermore, as x -> 0 the solution u{x) behaves like c + xb for any 0 < b < 1 (see [13]). (Actually, it appears (see [2]) that U(x) ~ c + x logx. )…”

Convergence of product integration rules over (0,∞) for functions with weak singularities at the origin

“…In particular, to test our bounds (2.12), we have applied the «-point Gauss-Laguerre formula to the following integral: (3.3) h= / e-xEx(\x-y\)xbdx, Jo where Ex(t) denotes the exponential integral (see [1]). Their kernels satisfy condition (2.14) with a = 0 and appear in certain well-known Wiener-Hopf integral equations (see [2,13,22]). We recall (see [10]) that the coefficients wni(y) of (2.13) can be represented as follows: We consider first the case K(x, y) = e~\x~y\.…”

“…Finally, we consider one further application of the product rule used in the last example. We use it to construct a Nyström interpolant for the following case of the well-known linear transport equation (see [2,13]): (3.21) u(y) -± jH Ex(\x -y\)u(x) dx = \ , whose solution at x = 0 assumes the known value u(0) = ^ . Since it is known (see [13]) that lim;c_,00 u(x) = 1 and u(x) -1 = o(e~ßX) as x -> oo, for any 0 < p < p*, p* = 0.957504, we set u(x) = u(x) -1 and rewrite (3.21) in the following form: We recall that ß e Cq°[0, oo) and, furthermore, as x -> 0 the solution u{x) behaves like c + xb for any 0 < b < 1 (see [13]).…”