These

boundary values are fed into the NEW android app.

Let u, v be two solutions of (1) with the partial homogeneous

boundary values (4) and with the initial values [u.sub.0], [v.sub.0], respectively.

In Section 3, we establish analogues of results from Section 2, in terms of smoothness with respect to

boundary values, for solutions of the nonlocal problem (1.1), (1.2).

The numerical results obtained were everywhere finite, free of singularities, and confirmed the well-known prediction that [u.sub.RS.sup.(p)] and [u.sub.RS.sup.(s)] reproduce the

boundary values assumed in their derivation ([partial derivative][u.sub.RS.sup.(p)]/[partial derivative]z [right arrow] ik and [u.sub.RS.sup.(s)] [right arrow] 1 as z [right arrow] 0) but not the compatible values ([u.sub.RS.sup.(p)] [right arrow] 1 and [u.sub.RS.sup.(s)]/[partial derivative]z [right arrow] ik) which are implied in the classical postulate that the aperture field is the same as the unperturbed geometrical field incident on the screen.

Data from our previous work was included in the more recent paper because we needed "

boundary values" between or below which we were looking for a threshold or a critical level.

Liu, "Positive solution to nonzero

boundary values problem for a coupled system of nonlinear fractional differential equations," International Journal of Differential Equations, vol.

These

boundary values are step functions that violate the wave equation and are the probable cause of the fact, shown in Appendix B, that the Rayleigh-Sommerfeld integrals also do not obey the wave equation in the immediate proximity of the aperture plane.

On the other hand, because of the wide mathematical and physical background, the existence of positive solutions for nonlinear integer-order

boundary values problems with p-Laplacian operator has received wide attention (see [8, 12, 13, 21, 24-29]).