Hence, by Theorem 2.5 in [22], p.112, we find that the function (F_{alpha} z)) has non-zero, finite nontangential boundary values at all points (e^{ivarphi}) ((varphiin[0,2pi])), except in a set (Ssubset[0,2pi]) of zero α-capacity.
A complete basis of homogeneous solutions for the interior and exterior regions, corresponding to all possible Dirichlet boundary values at the interface, are calculated in a preprocessing step.
At outflow points the boundary values at the upper time-level are obtained from data at the present time-level within and on the boundary via the analytical solutions, while the boundary values at inflow points remain constant in time.
The boundary values at the impacted end of the column are obtained from Hamilton's principle for the column and from the dynamics of the impacting mass.
In particular, the harmonic function (ln |h|) has to have boundary value at (z_0) equal to (-ln |z_0|).
By using Leggett-Williams norm-type theorems due to O'Regan and Zima [24], the existence of positive solutions for the boundary value problems at resonance with a linear derivative operator has been investigated (see [25 28]).
But in [8], Wang obtained the minimal and maximal nonnegative solutions for a second-order m-point boundary value problem at resonance by using a new fixed point theorem of increasing operators, and in this paper we use this method of Wang to establish the existence theorem of equations (1.1) and (1.2).
The one-dimensional initial boundary value problem has a fixed boundary at one end and a free boundary at the other.
The boundary value problem has a singularity at some (x=0) and hence, the scheme fails at (j=1).
Both use a time-stepping procedure, but one solves the boundary value problem at each time step by a boundary integral equation and the other uses a high-order spectral method.
