Second, a nonlinear term permits singularities with respect to both the time and the space variables.
At the same time, the nonlinearity f in this paper permits singularities with respect to both the time and the space variables which is seldom considered at present.
In particular, our framework includes classical equations with delay (or advance), singularities with respect to the independent variable, and implicit functional boundary conditions.
For nonlinear differential equations with singularities with respect to phase variables, Kiguradze established optimal sufficient conditions for the solvability of the Cauchy problem [148], nonlocal problems [149], [150] and periodic type problems [151].
Kiguradze obtained a number of a priori estimates for one-sided differential inequalities subject some boundary value conditions; these estimates form a basis for the construction of a complete theory of initial and boundary value problems for differential equations and systems with nonintegrable singularities with respect to the time variable.
Moreover, he obtained necessary and sufficient conditions for the existence of Kneser solutions blowing-up and vanishing at infinity for higher-order nonlinear differential equations with singularities with respect to the time and phase variables, as well as global two-sided estimates for such solutions [105], [107].
N 15 20 30 40 45 50 60 (lambda_{1delta}) 0.6976 0.8019 0.9054 0.9893 0.9993 1 1. Figure 6 shows the error curves in a logarithmic scale of the numerical solution of the discrete problem (8) (respectively the leading singularity coefficient) with respect to (log(N)) in the case of (omega= {2pi}).
We study now two examples of the numerical calculation of the discrete leading singularity coefficient (lambda_{1delta}) with respect to N. (varphi_{delta}^ x,y)=sin^{2}pi x^{2}sin^{2}pi y^{2}) and (omega={3pi over 2}).
For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order.
This method could be shown to be robust with respect to singularities in time and retains its high convergence order in case that the analytical solution is appropriately smooth.
