This "photonic calculus" would work by encoding parameters into the properties of an incoming electromagnetic wave and sending it through a metamaterial device; once inside, the device's unique structure would manipulate the wave in such a way that it would exit encoded with the solution to a pre-set integral equation
for that arbitrary input.
It is well known [6, 12,26,32] (see also [11, 15,20]) that the derivatives of a solution to a weakly singular integral equation
such as (1.1), in general, have certain boundary singularities.
A general form of an integral equation
in [phi](x) is of the form
In this case, the integral equation
for [[sigma].sub.2] takes the form :
For a nonperiodic indenter the integral equation
with Cauchy kernel is used :
Recently many authors proposed various numerical methods for solving one-dimensional fuzzy integral equations
Many engineering and physical problems result in the analysis of the nonlinear weakly singular Volterra integral equations
This is because of the inherent quality of the Fredholm integral equation
of the second kind, where the boundary conditions are naturally imbedded.
The method is based on the reduction of the problem to the Fredholm integral equation
of the second kind for the boundary values of the conjugate harmonic function.
(10) In the paper titled "Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations
with Weakly Singular Kernels," the authors developed a generalized Jacobi-Galerkin method for the second kind Volterra integral equations
with weakly singular kernels.
Pachpatte, "On Volterra-Fredholm integral equation
in two variables," Demonstratio Mathematica, vol.
and if k(t,x,y) = q(t) and f (t,x,y) = 1 for all t [member of] J and x, y [member of] R, it is reduced to nonlinear usual Volterra integral equation