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 [12]:
For a nonperiodic indenter the
integral equation with Cauchy kernel is used [10]:
Recently many authors proposed various numerical methods for solving one-dimensional fuzzy
integral equations [19].
Many engineering and physical problems result in the analysis of the nonlinear weakly singular Volterra
integral equations (WSVIEs).
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 with maxima