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Summary: TEHRAN (FNA)- Engineers have designed a metamaterial device that can solve integral equations. The device works by encoding parameters into the properties of an incoming electromagnetic wave; once inside, the device's unique structure manipulates the wave in such a way that it exits encoded with the solution to a pre-set integral equation for that arbitrary input.
Consider the weakly singular Fredholm integral equation of the second kind
An integral equation is defined as an equation in which the unknown function [phi](x) to be determined appear under the integral sign.
Transformation of the original integral equation taking into account the peculiarities of the magnetization of the core.
For the problem with periodic boundary conditions the integral equation with Hilbert kernel is used [11]:
Now, we represent the main theorem of this study, through which a weakly singular Volterra integral equation can be expressed as a series of fractional differential transform for
Generally, these problems can be solved by an integral equation using method of moments (MoM) [1] because it has lesser degrees of freedom than differential equation methods.
Many authors (e.g., [1-4]) introduced integral equation methods for the two-dimensional Laplace equation solution in order to calculate the potential field.
(15) In the paper titled "Multiple Positive Solutions for Quadratic Integral Equations of Fractional Order," the existence of multiple positive solutions for a class of quadratic integral equation of fractional order was obtained by utilizing Avery-Henderson and Leggett-Williams multiple fixed-point theorems on cones.
Zajap, "Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity," Nonlinear Analysis: Theory, Methods & Applications, vol.
In this paper we prove the existence as well as approximation of the solutions of a certain generalized quadratic integral equation with maxima via an algorithm based on successive approximations dveloped in Dhage iteration method under weak partial Lipschitz and compactness type conditions.