A reduction of the Maxwell’s system to a Fredholm second-kind integral equation with weakly singular kernel is given for electromagnetic (EM) wave scattering by one and many small bodies. This equation is solved asymptotically as the characteristic size of the bodies tends to zero. The technique developed is used for solving the manybody EM wave scattering problem by rigorously reducing it to s...

In this work, we are interested in the small time local null controllability for the viscous Burgers’ equation yt − yxx + yyx = u(t) on the line segment [0, 1], with null boundary conditions. The second-hand side is a scalar control playing a role similar to that of a pressure. In this setting, the classical Lie bracket necessary condition [f1, [f1, f0]] introduced by Sussmann fails to conclude...

In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...