In this thesis, the author presents several numerical methods for solving scalar and electromagnetic wave scattering problems. These methods are taken from the papers of Professor Alexander Ramm and the author, see [1] and [2].

In Chapter 1, scalar wave scattering by many small particles of arbitrary shapes with impedance boundary condition is studied. The problem is solved asymptotically and numerically under the assumptions a << d << λ, where k = 2π/λ is the wave number, λ is the wave length, a is the characteristic size of the particles, and d is the smallest distance between neighboring particles. A fast algorithm for solving this wave scattering problem by billions of particles is presented. The algorithm comprises the derivation of the (ORI) linear system and makes use of Conjugate Orthogonal Conjugate Gradient method and Fast Fourier Transform. Numerical solutions of the scalar wave scattering problem with 1, 4, 7, and 10 billions of small impedance particles are achieved for the first time. In these numerical examples, the problem of creating a material with negative refraction coefficient is also described and a recipe for creating materials with a desired refraction coefficient is tested.

In Chapter 2, electromagnetic (EM) wave scattering problem by one and many small perfectly conducting bodies is studied. A numerical method for solving this problem is presented. For the case of one body, the problem is solved for a body of arbitrary shape, using the corresponding boundary integral equation. For the case of many bodies, the problem is solved asymptotically under the physical assumptions a << d << λ, where a is the characteristic size of the bodies, d is the minimal distance between neighboring bodies, λ = 2π/k is the wave length and k is the wave number. Numerical results for the cases of one and many small bodies are presented. Error analysis for the numerical method are also provided.