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.