The mathematical model of some environmental physics problems is represented by a
singular integral equation with an oscillatory kernel. We investigate a method for the numerical evaluation of Cauchy principal value integrals of oscillatory functions. The method is based on an interpolatory procedure at the zeros of the orthogonal
polynomials with respect to a Jacobi weight. In this way, we obtain a procedure that is numerically stable and the algorithm can be implemented in a fast way yielding satisfactory numerical results. Bounds of the error and of the amplification factor are also provided.
Using the de la Vallèe Poussin interpolation at the Chebyshev zeros, the authors construct
polynomial interpolating wavelets and give the corresponding decomposition and reconstruction algorithms. The involved
matrices can be
diagonalized by sine and cosine orthogonal matrices. So the algorithms can be realized using fast sine and cosine transforms.
We consider a Cauchy singular integral equation on the real line. A direct numerical mehod
for solving this integral equation is given. We prove the convergence of the proposed method.
The numerical resolution of a Cauchy singular integral equation on the real line is stricly
related with the good approximation of the Hilbert transform. In this paper we consider a numerical method besed on an
approximation of
the Hilbert transform and for this we prove the convergence in weighted uniform spaces.
We deal with the numerical evaluation of the Hilbert transform on the real line by a Gauss
type quadrature rule. The convergence and the stability of the method are investigated.
In this paper we consider a class of strongly singular, nonlinear integral equations. A
numerical method is proposed and its convergence is proved in weighted Sobolev spaces.
An algorithm for the approximate evaluation of the Hilbert transform has been prposed. The
convergence of the procedure is proved. The stability of the algorithm is considered and some numerical examples are given.
The authors study the convergence and the stability of a collocation and a discrete collocation method for Cauchy singular
integral equations with weakly singular perturbation kernel in some weighted uniform norms. Uniform error estimates are also
given.
The authors consider some procedure for solving a class of singular integral equations of Cauchy type by the collocation
method on the zeros of orthogonal polynomials with respect to a weight function similar to that of Jacobi. Uniform convergence
theorems are proved.
