Inversion of the Laplace transform from the real axis using an adaptive iterative method

In this paper a new method for inverting the Laplace transform from the real axis is formulated. This method is based on a quadrature formula. We assume that the unknown function $f(t)$ is continuous with (known) compact support. An adaptive iterative method and an adaptive stopping rule, which yield the convergence of the approximate solution to $f(t)$, are proposed in this paper.Comment: 38 pages, 12 figure

Creating materials with a desired refraction coefficient: numerical experiments

A recipe for creating materials with a desired refraction coefficient is implemented numerically. The following assumptions are used: \bee \zeta_m=h(x_m)/a^\kappa,\quad d=O(a^{(2-\kappa)/3}),\quad M=O(1/a^{2-\kappa}),\quad \kappa\in(0,1), \eee where $\zeta_m$ and $x_m$ are the boundary impedance and center of the $m$-th ball, respectively, $h(x)\in C(D)$, Im$h(x)\leq 0$, $M$ is the number of small balls embedded in the cube $D$, $a$ is the radius of the small balls and $d$ is the distance between the neighboring balls. An error estimate is given for the approximate solution of the many-body scattering problem in the case of small scatterers. This result is used for the estimate of the minimal number of small particles to be embedded in a given domain $D$ in order to get a material whose refraction coefficient approximates the desired one with the relative error not exceeding a desired small quantity.Comment: 24 page

Dynamical Systems Method for solving ill-conditioned linear algebraic systems

A new method, the Dynamical Systems Method (DSM), justified recently, is applied to solving ill-conditioned linear algebraic system (ICLAS). The DSM gives a new approach to solving a wide class of ill-posed problems. In this paper a new iterative scheme for solving ICLAS is proposed. This iterative scheme is based on the DSM solution. An a posteriori stopping rules for the proposed method is justified. This paper also gives an a posteriori stopping rule for a modified iterative scheme developed in A.G.Ramm, JMAA,330 (2007),1338-1346, and proves convergence of the solution obtained by the iterative scheme.Comment: 26 page

An iterative method for solving Fredholm integral equations of the first kind

The purpose of this paper is to give a convergence analysis of the iterative scheme: \bee u_n^\dl=qu_{n-1}^\dl+(1-q)T_{a_n}^{-1}K^*f_\dl,\quad u_0^\dl=0,\eee where $T:=K^*K,\quad T_a:=T+aI,\quad q\in(0,1),\quad a_n:=\alpha_0q^n, \alpha_0>0,$ with finite-dimensional approximations of $T$ and $K^*$ for solving stably Fredholm integral equations of the first kind with noisy data.Comment: 29 page

A collocation method for solving some integral equations in distributions

A collocation method is presented for numerical solution of a typical integral equation Rh :=\int_D R(x, y)h(y)dy = f(x), x {\epsilon} D of the class R, whose kernels are of positive rational functions of arbitrary selfadjoint elliptic operators defined in the whole space R^r, and D \subset R^r is a bounded domain. Several numerical examples are given to demonstrate the efficiency and stability of the proposed method.Comment: 26 pages, 4 figure