a chebyshev approximation for solving nonlinear integral equations of hammerstein type abstract a numerical method for solving fredholm volterra hammerstein integral equations is presented this method is based on replacement of the unknown function by truncated series of well known chebyshev expansion of functions the quadrature formula which we use to calculate integral terms can be estimated by fast fourier transform fft the numerical examples and the number of operations show the advantages of this method to some other usual methods

an approximation method for the solution of nonlinear integral equations a chebyshev collocation method has been presented to solve nonlinear integral equations in terms of chebyshev polynomials this method transforms the integral equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown chebyshev coefficients finally some examples are presented to illustrate the method and results discussed

practical computation of matrix functions abstract several new representations for an analytic function f a of a complex matrix a and in particular for e at and a t are derived which also are numerically useful in that they avoid the computation of eigenvalues of a

computability on the probability measures on the borel sets of the unit interval while computability theory on many countable sets is well established and for computability on the real numbers several mutually non equivalent definitions are applied for most other uncountable sets in particular for measures no generally accepted computability concepts at all ha ve been available until now in this contribution we introduce computability on the set m of probability measures on the borel subsets of the unit interval 0 1 its main purpose is to demonstrate that this concept of computability is not merely an ad hoc definition but has very natural properties although the definitions and many results can of course be transferred to more general spaces of measures we restrict our attention to m in order to keep the technical details simple and concentrate on the central ideas in particular we show that simple obvious reqirements exclude a number of similar definitions that the definition leads to the expected computability results that there are other natural definitions inducing the same computability theory and that the theory is embedded smoothly into classical measure theory as background we consider tte type 2 theory of effectivity kw84 kw85 which provides a frame for very realistic computability definitions in this approach computability is defined on finite and infinite sequences of symbols explicitly by turing machines and on other sets by means of notations and representations canonical representations are derived from information structures wei97 we introduce a standard representation delta m subseteq sum omega to m via some natural information structure defined by a subbase the atomic properties of some topology on m and a standard notation of while several modifications of m suggesting themselves at first glance violate simple and obvious requirements m has several very natural properties and hence should induce an important computability theory many interesting functions on measures turn out to be computable in particular linear combination integration of continuous functions and any transformation defined by a computable iterated function system with probabilities some other natural representations of m are introduced among them a cauchy representation associated with the hutchinson metric and proved to be equivalent to m as a corollary the final topology of m is the well known weak topology on m

dynamical systems measures and fractals via domain theory abstract we introduce domain theory in dynamical systems iterated function systems fractals and measure theory for a discrete dynamical system given by the action of a continuous map f x x on a metric space x we study the extended dynamical systems vx vf ux uf and lx lf where v u and l are respectively the vietoris hyperspace the upper hyperspace and the lower hyperspace functors we show that if x f is chaotic then so is ux uf when x is locally compact ux is a continuous bounded complete dcpo if x is second countable as well then ux will be continuous and can be given an effective structure we show how strange attractors attractors of iterated function systems fractals and julia sets are obtained effectively as fixed points of deterministic functions on ux or fixed points of non deterministic functions on cux where c is the convex plotkin power domain we also show that the set m x of finite borel measures on x can be embedded in pux where p is the probabilistic power domain this provides an effective framework for measure theory we then prove that the invariant measure of an hyperbolic iterated function system with probabilities can he obtained as the unique fixed point of an associated continuous function on pux

superconvergence of the iterated galerkin methods for hammerstein equations in this paper the well known iterated galerkin method and iterated galerkin kantorovich regularization method for approximating the solution of fredholm integral equations of the second kind are generalized to hammerstein equations with smooth and weakly singular kernels the order of convergence of the galerkin method and those of superconvergence of the iterated methods are analyzed numerical examples are presented to illustrate the superconvergence of the iterated galerkin approximation for hammerstein equations with weakly singular kernels

fast numerical solution of weakly singular volterra integral equations abstract we give a fortran program for the numerical solution of abel volterra convolution equations of the second and first kind we use a fractional linear multistep method bdf4 1 2 and use fast fourier transform techniques to exploit the convolution structure a detailed description of the computation of the quadrature weights using fft is given numerical examples are included

a computable version of the daniell stone theorem on integration and linear functionals for every measure the integral i f f d is a linear functional on the set of real measurable functions by the daniell stone theorem for every abstract integral f r on a stone vector lattice f of real functions f r there is a measure such that f d f for all f f in this paper we prove a computable version of this theorem

improved newton s method with exact line searches to solve quadratic matrix equation in this paper we study the matrix equation ax 2 bx c 0 where a b and c are square matrices we give two improved algorithms which are better than newton s method with exact line searches to calculate the solution some numerical examples are reported to illustrate our algorithms

product integration methods for solving a system of nonlinear volterra integral equations in this paper the technique of subtracting out singularities is used to derive explicit and implicit product euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of volterra integral equations with a weakly singular kernel the convergence proofs of the numerical schemes are presented these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global lipschitz condition

on the frechet derivative of matrix functions abstract in 1956 rinehart 4 discussed the derivatives of matrix functions by considering differences a e a for matrices e commuting with a in that case the derivative turned out to be a in this paper the case of noncommutative a and e is treated this leads to the frechet derivative of the matrix function an explicit integral representation is obtained using an approach that is similar to the one in 5 a finite sum in polynomials of a is obtained the coefficients may be computed recursively this is useful in computing the frechet derivative which is needed for newton s method to solve nonlinear matrix equations

on monotone abel volterra integral equations on the half line in this paper we reanalyze the trapezoidal method for the solution of nonlinear abel volterra integral equations on the half line we prove the convergence of the method in the uniform norm provided the nonlinearity is lipschitz continuous and strictly monotone

variable transformations in the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels extensions to fredholm integral equations abstract a number of techniques that use variable transformations in numerical integration have been developed recently cf sidi numerical integration iv h brass g hammerlin eds birkhauser basel 1993 pp 359 373 laurie j comput appl math 66 1996 337 344 the use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule in this paper the application of variable transformation techniques of sidi and laurie type to the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels is considered since the transformations are such that the end points of integration need not be used as mesh points the methods introduced can be used for vie with both continuous and weakly singular kernel in a uniform way the methods have also the advantages of simplicity of application and of achieving high order of convergence the application of the idea to fredholm integral equations with continuous and weakly singular equations is also considered numerical results are included and they verify the expected increased order of convergence they were obtained by using the trapezoidal formula for the evaluation of the transformed integrals

high order methods for weakly singular integral equations with nonsmooth input functions to solve one dimensional linear weakly singular integral equations on bounded intervals with input functions which may be smooth or not we propose to introduce first a simple smoothing change of variable and then to apply classical numerical methods such as product integration and collocation based on global polynomial approximations the advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well conditioned

computability on random events and variables in a computable probability space we study aspects of computability concerning random events and variables in a computable probability space which fulfills certain computability axioms to this end we introduce two multirepresentations of random events and random variables respectively employing the frechet nikodym metric and the ky fan metric they are shown to be recursively complete in guaranteeing computability of basic operations on random events and random variables some natural variations of the multirepresentation of random variables are defined for the integrable variables to explore computability of integration

a collocation method to find solutions of linear complex differential equations in circular domains in this study we introduce a collocation approach for solving high order linear complex differential equations in circular domain by using collocation points defined in a circular domain and bessel functions of the first kind this method transforms the linear complex differential equations into a matrix equation the matrix equation corresponds to a system of linear equations with the unknown bessel coefficients proposed method gives the analytic solution when the exact solutions are polynomials numerical examples are given to demonstrate the validity and applicability of the technique and the comparisons are made with existing results the results obtained from the examples demonstrate the efficiency and accuracy of the present work all of the numerical computations have been computed on computer using a code written in matlab

maximum and minimum entropy states yielding local continuity bounds given an arbitrary quantum state we obtain an explicit construction of a state e respectively e which has the maximum respectively minimum entropy among all states which lie in a specified neighborhood e ball of computing the entropy of these states leads to a local strengthening of the continuity bound of the von neumann entropy i e the audenaert fannes inequality our bound is local in the sense that it depends on the spectrum of the states e and e depend only on the geometry of the e ball and are in fact optimizers for a larger class of entropies these include the renyi entropy and the minimum and maximum entropies providing explicit formulas for certain smoothed quantities this allows us to obtain local continuity bounds for these quantities as well in obtaining this bound we first derive a more general result which may be of independent interest namely a necessary and sufficient condition under which a state maximizes a concave

the numerical solution of weakly singular volterra integral equations by collocation on graded meshes since the solution of a second kind volterra integral equation with weakly singular kernel has in general unbounded derivatives at the left endpoint of the interval of integration its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates in this paper we investigate the convergence rates with respect to graded meshes and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation

computability on random variables abstract in this paper we study aspects of computability concerning random variables under the background of type 2 theory of effectivity tte we show that the resulting definitions are natural as they suffice to successfully discuss questions of computability of basic queueing systems e g of the so called m g 1 system

periodizing transformations for numerical integration the order of the trapezoidal rule can be raised by making a substitution that transforms the integrand into a new integrand which has a smooth periodic extension the exponent in the rate at which the transformed weights approach zero near the endpoints is called the damping power of the transformation currently used polynomial transformations of damping power m have order m 1 if m is odd and order m 2 if m is even we prove that the highest possible order is m 1 if m is odd but 2m 2 if m is even and give new polynomial transformations that achieve the optimal order

bessel polynomial solutions of high order linear volterra integro differential equations abstract in this study a practical matrix method which is based on collocation points is presented to find approximate solutions of high order linear volterra integro differential equations vides under the mixed conditions in terms of bessel polynomials numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results the results show the efficiency and accuracy of the present work all of the numerical computations have been performed on the computer using a program written in matlab v7 6 0 r2008a

on the solutions of a system of linear retarded and advanced differential equations by the bessel collocation approximation in this study we present a numerical approximation for the solutions of the system of high order linear retarded and advanced differential equations with variable coefficients under the mixed conditions this method is based on taking the truncated bessel expansion of the functions in the retarded and advanced differential equation system by the aid of the matrix operations and the collocation points the problem is transformed into a matrix equation with the unknown bessel coefficients by solving this matrix equation the unknown coefficients of the approximate solutions are computed in addition examples that illustrate the pertinent features of the method are presented and the results of this investigation are discussed

a family of methods for abel integral equations of the second kind abstract a class of methods depending on some parameters are introduced for the numerical solution of the abel integral equations of the second kind some bounds on the parameters are determined so that the corresponding methods have infinite stability intervals

a computable approach to measure and integration theory we introduce a computable framework for lebesgue s measure and integration theory in the spirit of domain theory for an effectively given second countable locally compact hausdorff space and an effectively given finite borel measure on the space we define a recursive measurable set which extends the corresponding notion due to s anin for the lebesgue measure on the real line we also introduce the stronger notion of a computable measurable set where a measurable set is approximated from inside and outside by sequences of closed and open subsets respectively the more refined property of computable measurable sets give rise to the idea of partial measurable subsets which naturally form a domain for measurable subsets we then introduce interval valued measurable functions and develop the notion of recursive and computable measurable functions using interval valued simple functions this leads us to the interval versions of the main results in classical measure theory the lebesgue integral is shown to be a continuous operator on the domain of interval valued measurable functions and the interval valued lebesgue integral provides a computable framework for integration

product integration methods for second kind abel integral equations the construction and convergence of high order product integration methods for the second kind abel equation are discussed and the results of de hoog and weiss are generalised backward difference methods are introduced and numerical results are presented which verify the theoretical rates of convergence