A numerical method for solving Fredholm-Volterra integral equations in two-dimensional spaces using block pulse functions and an operational matrix

In this paper, the block pulse functions (BPFs) and their operational matrix are used to solve two-dimensional Fredholm-Volterra integral equations (F-VIE). This method converts F-VIE to systems of linear equations whose solutions are the coefficients of block pulse expansions of the solutions of F-VIE.Finally some numerical examples are presented to show the efficiency and accuracy of the method.     

On the regularity of solutions to integral equations with nonsmooth kernels on a union of open intervals

The behaviour of a solution to a Fredholm integral equation of the second kind on a union of open intervals is examined. The kernel of the corresponding integral operator may have diagonal singularities, information about them is given through certain estimates. The weighted spaces of smooth functions with boundary singularities containing the solution of the integral equation are described.     

On the Spectrum of the Reflection Operator on Conical Surfaces

We study the mapping properties of the reflection operator on a conical surface. This allows us to derive regularity results for the solution of the radiosity equation on conical surfaces in a scale of weighted Sobolev spaces. To motivate the calculations we first study the operator on a cylinder. Here we estimate the asymptotic behavior of the spectrum of the reflection operator by partial integration. This method works also for the conical case, but first we have to find a simple representation for some hypergeometric functions.     

Properties of an integral operator preserved under pointwise multiplication of its kernel by a function

Let K be a kernel that determines an integral operator on some space of functions, and let H be a function. This paper investigates conditions under which certain properties of the integral operator determined by K (especially compactness properties) also hold for the integral operator determined by HK.     

A new explicit solution method for the diffraction through a slit – Part 2

In the paper [N. Gorenflo, A new explicit solution method for the diffraction through a slit, ZAMP 53 (2002), 877–886] the problem of diffraction through a slit in a screen has been considered for arbitrary Dirichlet data, prescribed in the slit, and under the assumption that the normal derivative of the diffracted wave vanishes on the screen itself. For this problem certain functions with the following properties have been constructed: Each function f is defined on the whole of R and on the screen the values f(x), |x| ≥ 1, are the Dirichlet data of the diffracted wave which takes on the Dirichlet data f(x), |x| ≤ 1, in the slit. The problem of expanding arbitrary Dirichlet data, prescribed in the slit, into a series of functions of the considered form has been addressed, but not solved in a satisfactory way (only the application of the Gram-Schmidt orthogonalization process to such functions has been proposed). In this continuation of the aforementioned paper we choose the remaining degrees of freedom in the earlier given representations of such functions in a certain way. The resulting concrete functions can be expressed by Hankel functions and explicitly given coefficients. We suggest the expansion of arbitrary Dirichlet data, prescribed in the slit, into a series of these functions, here the expansion coefficients can be expressed explicitly by certain moments of the expanded data. Using this expansion, the diffracted wave can be expressed in an explicit form. In the future it should be examined whether similar techniques as those which are presented in the present paper can be used to solve other canonical diffraction problems, inclusively vectorial diffraction problems.     

Zeros of entire functions and a problem of Ramanujan

We derive representations for certain entire q-functions and apply our technique to the Ramanujan entire function (or q-Airy function) and q-Bessel functions. This is used to show that the asymptotic series of the large zeros of the Ramanujan entire function and similar functions are also convergent series. The idea is to show that the zeros of the functions under consideration satisfy a nonlinear integral equation.     

An approximation method using approximate approximations

The aim of this article is to extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present article we develop an approximation procedure for the solution of the interior Dirichlet problem for the Laplace equation in two dimensions using approximate approximations. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In the first step, the unknown source density in the potential representation of the solution is replaced by approximate approximations. In the second, the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in the third, Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.     

The existence problem for a nonlinear Abel equation on the half-line

We discuss a nonlinear Abel equation on the half-line (−∞,c), c>0. The basic results provide criteria for the existence of nontrivial everywhere positive solutions. They are expressed in terms of the generalized Osgood condition.     

Bounded mild solutions of perturbed Volterra equations with infinite delay

We study existence and regularity of bounded mild solutions on the real line to perturbed integral equations with infinite delay in the space of almost periodic functions (in the Bohr sense), the space of compact almost automorphic functions, the space of almost automorphic functions and the space of asymptotically almost automorphic functions.     

Zonal spherical functions on the complex reflection groups and (n+1,m+1)-hypergeometric functions

The pair of groups, complex reflection group G(r,1,n) and symmetric group S_n, is a Gelfand pair. Its zonal spherical functions are expressed in terms of multivariate hypergeometric functions called (n+1,m+1)-hypergeometric functions. Since the zonal spherical functions have orthogonality, they form discrete orthogonal polynomials. Also shown is a relation between monomial symmetric functions and the (n+1,m+1)-hypergeometric functions.     

On the inversion of the Laplace transform for resolvent families in <Emphasis Type="Italic">UMD</Emphasis> spaces

We analyze the inversion of the Laplace transform in UMD-spaces for resolvent families associated to an integral Volterra equation of convolution type.Received: 25 March 2002     

Solutions of some singular integral equation

A singular integral equation with a Holderian second member function on [a,b] is considered and solved for four different type of kernels in the class of functions that are unbounded at the end points of the interval.     

p-Harmonic functions with boundary data having jump discontinuities and Baernstein's problem

For p-harmonic functions on unweighted R², with 1<p<∞, we show that if the boundary values f has a jump at an (asymptotic) corner point z₀, then the Perron solution Pf is asymptotically a+barg(z−z₀)+o(|z−z₀|). We use this to obtain a positive answer to Baernstein's problem on the equality of the p-harmonic measure of a union G of open arcs on the boundary of the unit disc, and the p-harmonic measure of . We also obtain various invariance results for functions with jumps and perturbations on small sets. For p>2 these results are new also for continuous functions. Finally we look at generalizations to Rn and metric spaces.     

Cut-and-paste of quadriculated disks and arithmetic properties of the adjacency matrix

We define cut-and-paste, a construction which, given a quadriculated disk obtains a disjoint union of quadriculated disks of smaller total area. We provide two examples of the use of this procedure as a recursive step. Tilings of a disk Δ receive a parity: we construct a perfect or near-perfect matching of tilings of opposite parities. Let B_Δ be the black-to-white adjacency matrix: we factor , where L and U are lower and upper triangular matrices, is obtained from a larger identity matrix by removing rows and columns and all entries of L, and U are equal to 0, 1 or -1.     

Convergence theorems for HL integrable vector-valued functions with applications

In this paper we prove dominated and monotone convergence theorems for HL integrable Banach-valued functions. These results and a fixed point theorem in ordered spaces are then applied to prove existence and comparison results for integral equations of Fredholm type in ordered Banach spaces involving Kurzweil integrals or improper integrals. Results are used also to solve concrete second-order functional boundary value problems involving discontinuities and singularities.     

Chromatic sums of general maps on the sphere and the projective plane

In this paper, we study the chromatic sum functions of rooted general maps on the sphere and the projective plane. The chromatic sum function equations of such maps are obtained. From the chromatic sum equations of such maps, the enumerating function equations of rooted loopless maps, bipartite maps and Eulerian maps are also derived. Moreover, some explicit expressions of enumerating functions are also derived.