Hypersingular Integral Equations Arising in the Boundary Value Problems of the Elasticity Theory

Summary The exact solutions of a class of hypersingular integral equations with kernels $\left( {s-x} \right)^{-2}$, $\left( {\sin \frac{s-x}{2}} \right)^{-2}$, $\left( {\sinh \frac{s-x}{2}} \right)^{-2},\cos \frac{s-x}{2}\left( {\sin \frac{s-x}{2}} \right)^{-2}$, $\cosh \frac{s-x}{2}\left( {\sinh \frac{s-x}{2}} \right)^{-2}$ are obtained where the integrals must be interpreted as Hadamard finite-part integrals. Problems of cracks in elastic bodies of various canonical forms under antiplane and plane deformations, where the crack edges are loaded symmetrically, lead to such equations. These problems, in turn, lead to mixed boundary value problems of the mathematical theory of elasticity for a half-plane, a circle, a strip and a wedge.

Jacobi Series for General Parameters via Hadamard Finite Part and Application to Nonhomogeneous Hypergeometric Equations

The representation of analytic functions as convergent series in Jacobi polynomials Pn(α,β) is reformulated using the Hadamard principal part of integrals for all α,β∈C∖{0,-1,-2,…}, α+β≠-2,-3,…. The coefficients of the series are given as usual integrals in the classical case (when Rα,Rβ>-1) or by their Hadamard principal part when they diverge. As an application it is shown that nonhomogeneous differential equations of hypergeometric type do generically have a unique solution which is analytic at both singular points in C.

Integro-differential equations with singular and hypersingular integrals

Quadrature linear integro-differential equations on a closed curve located in the complex plane are solved. The equations contain singular integrals which are understood in the sense of the main value and hypersingular integrals which are understood in the sense of the Hadamard finite part. The coefficients of the equations have a special structure.

A Further Property of Functions in Class B(m): An Application of Bell Polynomials

We say that a function $\alpha(x)$ belongs to the set ${\bf A}^{(\gamma)}$ if it has an asymptotic expansion of the form $\alpha(x)\sim \sum^\infty_{i=0}\alpha_ix^{\gamma-i}$ as $x\to\infty$, which can be differentiated term by term infinitely many times. A function  $f(x)$ is in the class ${\bf B}^{(m)}$ if it satisfies a linear homogeneous differential equation of the form $f(x)=\sum^m_{k=1}p_k(x)f^{(k)}(x)$, with $p_k\in {\bf A}^{(i_k)}$, $i_k$ being integers satisfying $i_k\leq k$. These functions appear in many problems of applied mathematics and other scientific disciplines. They have been shown to have many interesting properties,  and their integrals $\int^\infty_0 f(x)\,dx$, whether convergent or divergent,  can be evaluated very efficiently via the Levin--Sidi $D^{(m)}$-transformation,  a most effective convergence acceleration method. (In case of divergence, these integrals  are defined in some summability sense, such as Abel summability or Hadamard finite part or a mixture of these two.) In this note, we show that if $f(x)$ is in ${\bf B}^{(m)}$, then so is $(f\circ g)(x)=f(g(x))$, where $g(x)>0$ for all large $x$ and $g\in {\bf A}^{(s)}$,  $s$ being a positive integer. This enlarges the scope of the $D^{(m)}$-transformation considerably to include functions of complicated arguments. We demonstrate  the validity of our result with an application of the $D^{(3)}$ transformation to two integrals $I[f]$ and $I[f\circ g]$, for some $f\in{\bf B}^{(3)}$ and $g\in{\bf A}^{(2)}$. The Fa\`{a} di Bruno formula and Bell polynomials play a central role in our study.

Error bounds for a Gauss-type quadrature rule to evaluate hypersingular integrals

In the present paper we consider hypersingular integrals of the following type =?+?,0 f(x)/(x-t)p+1 w?(x)dx, where the integral is understood in the Hadamard finite part sense, p is a positive integer, w?(x) = e-xx? is a Laguerre weight of parameter ? ? 0 and t > 0. In [6] we proposed an efficient numerical algorithm for approximating (1), focusing our attention on the computational aspects and on the efficient implementation of the method. Here, we introduce the method discussing the theoretical aspects, by proving the stability and the convergence of the procedure for density functions f s.t. f(p) satisfies a Dini-type condition. For the sake of completeness, we present some numerical tests which support the theoretical estimates.