The exact order of information-based complexity of weakly singular integral equations with periodic analytic coefficients

The exact order of complexity of weakly singular integral equations including logarithmic singularities with periodic and analytic coefficients is found. This class of equations contains the boundary equations of exterior boundary value problems for the two-dimensional Helmholtz equation. Translated fromMatematicheskie Zarnetki, Vol. 62, No. 5, pp. 643–656, November, 1997. Translated by O. V. Sipacheva     

Spectral singularities of Klein-Gordon s-wave equations with an integral boundary condition

We investigate the spectral singularities and the eigenvalues of the boundary value problem $$\begin{gathered} y'' + \left[ {\lambda - Q\left( x \right)} \right]^2 y = 0,x \in R_ + = [0,\infty ), \hfill \\ \quad \int\limits_0^\infty {K\left( x \right)y\left( x \right)dx + \alpha y'\left( 0 \right) - \beta y\left( 0 \right) = 0,} \hfill \\ \end{gathered}$$ where Q and K are complex valued functions, K∈L ²(R ₊), α,β∈C with |α|+|β|≠0 and λ is a spectral parameter.     

On the singularities of solutions of partial differential equations with constant coefficients

LetP be a differential operator with constant coefficients in ℝ ⁿ . Ifu is a distribution, the singular support ofu is the complement of the largest set whereu ∈C ^∞. Necessary and sufficient conditinos are obtained for a closed convex set Γ to be equal to the singular support ofu for someu withPu ∈C ^∞ or, equivalently, for Γ to contain the singular support ofu for someu withPu ∈C ^∞ butu ∉C ^∞. Related local uniqueness theorems analogous to the Holmgren theorem with supports replaced by singular supports are also given, as well as applications concerningP-convexity with respect to singular supports.     

On boundary integral equations in electroelasticity

A class of plane electroelasticity problems of the steady vibrations of bodies with a smooth boundary is studied. A system of boundary integral equations for the components of the displacement vector and the potential is formulated on the basis of the fundamental solution constructed and its analysis.     

On polynomial collocation for Cauchy singular integral equations with fixed singularities

In this paper we consider a polynomial collocation method for the numerical solution of Cauchy singular integral equations with fixed singularities over the interval, where the fixed singularities are supposed to be of Mellin convolution type. For the stability and convergence of this method in weightedL ² spaces, we derive necessary and sufficient conditions.     

Numerical solution of Volterra integral equations with singularities

The numerical solution of linear Volterra integral equations of the second kind is discussed. The kernel of the integral equation may have weak diagonal and boundary singularities. Using suitable smoothing techniques and polynomial splines on mildly graded or uniform grids, the convergence behavior of the proposed algorithms is studied and a collection of numerical results is given.     

On the extraction technique in boundary integral equations

In this paper we develop and analyze a bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the Cauchy data of regular three-dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations. The method rests on the derivatives of the generalized Green's representation formula, which are expressed in terms of singular boundary integrals as Hadamard's finite parts. Their regularization, together with asymptotic pseudohomogeneous kernel expansions, yields a constructive method for obtaining generalized jump relations. These expansions are obtained via composition of Taylor expansions of the local surface representation, the density functions, differential operators and the fundamental solution of the original problem, together with the use of local polar coordinates in the parameter domain. For boundary integral equations obtained by the direct method, this method allows the recursive numerical extraction of potentials and their derivatives near and up to the boundary surface.

     

Splitting extrapolation algorithm for first kind boundary integral equations with singularities by mechanical quadrature methods

The accuracy of numerical solutions near singular points is crucial for numerical methods. In this paper we develop an efficient mechanical quadrature method (MQM) with high accuracy. The following advantages of MQM show that it is very promising and beneficial for practical applications: (1) the O(h_max³) O(h_{\rm {max}}^{3}) convergence rate; (2) the O(h_max⁵)O(h_{\rm {max}}^{5}) convergence rate after splitting extrapolation; (3) Cond = O(h_min^-1)O(h_{\rm {min}}^{-1}); (4) the explicit discrete matrix entries. In this paper, the above theoretical results are briefly addressed and then verified by numerical experiments. The solutions of MQM are more accurate than those of other methods. Note that for the discontinuous model in Li et al. (Eng Anal Bound Elem 29:59–75, 2005), the highly accurate solutions of MQM may even compete with those of the collocation Trefftz method.     

On the supports of solutions of linear partial differential equations with analytic coefficients

Sunto In questo lavoro vengono dimostrati, con un metodo basato sui funzionali analitici, il teorema di Holmgren e varie sue estensioni. Il punto essenziale è che nelle dimostrazioni sono usate soltanto transformazioni lineari. Così è messo in luce che la variabile ? tempo ? ha un ruolo completamente diverso dal ruolo delle variabili ? spaziali ? per l'unicità del problema lineare di Cauchy. Viene anche studiato il problema globale di Cauchy in un semispazio.

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Entrato in Redazione il 1° aprile 1971.     

On numerical solving integral equations of the third kind with singularities in a kernel

In the present paper we study a linear integral equation of the third kind with fixed singularities in a kernel. We propose and investigate certain special generalized methods for the approximate solving of these equations in the space of generalized functions.     

Solving integral equations with logarithmic kernels by Chebyshev polynomials

In this paper, a finite Chebyshev expansion is developed to solve Volterra integral equations with logarithmic singularities in their kernels. The error analysis is derived. Numerical results are given showing a marked improvement in comparison with the piecewise polynomial collocation method given in literature.     

Solution of singular integral equations with logarithmic and Cauchy kernels

A direct method of solution is presented for singular integral equations of the first kind, involving the combination of a logarithmic and a Cauchy type singularity. Two typical cases are considered, in one of which the range of integration is a single finite interval and, in the other, the range of integration is a union of disjoint finite intervals. More such general equations associated with a finite number (greater than two) of finite, disjoint, intervals can also be handled by the technique employed here.     

On lower bounds on the complexity of solution of integral equations

We obtain lower bounds for the number of arithmetic operations required for the solution of Fredholm integral equations with accuracy ? under the assumption that the set of simplest operations is Φ = {arithmetic operations, evaluation of functionals}.     

On polynomial collocation for second kind integral equations with fixed singularities of Mellin type

Summary.  We consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of Mellin convolution type. Using a stability result by Junghanns and one of the authors, we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and, using the asymptotics of the solution, we derive the rates of convergence. Finally, we describe an algorithm to compute the stiffness matrix based on simple Gau? quadratures and an alternative algorithm based on a recursion in the spirit of Monegato and Palamara Orsi. All together an almost best approximation to the solution of the integral equation can be computed with 𝒪(n ²[log n]²) resp. 𝒪(n ²) operations, where n is the dimension of the polynomial trial space. Received February 18, 2002 / Revised version received May 15, 2002 / Published online October 29, 2002 RID="⋆" ID="⋆" Correspondence to: A. Rathsfeld Mathematics Subject Classification (1991): 65R20