Lectures on Cauchy’s Problem in Linear Partial Differential Equations. By J. Hadamard. Pp. viii+316. 15s.net. 1923. (Per Oxford University Press.)

In this paper some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m or = 1 Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations A mixed boundary value problem from the theory of elasticity is considered as an example Particularly for integral equations where the kernel contains, in addition to the dominant term (t,x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results

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On the solution of integral equations with strong ly singular kernels

Numerical evaluation of hypersingular integrals

We investigate the error term of the dth degree compound quadrature formulae for finite-part integrals of the form f(x -p f(x)dx where p ∈ R and p ≥ 1. We are mainly interested in error bounds of the form (R[f]( ≤ c∥f (s) ∥∞ with best possible constants c. It is shown that, for p ∈ N and n uniformly distributed nodes, the error behaves as O(n p-s-1 ) for f ∈ C 5 [0,1], p - 1 < s ≤ d + 1. In a previous paper we have shown that this is not true for p ∈ N. As an improvement, we consider the case of non-uniformly distributed nodes. Here, we show that for all p ≥ 1 and f ∈ C 5 [0,1], an O(n -s ) error estimate can be obtained in theory by a suitable choice of the nodes. A set of nodes with this property is stated explicitly. In practice, this graded mesh causes stability problems which are computationally expensive to overcome.

Generalized compound quadrature formulae for finite-part integrals

We show that the error term of every modified compound quadrature rule for Cauchy principal value integrals with degree of exactnesss is of optimal order of magnitude in the classesC k [−1,1],k=1,2,...,s, but not inCs+1[−1,1]. We give explicit upper bounds for the error constants of the modified midpoint rule, the modified trapezoidal rule and the modified Simpson rule. Furthermore, the results are generalized to analogous rules for Hadamard-type finite part integrals.

Modified compound quadrature rules for strongly singular integrals

In this paper we deal with the numerical calculation of singular integrals in the sense of Hadamard by simple integration methods. Error estimates and numerical examples are given. The optimality of the quadrature rules will be examined.

H. W. Stolle

Papers

On the numerical integration of certain singular integrals