Abstract
In this paper, multidimensional weakly singular integrals are solved by using rectangular quadrature rules which base on quadrature rules of one dimensional weakly singular and multidimensional regular integrals with their Euler-Maclaurin asymptotic expansions of the errors. The presented method is suit for solving multidimensional and singular integrals by comparing with Gauss quadrature rule. The error asymptotic expansions show that the convergence order of the initial quadrature rules is , where . The order of accuracy can reach to by using extrapolation and splitting extrapolation, where h0 is the maximum mesh width. Some numerical examples are constructed to show the efficiency of the method.
Highlights
It is well known that multidimensional singular integrals are models arising in diverse engineering problems and mathematical applications
Multidimensional weakly singular integrals are solved by using rectangular quadrature rules which base on quadrature rules of one dimensional weakly singular and multidimensional regular integrals with their Euler-Maclaurin asymptotic expansions of the errors
The Gauss-quadrature rules are considered to be a good choice for solving high dimensional integrals because they were accurate for polynomial approximation and the cost is low
Summary
It is well known that multidimensional singular integrals are models arising in diverse engineering problems and mathematical applications. In the boundary element fracture analysis problem, elasticity problem [1], bimaterial interfacial cracks [2] and wedge-sharped bimaterial interface [3], etc Few of these integrals and equations can be solved explicitly, it is necessary to find a good numerical method. The structure of this paper is as follows: In Section 2, we give quadrature rules for weakly singular integral with multivariate errors asymptotic expansions. 2. Multi-Parameters Asymptotic Expansions of the Errors for Weakly Singular Integrals. Multi-Parameters Asymptotic Expansions of the Errors for Weakly Singular Integrals In this part, we mainly consider multidimensional weakly singular integrals. We give the corresponding results of multidimensional weakly singular integrals according to the quadrature formula and asymptotic expansions of the errors of one-dimensional integrals. Proof: We prove the theorem by the mathematic induction method.
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