## Multidimensional Weakly Singular Integral EquationsThe final aim of the book is to construct effective discretization methods to solve multidimensional weakly singular integral equations of the second kind on a region of Rn e.g. equations arising in the radiation transfer theory. To this end, the smoothness of the solution is examined proposing sharp estimates of the growth of the derivatives of the solution near the boundary G. The superconvergence effect of collocation methods at the collocation points is established. This is a book for graduate students and researchers in the fields of analysis, integral equations, mathematical physics and numerical methods. No special knowledge beyond standard undergraduate courses is assumed. |

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### Contents

SOME PROBLEMS LEADING TO MULTIDIMENSIONAL WEAKLY | 1 |

PRELIMINARIES | 10 |

SMOOTHNESS OF THE SOLUTION | 24 |

Copyright | |

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accuracy Algebraic Algorithm 6.1 Analysis arithmetical operations Assume Banach space BC(G boundary dG bounded set cells Chapter collocation method collocation points compact convergence compactly connectivity components const continuous functions continuously differentiable cubature cubature formula defined derivatives dimensional discrete convergence discretely compact evaluation fulfilled function Galerkin method GcRn Gj.h Holder inequality holds inequality Initial guess integral equation 3.1 integral operator iteration methods kernel K(x,y Lemma linear log|x-y logh LP(G norm obtain open bounded set partitions of G PCCM piecewise smooth polynomials of degree problem 1.1 Proceedings proof of Lemma proof of Theorem prove quadrature formula radiation transfer respect satisfies 3.2 Section singular integral equations singular Integral operators superconvergence Th-Th Theorem 3.1 Theory Thuh ueBC(G uh-phu uh(x uheEh uniquely solvable Vainikko vector field VIII weakly singular integral whereby x1 and x2 XeAh