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ON ANALYSIS AND DISCRETIZATION OF NONLINEAR ABEL INTEGRAL EQUATIONS OF FIRST KIND

RUDOLF GORENFLO, ANDREAS PFEIFFER

Abstract

For 0≤x≤B, 0<β<1, we consider the integral equation ∫0x(x−t)−βK(x,t,y(y))dt=f(x) under appropriate Lipschitz-like conditions on the function K and some of its derivatives, the most essential condition being Ku(x,t,u)≥c>0 for 0≤t≤x≤B, u∈R. After a survey on theorems of existence, uniqueness and stability of the solution we generalize a numerical method, proposed and investigated 1976 by H. W. Branca for the particular case β=1/2, to all β∈(0,1) and show it to be O(h2) convergent for all β∈[0.2118,1) if the solution y is sufficiently smooth. The method is based on piecewise linear interpolation, one-point weighted Gauss quadrature on parition intervals of equal lenght h, and collocation.

Viện Toán học, Viện Hàn lâm Khoa học và Công nghệ Việt Nam

Địa chỉ: Số 18 đường Hoàng Quốc Việt, quận Cầu Giấy, Hà Nội

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Email: vientoan@math.ac.vn

Viện nghiên cứu cao cấp về toán Hội Toán học Mỹ MathScinet Thư viện số (VAST) Arxiv
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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