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A meshless based method for solution of integral equations. (English) Zbl 1202.65174

The authors propose a numerical method based on the moving least squares method (MLS). The fact that for two dimensional problems the domain could have a non-rectangular geometry is viewed as a significant advantage by the authors. In addition the method does not require domain elements or background cells for interpolation or approximation.
The MLS approximation method is discussed in section 2. In section 3 the authors introduce their approach by applying it to a one-dimensional Fredholm integral equation. A linear system of equations is obtained and is solved using a numerical method such as Gauss elimination or LU factorization. The two-dimensional Fredholm integral equation with domains of the first, second and third kinds, is then considered. Again a linear system of equations is obtained.
Section 4 focuses on the convergence analysis of the proposed method. The error estimate of the method is based on those obtained by C. Zuppa [Bull. Braz. Math. Soc. 34, 231–249 (2003; Zbl 1056.41007)]. Properties of a partition of unity, the global assumptions about parameters made in [loc. cit.] and other theoretical results relevant to the error analysis presented in the paper are included. In section 5 the method is applied to Volterra integral equations in both one and two dimensions.
Illustrative numerical examples, confirming the theoretical error estimates, are presented in section 6. Routines were written in Fortran 6.1. Values of maximum errors, rates of convergence, ratio of errors and CPU times are tabulated as appropriate to each example. The examples include both Fredholm and Volterra integral equations in both one and two dimensions. The use of a non-rectangular domain and of different kinds of domain is demonstrated.
Concluding remarks are given in Section 7, along with a list of proposed ideas for future research.
Reviewer: Pat Lumb (Chester)

MSC:

65R20 Numerical methods for integral equations
45B05 Fredholm integral equations
45D05 Volterra integral equations

Citations:

Zbl 1056.41007
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References:

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