Constrained Approximation Techniques for Solving Integral Equations

1985 ◽  
pp. 68-73
Author(s):  
M. Brannigan
Keyword(s):  
Integral Equations ◽  
Approximation Techniques ◽  
Constrained Approximation

scholarly journals A Numerical Method for Weakly Singular Nonlinear Volterra Integral Equations of the Second Kind

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1862
Author(s):  
Sanda Micula
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Error Estimates ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Integration Scheme ◽  
True Solution ◽  
Approximation Techniques ◽  
Weakly Singular ◽  
Numerical Integration Scheme

This paper presents a numerical iterative method for the approximate solutions of nonlinear Volterra integral equations of the second kind, with weakly singular kernels. We derive conditions so that a unique solution of such equations exists, as the unique fixed point of an integral operator. Iterative application of that operator to an initial function yields a sequence of functions converging to the true solution. Finally, an appropriate numerical integration scheme (a certain type of product integration) is used to produce the approximations of the solution at given nodes. The resulting procedure is a numerical method that is more practical and accessible than the classical approximation techniques. We prove the convergence of the method and give error estimates. The proposed method is applied to some numerical examples, which are discussed in detail. The numerical approximations thus obtained confirm the theoretical results and the predicted error estimates. In the end, we discuss the method, drawing conclusions about its applicability and outlining future possible research ideas in the same area.

scholarly journals A comparison between current-based integral equations approaches for eddy current problems

2021 ◽  
Vol 2090 (1) ◽  
pp. 012137
Author(s):  
F Lucchini ◽  
N Marconato
Keyword(s):  
Integral Equations ◽  
Eddy Current ◽  
Large Scale ◽  
Low Rank ◽  
Loop Current ◽  
Problem Size ◽  
Low Rank Approximation ◽  
Approximation Techniques ◽  
Current Formulation ◽  
Simply Connected

Abstract In this paper, a comparison between two current-based Integral Equations approaches for eddy current problems is presented. In particular, the very well-known and widely adopted loop-current formulation (or electric vector potential formulation) is compared to the less common J-φ formulation. Pros and cons of the two formulations with respect to the problem size are discussed, as well as the adoption of low-rank approximation techniques. Although rarely considered in the literature, it is shown that the J-φ formulation may offer some useful advantages when large problems are considered. Indeed, for large-scale problems, while the computational efforts required by the two formulations are comparable, the J-φ formulation does not require any particular attention when non-simply connected domains are considered.

SOLUTION TO THE MODEL PROBLEM OF EXCITATION OF LOADED CONIC SLOT ANTENNA BY METHOD OF SINGULAR INTEGRAL EQUATIONS

2016 ◽  
Vol 75 (20) ◽  
pp. 1799-1812
Author(s):  
V. A. Doroshenko ◽  
S.N. Ievleva ◽  
N.P. Klimova ◽  
A. S. Nechiporenko ◽  
A. A. Strelnitsky
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Model Problem ◽  
Singular Integral Equations ◽  
Slot Antenna
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