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Generalized quadrature for solving singular integral equations of Abel type in application to infrared tomography. (English) Zbl 1416.65552
Summary: We propose the generalized quadrature methods for numerical solution of singular integral equation of Abel type. We overcome the singularity using the analytic computation of the singular integral. The problem of solution of singular integral equation is reduced to nonsingular system of linear algebraic equations without shift meshes techniques employment. We also propose generalized quadrature method for solution of Abel equation using the singular integral. Relaxed errors bounds are derived. In order to improve the accuracy we use Tikhonov regularization method. We demonstrate the efficiency of proposed techniques on infrared tomography problem. Numerical experiments show that it makes sense to apply regularization in case of highly noisy (about 10%) sources only. That is due to the known fact that Volterra equations of the first kind enjoy selfregularization property.

##### MSC:
 65R20 Numerical methods for integral equations 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 92C55 Biomedical imaging and signal processing
EDA
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