Generalized quadrature for solving singular integral equations of Abel type in application to infrared tomography

Abstract

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 make sense to apply regularization in case of highly noisy (about 10%) sources only. That is due to the fact that singular integral equations enjoy selfregularization property.

On a discussion of Volterra–Fredholm integral equation with discontinuous kernel

“…Equations (19) and (22) represent the Volterra integral equations of the second kind that have the same continuous kernel (t, τ ) ∈ C([ 0, T] ×[ 0, T] ), and each of them has a unique solution in the class C[ 0, T]; the books edited by Linz [28] and Burton [29] contain many different methods to solve the integral Eqs. (19) and (22).…”

“…Due to this, it is required to obtain an efficient numerical solution [15]. There are numerous studies in literature concerning the numerical solution of integral equations such as [16][17][18][19][20][21][22].…”

Direct and inverse problems of infrared tomography

“…(9) by the generalized quadrature method according to Eq. 17were obtained in [25,37]. Specificity of these estimations consists in the fact that they are obtained without knowledge of the exact solutionk (cf.…”

“…18] are not overstated. The estimations of the solution error are obtained in [37], also taking into account the errors δ in the right-hand side q of Eq. (9), not only for the quadrature error.…”

Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations

“…Over several decades, numerical methods in electromagnetic problems have been one of the most important subjects of extensive researches [1][2][3][4]. On the other hand, many problems in electromagnetics can be modeled by integral equations mentioned in [24][25][26], for example, electric field integral equation (EFIE) and magnetic field integral equation (MFIE). In recent years, several numerical methods for solving linear and nonlinear integral equations have been presented.…”