PSI - Issue 3

Nataly Vaysfeld et al. / Procedia Structural Integrity 3 (2017) 526–544 Author name / Structural Integrity Procedia 00 (2017) 000–000

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It is known from general theory of full elliptical integrals, that integral   K x has a logarithmic singularity at 1.  x The singular kernels of system (19) with regard of it are represented as   1 sech sech ln , 2 2                       i i i K l where the functions   i l x are even, continuous with their derivative and   0 lim ln8 .    i i x l x The system of integral equations for the estimation of functions     i is written finally 1,      i N All transformations for the Problem №2 should be done in an analogical way. These transformations lead at first to the system(18) where in the formulas for     i f ratio     0 2    k k I I should be changed to           1 0 2 0 2 ,                k k k k k I K K I and in the formula for   ,   ij R t ratio         2 0 0 2      k k k k K I I t I should be changed too by the expression               2 0 0 2 0 2                 k k k k k k I K t K I I K k               2 0 0 2 0 2 .                  k k k k k k K I t K I I K k The system of integral equations of type (20) is constructed after changing the variables. So, both problems lead to the system of the integral equations of type (20). The structure of equation singular kernels in system (20) and availability of the spectral correspondence Popov (1982)     1 2 1 ln 2, 0 ln 1 , , 1 1                       n n n n n T d T n n allow the use of the orthogonal polynomial method to solve this system. Accordingly to the scheme of the method the solution of the system is searched as the series expansion by Chebyshev polynomials of І-st order   n T x     2 0 , 1, . 1            n i i n n T i N (21)      d     , M L     ,            d g      1  1  1 1 1 ln 1 ,                 N i i ij ij j i i i C h j l (20) 1 1,