On the Solvability of the Peridynamic Equation with a Singular Kernel

Silling, S.A., Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 2000, vol. 48, no. 1, pp. 175–209. Silling, S.A., Zimmermann, M., and Abeyaratne, R., Deformation of a peridynamic bar, J. Elasticity, 2003, vol. 73, pp. 173–190. Seleson, P., Parks, M.L., Gunzburger, M., and Lehoucq, R.B., Peridynamics as an upscaling of molecular dynamics, Multiscale Model. Simul., 2009, vol. 8, no. 1, pp. 204–227. Du, Q., Kamm, J.R., Lehoucq, R.B., and Parks, M.L., A new approach for a nonlocal, nonlinear conservation law, SIAM J. Appl. Math., 2012, vol. 72, no. 1, pp. 464–487. Emmrich, E., Lehoucq, R., and Puhst, D., A nonlocal continuum theory, Lect. Notes Comput. Sci. Eng., 2013, vol. 89, pp. 45–65. Alimov, S.A., Cao, Y., and Ilhan, O.A., On the problems of peridynamics with special convolution kernels, J. Integral Equat. Appl., 2014, vol. 26, no. 3, pp. 301–321. Alimov, S.A. and Sheraliev, S., On the solvability of the singular equation of peridynamics, Complex Var. Elliptic Equat., 2019, vol. 64, no. 5, pp. 873–887. Berezanskii, Yu.M., Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov (Expansion in Eigenfunctions of Self-Adjoint Operators), Kiev: Naukova Dumka, 1965. Il’in, V.A., Spektral’naya teoriya differentsial’nykh operatorov. Samosopryazhennye differentsial’nye operatory (Spectral Theory of Differential Operators. Self-Adjoint Differential Operators), Moscow: Nauka, 1991. Il’in, V.A. and Alimov, Sh.A., On the divergence of mean Riesz fractional-order kernels on a set of positive measure, Differ. Uravn., 1972, vol. 8, no. 2, pp. 372–373. Alimov, Sh.A. and Rakhimov, A.A., On the localization of spectral expansions of distributions, Differ. Uravn., 1996, vol. 32, no. 6, pp. 792–796. Nikol’skii, S.M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya (Approximation of Multivariate Functions and Embedding Theorems), Moscow: Nauka, 1977.