The integro-differential equation of peridynamics with hyper-singular kernel is considered. The existence and uniqueness of solution is proved.
1. Alimov S.A., Cao Y., and Ilhan O.A. On the problems of peridynamics with special convolution kernels. J. of Integral Equations and Applications, Vol. 26, Issue 3, 301–321 (2004).
2. Alimov S.A. and Sheraliev S. On the solvability of the singular equation of peridynamics. Complex Variables and Elliptic Equations Vol. 64, No.5, 873–887 (2019).
3. Du Q., Kamm J.R., Lehoucq R.B. and Michael~L. Parks. A new approach for a nonlocal, nonlinear conservation law. SIAM J. Appl. Math. Vol. 72, Issue 1, 464–487 (2012).
4. Emmrich E., Lehoucq R., Puhst D. Peridynamics: A Nonlocal Continuum Theory. Lecture Notes in Computational Science and Engineering, Vol. 89, 45–65 (2013).
5. Gunzburger M., Lehoucq R. A nonlocal vector calculus with application to nonlocal boundary value problems. Multiscale Model. Simul., Vol. 8, Issue 5, 1581–1598 (2010).
6. Kantorovich L.V., Akilov G.P. Functional Analysis (Russian). "Nauka", Moscow, (1977).
7. Nikol'skiy S.M. Approximation of functions of several variables and imbedding theorems. Grundl. Math. Wissensch., 205, Springer-Verlag, New York, viii+418 pp. (1975).
8. Seleson P., Michael L. Parks, Gunzburger M. and Lehoucq R.B. Peridynamics as an upscaling of molecular dynamics. Multiscale Model. Simul, Vol. 8, Issue 1, 204–227 (2009).
9. Silling S.A. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids, Vol. 48, Issue 1, 175–209 (2000).
10. Titchmarsh E.C. Eigenfunction expansions associated with second-order differential equations, Vol. 2, Oxford, at the Clarendon Press, (1958).
11. Watson G.N. A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, England, (1944).
12. Zhou K. and Du Q. Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal., Vol. 48, Issue 5, 1759–1780 (2010).
Alimov, Shavkat and Sheraliev, Shukhrat
"On the solvability of hypersingular equation of peridynamics,"
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 3
, Article 1.
Available at: https://uzjournals.edu.uz/mns_nuu/vol3/iss3/1