References
- Kirchhoff G. Forlesungen über machanik. Leipzig: Teubner; 1883.
- Bernstein S. Sur une classe dëquations fonctionelles aux derivees partielles. Isv SSSR, Serie Math. 1940;4:17–26.
- Pohožaev SI. On a class of quasilinear hyperbolic equations. Math USSR Sbornik. 1975;25:145–158. doi: https://doi.org/10.1070/SM1975v025n01ABEH002203
- Lions JL. On some questions in boundary value problems of mathematical physics. Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Rio de Janeiro, 1977), North-Holland Math. Stud., pp. 284-346. North-Holland, Amsterdam, 1978.
- Hirosawa F. A class of non-analytic function for the global solvability of Kirchhoff equation. Nonlinear Anal. 2015;116:37–63. doi: https://doi.org/10.1016/j.na.2014.12.016
- Medeiros LA, Ferrel JL, Menezes SB. Vibrations of elastic strings: mathematical aspects part one. J Comput Anal Appl. 2002;4(2):91–127.
- Chaudhary S, Srivastava V, Srinivas Kumar VVK, et al. Finite element approximation of nonlocal parabolic problem. Numer Methods Partial Differ Equ. 2017;33(3):786–813. doi: https://doi.org/10.1002/num.22123
- Chaudhary S. Finite element analysis of nonlocal coupled parabolic problem using Newton's method. Comput Math Appl. 2017;75(3):981–1003. doi: https://doi.org/10.1016/j.camwa.2017.10.034
- Chaudhary S. Finite element scheme with Crank-Nicolson method for parabolic nonlocal problems involving the Dirichlet energy. Int J Comp Meth. 2017;14(5):1750053.
- Robalo RJ, Almeida RMP, do C. Coimbra M, et al. A reaction diffusion model for a class of nonlinear parabolic equations with moving boundaries: existence, uniqueness, exponential decay and simulation. Appl Math Model. 2014;38(23):5609–5622. doi: https://doi.org/10.1016/j.apm.2014.04.045
- Gudi T. Finite element method for a nonlocal problem of Kirchhoff type. SIAM J Numer Anal. 2012;50(2):657–668. doi: https://doi.org/10.1137/110822931
- Christie I, Sanz-Serna JM. A Galerkin method for a nonlinear integro-differential wave system. Comput Methods Appl Mech Engrg. 1984;44:229–237. doi: https://doi.org/10.1016/0045-7825(84)90144-0
- Suchde P, Vasudeva Murthy AS. On the wave equations of Kirchhoff-Narasimha and carrier. In: Cushing J, Saleem M, Srivastava H, Khan M, Merajuddin M, editors. Applied analysis in biological and physical sciences. Springer proceedings in mathematics & statistics. New Delhi: Springer; 2016, Vol. 186.
- Wheeler MF. A priori l2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J Numer Anal. 1973;10(4):723–759. doi: https://doi.org/10.1137/0710062
- Rannacher R, Scott R. Some optimal error estimates for piecewise linear finite element approximation. Math Comp. 1982;38:437–445. doi: https://doi.org/10.1090/S0025-5718-1982-0645661-4
- Govaerts W, Pryce JD. Block elimination with one refinement solves bordered linear system accurately. BIT. 1990;30:490–507. doi: https://doi.org/10.1007/BF01931663