We propose two approaches which allow us to construct probabilistic representations of classical and viscosity solutions of the Cauchy problem for a system of quasilinear parabolic equations. Based on these representations, we develop two numerical algorithms to construct the required solution. The system under consideration arises as a mathematical model of parabolic conservation laws. Bibliography: 14 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ya. I. Belopolskaya and Yu. L. Dalecky, “Investigation of the Cauchy problem for quasilinear systems with finite and infinite number of arguments by means of Markov random processes,” Izv. VUZ Mat., No. 12, 6–17 (1978).
Ya. Belopolskaya and Yu. Dalecky, Stochastic Equations and Differential Geometry, Kluwer Academic Publishers (1990).
Ya. I. Belopolskaya and Z. I. Nagolkina, “A class of stochastic equations with partial derivatives,” Teor. Veroyatn. Primen., 27, 551–559 (1982).
G. N. Milstein, “The probability approach to numerical solution of nonlinear parabolic equations,” Numerical Methods for Partial Differential Equations, 18, 490–522 (2002).
G. N. Milstein and M. V. Tretyakov, “Numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations,” Mathematics of Computation, 69, 237–267 (1999).
J. Marsden, “On product formulas for nonlinear semigroups,” J. Funct. Anal., 13, 51–72 (1973).
Yu. L. Dalecky, “A representation of operator equation solutions in the form of continual integrals,” Dokl. Akad. Nauk SSSR, 134, 1013–1016 (1960).
H. Trotter, “On the product of semigroups of operators,” Proc. Amer. Math. Soc., 10, 545–551 (1959).
E. Pardoux and S. Peng, “Backward stochastic differential equations and quasilinear parabolic partial differential equations,” Lect. Notes CIS, 176, 200–217 (1992).
Ya. I. Belopolskaya, “Forward-backward stochastic equations associated with systems of quasilinear parabolic equations and comparison theorems,” J. Math. Sci., 204, 7–27 (2015).
M. Crandall, H. Ishii, and P. Lions, “User’s guide to viscosity solutions of second order partial differential equations,” Bull. Amer. Math. Soc., 27, 1–67 (1992).
E. Pardoux, “Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order,” Stochastic Analysis and Relates Topics: The Geilo Workshop, Birkhäuser, 79–127 (1996).
E. Pardoux and S. Tang, “Forward-backward stochastic differential equations and quasilinear parabolic PDEs,” Probab. Theory Related Fields, 114, 123–150 (1999).
S. Peng and M. Xu, “Numerical algorithms for backward stochastic differential equations with 1-d Brownian motion. Convergence and simulations,” ESAIM: Mathematical Modelling and Numerical Analysis, 45, 225–360 (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 442, 2015, pp. 18–47.
Rights and permissions
About this article
Cite this article
Belopolskaya, Y.I., Nemchenko, E.I. Probabilistic Representations and Numerical Algorithms for Classical and Viscosity Solutions of the Cauchy Problem for Quasilinear Parabolic Systems. J Math Sci 225, 733–750 (2017). https://doi.org/10.1007/s10958-017-3490-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-017-3490-5