Probabilistic representations and numerical algorithms to construct classical and viscosity solutions of the Cauchy problem for systems of quasilinear parabolic equations
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 23, Tome 442 (2015), pp. 18-47 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We propose two approaches that allow to construct probabilistic representationы of classical and viscosity solutions of the Cauchy problem for systemы of quasilinear parabolic equations. Based on this representations we develop two numerical algorithms to construct the required solutions.
@article{ZNSL_2015_442_a1,
     author = {Ya. I. Belopolskaya and E. I. Nemchenko},
     title = {Probabilistic representations and numerical algorithms to construct classical and viscosity solutions of the {Cauchy} problem for systems of quasilinear parabolic equations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {18--47},
     year = {2015},
     volume = {442},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_442_a1/}
}
TY  - JOUR
AU  - Ya. I. Belopolskaya
AU  - E. I. Nemchenko
TI  - Probabilistic representations and numerical algorithms to construct classical and viscosity solutions of the Cauchy problem for systems of quasilinear parabolic equations
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2015
SP  - 18
EP  - 47
VL  - 442
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2015_442_a1/
LA  - ru
ID  - ZNSL_2015_442_a1
ER  - 
%0 Journal Article
%A Ya. I. Belopolskaya
%A E. I. Nemchenko
%T Probabilistic representations and numerical algorithms to construct classical and viscosity solutions of the Cauchy problem for systems of quasilinear parabolic equations
%J Zapiski Nauchnykh Seminarov POMI
%D 2015
%P 18-47
%V 442
%U http://geodesic.mathdoc.fr/item/ZNSL_2015_442_a1/
%G ru
%F ZNSL_2015_442_a1
Ya. I. Belopolskaya; E. I. Nemchenko. Probabilistic representations and numerical algorithms to construct classical and viscosity solutions of the Cauchy problem for systems of quasilinear parabolic equations. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 23, Tome 442 (2015), pp. 18-47. http://geodesic.mathdoc.fr/item/ZNSL_2015_442_a1/

[1] Ya. I. Belopolskaya, Yu. L. Daletskii, “Issledovanie zadachi Koshi dlya kvazilineinykh parabolicheskikh sistem pri pomoschi markovskikh sluchainykh protsessov”, Izv. vuzov. Matem., 1978, no. 2, 6–17 | MR | Zbl

[2] Ya. Belopolskaya, Yu. Dalecky, Stochastic equations and differential geometry, Kluwer Academic Publishers, 1990 | MR | Zbl

[3] Ya. I. Belopolskaya, Z. I. Nagolkina, “Ob odnom klasse stokhasticheskikh uravnenii s chastnymi proizvodnymi”, TViP, 27:3 (1982), 551–559 | MR | Zbl

[4] G. N. Milstein, “The Probability Approach to Numerical Solution of Nonlinear Parabolic Equations”, Numerical Methods for Partial Differential Equations, 18:4 (2002), 490–522 | DOI | MR | Zbl

[5] G. N. Milstein, M. V. Tretyakov, “Numerical algorithms for semilinear parabolic equations with small parameter base on approximation of stochastic equations”, Mathematics of Computation, 69:229 (1999), 237–267 | DOI | MR

[6] J. Marsden, “On Product Formulas for Nonlinear Semigroups”, J. Funct. Anal., 13 (1973), 51–72 | DOI | MR | Zbl

[7] Yu. L. Daletskii, “O predstavimosti reshenii operatornykh uravnenii v vide kontinualnykh integralov”, DAN SSSR, 134:5 (1960), 1013–1016

[8] H. Trotter, “On the Product of Semigroups of Operators”, Proc. Amer. Math. Soc., 10 (1959), 545–551 | DOI | MR | Zbl

[9] E. Pardoux, S. Peng, “Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations”, Lecture Notes in CIS, 176, Springer-Verlag, 1992, 200–217 | MR

[10] Ya. I. Belopolskaya, “Pryamye-obratnye stokhasticheskie uravneniya, svyazannye s sistemami kvazilineinykh parabolicheskikh uravnenii i teoremy sravneniya”, Veroyatnost i statistika. 19, Zap. nauchn. semin. POMI, 412, 2013, 15–46 | MR

[11] M. Crandall, H. Ishii, P. Lions, “User's guide to viscosity solutions of second order partial differential equations”, Bull. AMS, 27:1 (1992), 1–67 | DOI | MR | Zbl

[12] 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, 1996, 79–127 | MR

[13] E. Pardoux, S. Tang, “Forward – backward stochastic differential equations and quasilinear parabolic PDEs”, Probab. Theory Related Fields, 114:2 (1999), 123–150 | DOI | MR | Zbl

[14] S. Peng, M. Xu, “Numerical algorithms for backward stochastic differential equations with 1-d brownian motion: Convergence and simulations”, ESAIM: Mathematical Modelling and Numerical Analysis, 45:2 (2011), 335–360 | DOI | MR | Zbl

[15] I. Prigogine, R. Lefever, “Symmetry-breaking instabilities in dissipative systems”, J. Chem. Phys., 48 (1968), 1695–1700 | DOI

[16] A. Madzvamuse, R. Barreira, “Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces”, Phys. Rev. E, 90 (2014), 043307, 14 pp. | DOI