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We apply Newton’s method to hyperbolic stochastic functional partial differential equations of the first order driven by a multidimensional Brownian motion. We prove a first-order convergence and a second-order convergence in a probabilistic sense.
Wydawca
Czasopismo
Rocznik
Tom
Strony
51--64
Opis fizyczny
Bibliogr. 21 poz.
Twórcy
autor
- Institute of Mathematics, University of Gdańsk Wita Stwosza 57, 80-952 Gdańsk
Bibliografia
- [1] K. Amano, A note on Newton’s method for stochastic differential equations and its error estimate, Proc. Japan Acad., 84 (2008), Ser. A, 1-3.
- [2] K. Amano, Newton’s method for stochastic differential equations and its probabilistic second order error estimate, Electron. J. Differential Equations, 2012 (2012), No. 03, 1-8.
- [3] Y.-Z. Chen, The generalized quasilinearization method for partial differential equations of first order, J. Math. Anal. Appl. 197 (1996) 726-734.
- [4] T. Człapiński, On the mixed problem for quasilinear partial differential-functional equations of the first order, Z. Anal. Anwendungen 16, No. 2, (1997), 463478.
- [5] S. G. Deo, S. Sivasundaram Extension of the method of quasilinearization to hyperbolic partial differential equations of first order, Appl. Anal. 59 (1995), no. 1-4, 153162.
- [6] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, 2 edition, 2010.
- [7] T. Funaki Construction of a solution of random transport equation with boundary condition, J. Math. Soc. Japan. 31 (1979), 719-744.
- [8] H. Holden, B. Øksendal, J. Ubøe, T. Zhang, Stochastic partial differential equations. A modeling, white noise functional approach, Probability and its Applications, Birkhuser Boston.
- [9] O. Kallenberg, Foundations of Modern Probability, Springer, 1997.
- [10] E.A. Kalpinelli, N.E. Frangos, A. N. Yannacopoulos, A Wiener Chaos Approach to Hyperbolic SPDEs, Stochastic Analysis and Application, vol.29, no.2, (2011), 237-258.
- [11] Z. Kamont, First order partial functional differential equations with state dependent delays, Nonlinear Studies 12, 2005, 135-157.
- [12] S. Kawabata, and T. Yamada, On Newton’s method for stochastic differential equations, Seminaire de Probabilites, XXV, Lecture Notes in Math., 1485, Springer, Berlin, (1991), pp.121-137.
- [13] H. Leszczyński, On CC-solutions to the initial-boundary-value problem for systems of nonlinear first-order partial differential-functional equations, Rendiconti Mat. 15, Roma (1995), 173-209.
- [14] A.G. McKendrick, Applications of mathematics to medical problems, Pro. Edinburgh Math. Soc., 44:98130, 1926
- [15] T. M. Mestechkina, On the solution of the Cauchy problem for a class of first-order stochastic partial differential equations, Izv. Vyssh. Uchebn. Zaved. Mat., 1991, no. 8, 42-44; translation in SovietMath. (Iz. VUZ) 35 (1991), no. 8, 39-41.
- [16] S.E.A. Mohammed, Stochastic Functional Differential Equations, Pitman Publishing Program, Boston, 1984.
- [17] A.E. Rodkina, On existence and uniqueness of solution of stochastic differential equation with hereditary, Stochastics 12 (1984), 187-200.
- [18] C. Roth, Weak approximations of solutions of a first order hyperbolic stochastic partial differential equation, Monte Carlo Methods Appl. 13, no. 2, (2007), 117-133.
- [19] J. Turo, Mixed problems for quasilinear hyperbolic systems, Proceedings of the Second World Congress of Nonlinear Analysis, Part 4 (Athens, 1996). Nonlinear Anal. 30 No. 4, (1997), 23292340.
- [20] T.D. Van, M. Tsuji, N.D. Thai Son, The Characteristic Method and its Generalizations for First-Order Nonlinear Partial Differential Equations, Boca Raton, London, Chapman & Hall/CRC, 2000.
- [21] M. Wrzosek, Newton’s method for stochastic functional differential equations, Electron. J. Differential Equations, 2012, No. 130, (2012), 1-10.
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Bibliografia
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