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Mathematical Biosciences and Engineering

2017, Volume 14, Issue 1: 237-248. doi: 10.3934/mbe.2017015
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Newtons method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion

  • Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland
  • Received: 29 October 2015 Accepted: 12 April 2016 Published: 01 February 2017

  • MSC : Primary: 60H15; Secondary: 35R60

  • We consider nonlinear stochastic wave equations driven by one-dimensional white noise with respect to time. The existence of solutions is proved by means of Picard iterations. Next we apply Newton's method. Moreover, a second-order convergence in a probabilistic sense is demonstrated.

    Citation: Henryk Leszczyński, Monika Wrzosek. Newtons method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion[J]. Mathematical Biosciences and Engineering, 2017, 14(1): 237-248. doi: 10.3934/mbe.2017015

    Related Papers:

  • We consider nonlinear stochastic wave equations driven by one-dimensional white noise with respect to time. The existence of solutions is proved by means of Picard iterations. Next we apply Newton's method. Moreover, a second-order convergence in a probabilistic sense is demonstrated.


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    [1] [ K. Amano, Newton's method for stochastic differential equations and its probabilistic second-order error estimate, Electron. J. Differential Equations, 2012 (2012): 1-8.
    [2] [ Z. Brzeźniak,M. Ondreját, Weak solutions to stochastic wave equations with values in Riemannian manifolds, Commun. Part. Diff. Eq., 36 (2011): 1624-1653.
    [3] [ P. Caithamer, The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise, Stoch. Dynam., 5 (2005): 45-64.
    [4] [ P.-L. Chow, Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002): 361-381.
    [5] [ D. Conus,R. C. Dalang, The non-linear stochastic wave equation in high dimension, Electron. J. Probab., 13 (2008): 629-670.
    [6] [ R. C. Dalang, Extending martingale measure stochastic integral with applications to spatially homogeneous spdes, Electron. J. Probab., 4 (1999): 1-29.
    [7] [ R. C. Dalang, The stochastic wave equation, A Minicourse on Stochastic Partial Differential Equations, in Lecture Notes in Math., 1962 (2009), Springer Berlin, 39-71.
    [8] [ R. C. Dalang,C. Mueller,R. Tribe, A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other P.D.E.s, Trans. Amer. Math. Soc., 360 (2008): 4681-4703.
    [9] [ R. C. Dalang,M. Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., 199 (2009): 1-70.
    [10] [ E. Hausenblas, Weak approximation of the stochastic wave equation, J. Comput. Appl. Math., 235 (2010): 33-58.
    [11] [ J. Huang,Y. Hu,D. Nualart, On Hölder continuity of the solution of stochastic wave equations, Stoch. Partial Differ. Equ. Anal. Comput., 2 (2014): 353-407.
    [12] [ S. Kawabata and T. Yamada, On Newton's method for stochastic differential equations, in Séminaire de Probabilités XXV, Lecture Notes in Math., 1485 (1991), Springer Berlin, 121-137.
    [13] [ J. U. Kim, On the stochastic wave equation with nonlinear damping, Appl. Math. Optim., 58 (2008): 29-67.
    [14] [ C. Marinelli,L. Quer-Sardanyons, Existence of weak solutions for a class of semilinear stochastic wave equations, Siam J. Math. Anal., 44 (2012): 906-925.
    [15] [ A. Millet,M. Sanz-Solé, A stochastic wave equation in two space dimensions: Smoothness of the law, Ann. Probab., 27 (1999): 803-844.
    [16] [ M. Nedeljkov,D. Rajter, A note on a one-dimensional nonlinear stochastic wave equation, Novi Sad Journal of Mathematics, 32 (2002): 73-83.
    [17] [ S. Peszat, The Cauchy problem for a nonlinear stochastic wave equation in any dimension, J. Evol. Equ., 2 (2002): 383-394.
    [18] [ L. Quer-Sardanyons,M. Sanz-Solé, Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation, J. Funct. Anal., 206 (2004): 1-32.
    [19] [ L. Quer-Sardanyons,M. Sanz-Solé, Space semi-discretisations for a stochastic wave equation, Potential Anal., 24 (2006): 303-332.
    [20] [ M. Sanz-Solé,A. Suess, The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity, Electron. J. Probab., 18 (2013): 1-28.
    [21] [ J. B. Walsh, An introduction to stochastic partial differential equations, in: É cole d'été de Probabilités de Saint-Flour XIV, Lecture Notes in Math, 1180 (1986), Springer Berlin, 265-439.
    [22] [ J. B. Walsh, On numerical solutions of the stochastic wave equation, Illinois J. Math., 50 (2006): 991-1018.
    [23] [ M. Wrzosek, Newton's method for stochastic functional differential equations, Electron. J.Differential Equations, 2012 (2012): 1-10.
    [24] [ M. Wrzosek, Newton's method for parabolic stochastic functional partial differential equations, Functional Differential Equations, 20 (2013): 285-310.
    [25] [ M. Wrzosek, Newton's method for first-order stochastic functional partial differential equations, Commentationes Mathematicae, 54 (2014): 51-64.
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  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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