# American Institute of Mathematical Sciences

February  2017, 14(1): 237-248. doi: 10.3934/mbe.2017015

## Newton's 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  October 29, 2015 Accepted  April 12, 2016 Published  October 2016

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. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015
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##### References:
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