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The method of lines for hyperbolic stochastic functional partial differential equations. (English) Zbl 06890376
Summary: We apply an approximation by means of the method of lines for hyperbolic stochastic functional partial differential equations driven by one-dimensional Brownian motion. We study the stability with respect to small \(L^2\)-perturbations.
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
49M25 Discrete approximations in optimal control
Full Text: DOI
[1] Ashyralyev, A.; Koksal, M. E.; Agarwal, R. P., A difference scheme for Cauchy problem for the hyperbolic equation with self-adjoint operator, Math. Comput. Modelling 52 (2010), 409-424 · Zbl 1201.65150
[2] Ashyralyev, A.; Yurtsever, H. A., The stability of difference schemes of second-order of accuracy for hyperbolic-parabolic equations, Comput. Math. Appl. 52 (2006), 259-268 · Zbl 1137.65054
[3] Bahuguna, D.; Dabas, J.; Shukla, R. K., Method of lines to hyperbolic integro-differential equations in \(\mathbb{R}^n\), Nonlinear Dyn. Syst. Theory 8 (2008), 317-328 · Zbl 1206.45010
[4] Bátkai, A.; Csomós, P.; Nickel, G., Operator splittings and spatial approximations for evolution equations, J. Evol. Equ. 9 (2009), 613-636 · Zbl 1239.47031
[5] Berger, M. A.; Mizel, V. J., Volterra equations with Ito integrals I, J. Integral Equations 2 (1980), 187-245 · Zbl 0442.60064
[6] Prato, G. Da; Zabczyk, J., Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications 44, Cambridge University Press, Cambridge (1992) · Zbl 0761.60052
[7] Debbi, L.; Dozzi, M., On a space discretization scheme for the fractional stochastic heat equations, Avaible at https://arxiv.org/abs/1102.4689v1
[8] Friedman, A., Stochastic Differential Equations and Applications, Vol. 1, Probability and Mathematical Statistics 28, Academic Press, New York (1975) · Zbl 0323.60056
[9] Funaki, T., Construction of a solution of random transport equation with boundary condition, J. Math. Soc. Japan 31 (1979), 719-744 · Zbl 0405.60067
[10] Holden, H.; Øksendal, B.; Ubøe, J.; Zhang, T., Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach, Probability and Its Applications, Birkhäuser, Basel (1996) · Zbl 0860.60045
[11] Kamont, Z., Hyperbolic Functional Differential Inequalities and Applications, Mathematics and Its Applications 486, Kluwer Academic Publishers, Dordrecht (1999) · Zbl 0973.35188
[12] Kim, J. U., On the Cauchy problem for the transport equation with random noise, J. Funct. Anal. 259 (2010), 3328-3359 · Zbl 1203.35293
[13] Klebaner, F. C., Introduction to Stochastic Calculus with Applications, Imperial College Press, London (2005) · Zbl 1077.60001
[14] Kreiss, H.-O.; Scherer, G., Method of lines for hyperbolic differential equations, SIAM J. Numer. Anal. 29 (1992), 640-646 · Zbl 0754.65078
[15] Leszczyński, H., Quasi-linearisation methods for a nonlinear heat equation with functional dependence, Georgian Math. J. 7 (2000), 97-116 · Zbl 0964.35075
[16] Leszczyński, H., Comparison ODE theorems related to the method of lines, J. Appl. Anal. 17 (2011), 137-154 · Zbl 1276.34065
[17] McDonald, S., Finite difference approximation for linear stochastic partial differential equation with method of lines, MPRA Paper No. 3983. Avaible at http://mpra.ub.uni-muenchen.de/3983 (2006)
[18] Quer-Sardanyons, L.; Sanz-Solé, M., Space semi-discretisations for a stochastic wave equation, Potential Anal. 24 (2006), 303-332 · Zbl 1119.60061
[19] Reddy, S. C.; Trefethen, L. N., Stability of the method of lines, Numer. Math. 62 (1992), 235-267 · Zbl 0734.65077
[20] Rößler, A.; Seaïd, M.; Zahri, M., Method of lines for stochastic boundary-value problems with additive noise, Appl. Math. Comput. 199 (2008), 301-314 · Zbl 1142.65007
[21] Sharma, K. K.; Singh, P., Hyperbolic partial differential-difference equation in the mathematical modeling of neuronal firing and its numerical solution, Appl. Math. Comput. 201 (2008), 229-238 · Zbl 1155.65374
[22] Yoo, H., Semi-discretization of stochastic partial differential equations on \(\Bbb R^1\) by a finite-difference method, Math. Comput. 69 (2000), 653-666 · Zbl 0942.65006
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