×

zbMATH — the first resource for mathematics

Stochastic partial differential equations in continuum physics - on the foundations of the stochastic interpolation method for Ito’s type equations. (English) Zbl 0706.60063
Summary: This paper deals with the mathematical analysis of mathematical models in continuum physics which are described by partial differential equations, in one space dimension, with additional random noise. The first part of the paper provides the framework for the mathematical modelling of a large class of physical systems. Then, the so-called ‘stochastic interpolation method’ is applied to obtain quantitative solutions by transforming the original equation into a system of stochastic ordinary differential equations, and an analysis of the error estimates is developed in order to provide a bound estimate of the distance between the solution of the true original problem and the one obtained by solving the system of ordinary differential equations. As an application, a linear random model for the heat equation is finally considered following the analysis proposed here.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Acquistapace, P.; Terreni, B., Existence and sharp regularity for linear abstract parabolic integro-differential equations, Israel J. math., 53, 257-303, (1986) · Zbl 0603.45019
[2] Adomian, G., Stochastic sytems, (1983), Academic Press New York
[3] Arnold, L., Stochastic differential equations—theory and applications, (1974), Wiley London
[4] Bellman, R.; Adomian, G., Nonlinear partial differential equations, (1985), Reidel Dordrecht
[5] Bellman, R.; Kashef, B.; Casti, J., Differential quadrature, a technique for the rapid solution of nonlinear partial differential equations, J. comput. phys., 10, 40-52, (1972) · Zbl 0247.65061
[6] Bellomo, N.; de Socio, L.; Monaco, R., On the nonlinear random heat equation, Comp. math. appl., (1988), to appear · Zbl 0661.60082
[7] Bellomo, N.; Riganti, R., Nonlinear stochastic systems in physics and mechanics, (1987), World Scientific Publ. Co · Zbl 0623.60084
[8] Bharucha-Reid, T.; Sambandha, M., Random polynomials, (1986), Academic Press New York
[9] Flandoli, F., Dirichlet boundary value problem for stochastic parabolic equations: compatibility relations and regularity of solutions, (1988), preprint
[10] Ladyzenskaja, O.; Solonnikov, V.; Ural’ceva, N., Linear and quasilinear equations of parabolyc type, (1968), American Mathematical Society Providence, RI
[11] McShane, E.J., Stochastic calculus and models, (1974), Academic Press New York · Zbl 0292.60090
[12] Nelson, J., Quantum fluctuations, (1985), Princeton University Press Cambridge · Zbl 0563.60001
[13] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer Berlin · Zbl 0516.47023
[14] Sinestrari, E., On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. math. anal. appl., 107, 16-66, (1985) · Zbl 0589.47042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.