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Linear Partial Differential Equations with Random Forcing
Author(s) -
Wan Frederic Y. M.
Publication year - 1972
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1972512163
Subject(s) - forcing (mathematics) , mathematics , partial differential equation , stochastic partial differential equation , mathematical analysis , first order partial differential equation , exponential integrator , differential equation , ordinary differential equation , differential algebraic equation
where (0 l ) is the ensemble-averaging operation. The higher order moments can be determined by forming the ensemble average of different combinations of u(x, t) with the help of (1.4). Unfortunately, the Green’s function, G(x, t ; x’, t’), of (l.l), (1.2) and (1.3) cannot be obtained in terms of elementary or special functions except for the simplest cases. In general, we will have to obtain G by some numerical method. The desired statistics of the response will still have to be calculated by multiple integration. Even if we are willing to settle for the mean and the mean square value of u and ut