We also do not make use of the integral representation of fBm in terms of ordinary Brownian motion as in [7], where the hypothesis involves the transferring operator which is difficult to verify.
Our new algorithm is based on the use of some exact integrals that arise on judicious manipulation of the integral representation of Gqp and which reduce the overall problem to that of evaluation of a sequence of simpler integrals that can be effectively handled by standard quadrature formulas.
We discretize these equations using a new formulation that treats the electric field and magnetic induction using simple basis-function representations to obtain a discrete form of Faraday's law of induction, but that uses finite integral representations for the displacement current and magnetic field to obtain a discrete form of Ampere's law, as in the finite integration technique [T.
end{aligned} (23) In order to determine the dynamics of this reduced state, we use the integral representation of the partial trace over the harmonic oscillator [32]: begin{aligned} operatorname {tr}_{C}(rho) = frac{1}{pi^{vert C vert }} biggl[ int _{vec{alpha } in mathbb{C}^{vert C vert }} bigllangle vec{alpha}biglvert rho bigrvert vec{ alpha} bigrrangle, d^{2} vec {alpha} biggr].
For this purpose, the diffracted field of pressures and displacements, due to an initial pressure in the fluid, are expressed using boundary integral representations, which satisfy the equation of motion.
To establish asymptotic stability we prove the stability estimates by using integral representations of the solutions via asymptotic solutions, error estimates, and calculus on time scales.
Moreover, we will use the integral representation of a smooth function via the Green function (see [2]).
A significant advantage of this method is that the time history of the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae.
Also the reader can easily obtain various integral representations of these functions by using the integral representation of the new beta function (1) in the same way as in [1 6, 8 10].
We now use the path integral representation of the conditional probability (Equation 5.13) in order to estimate the time to extinction of a metastable nontrivial state.
