Numerical models have received a great attention in sciences and engineering in the recent years for modeling the differential equations. These models work on reducing the cost and time of computation especially for the physical phenomena that contain uncertain input. This phenomena can be studied by converting it into mathematical models of stochastic differential equations (SDE) and we can use the numerical methods 1,2 for overcoming these problems.
Numerical methods have been developed for simulating SDE such as moment equations, probability density method 3, etc. These methods are complicated in solving the nonlinear SDEs so the spectral decomposition techniques have received much attention in the recent years. The spectral decomposition technique was first suggested by the great mathematician Norbert Wiener 4. Wiener constructed an orthonormal random basis for expanding homogeneous chaos depending on white noise, and used it to study problems in statistical mechanics 5. The Hermite polynomial has been used to obtain the solution of SDE.
Mecham et al. 6 suggested the Wiener-Hermite expansion to study turbulence solution of Burger equation. In nonlinear stochastic differential equations, there exist always difficulties of solving the resultant set of deterministic integro-differential equations. The deterministic integro-differential equations got from the applications of a set of comprehensive averages on the stochastic integro-differential equation obtained after the direct application of WHE. Many authors introduced different methods to face these obstacles. Among them, the WHEP technique 7 was introduced using the perturbation technique to solve perturbed nonlinear problems. M. El-Tawil and his co-workers 7-11 used the WHE together with the perturbation theory (WHEP technique) to solve a perturbed nonlinear stochastic differential equation.
The WHEP technique is generalized to handle nth order polynomial nonlinearities, general order of WHE and general number of corrections 8. Cameron and martin 12 was developed a more explicit and intuitive formulation for the Hermite polynomial, which was called the Wiener-Chaos expansion. Their development is based on an explicit discretization of the white noise process through its Fourier expansion.
This approach is much easier to understand and more convenient to use, and hence replaced Wiener’s original formulation. Fourier chaos expansion has become a useful tool in stochastic analysis involving Brownian motion 13. Rozovskii et al. 14-16 derived Wiener chaos propagator equations for several important the stochastic partial differential equations (SPDEs) driven by Brownian motion forcing. Lototsky et al. 17, 18 proposed a new numerical method for solving the Zakai equation based on its Wiener chaos expansion. Using Fourier-Hermite expansion for modeling non-Gaussian processes is also investigated 19, 20.
Babuska et al. 21, Schwab et al. 22 and Keese el al. 23 developed and generalized Ghanem’s approach for solving stochastic elliptic equations.
Xiu and Karniadakis 24 generalized the Hermite polynomial expansion and used it to study flow-structure interactions. Zhang et al. 25 combined moment perturbation method with polynomial chaos expansion, and used it to study the saturation flows in heterogeneous porous media. The main goal of this paper is to use two stochastic spectral techniques, WCE and WHEP for solving the stochastic advection diffusion equation with multiplicative white noise and periodic boundary conditions. The two techniques convert SPDE into a system of DPDE. The DPDE can be solved using a proposed Eigen function expansion in both cases of WCE and WHEP techniques. The results will be studied through the mean and variance solutions. This paper is organized as follows: The formulation of the SADE is outlined in section 2.
The WCE technique is explained in section 3. In section 4, the algorithm of WHEP technique is introduced. In section 5 and 6, we apply the WCE and WHEP techniques respectively; the convergence analysis of the WCE is studied.
The proposed method for solving the resulting DPDEs; the numerical solutions of the WCE and WHEP techniques are introduced in section 7. The comparison and discussion of the results of the two techniques are in section 8. Finally, the conclusions are given in Section 9.
