Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener-Hermite Expansion

Abstract

In the current work, the Wiener-Hermite expansion (WHE) is used to solve the stochastic heat equation with nonlinear losses. WHE is used to deduce the equivalent deterministic system up to third order accuracy. The solution of the equivalent deterministic system is obtained using different techniques numerically and analytically. The finite-volume method (FVM) with Pickard iteration is used to solve the equivalent system iteratively. The WHE with perturbation technique (WHEP) is applied to deduce more simple and decoupled equivalent deterministic system that can be solved numerically without iterations. The system resulting from WHEP technique is solved also analytically using the eigenfunction expansion technique. The Monte-Carlo simulations (MCS) are performed to get the statistical properties of the stochastic solution and to verify other solution techniques. The results show that higher-order solutions are essential especially in case of nonlinearities where non-Gaussian effects cannot be neglected. The comparisons show the efficiency of the numerical WHE and WHEP techniques in solving stochastic nonlinear PDEs compared with the analytical solution and MCS.

A practical comparison between the spectral techniques in solving the SDEs

“…The stochastic quadratic nonlinear oscillatory equation using the WHEP was considered in the study by El-Beltagy and Al-Johani (2013). The stochastic heat equation with nonlinear losses using the WHEP was considered in the study by El-Beltagy and Al-Mulla (2014).…”

Random First-Order Linear Discrete Models and Their Probabilistic Solution: A Comprehensive Study

“…These types of models are extensively used in time series analysis in statistics [4]. Some recent interesting models based on Itô-type stochastic differential equations include [5][6][7], for instance. Complementary to these approaches, uncertainty can be directly introduced in differential and difference equations by assuming that coefficients, source term, and/or initial/boundary conditions are random variables and/or stochastic processes.…”