Abstract
<abstract><p>The purpose of this paper is to present a two-step approach for finding the series solution of the stochastic Newell-Whitehead-Segel (NWS) equation. The proposed two-step approach starts with the use of the Wiener-Hermite expansion (WHE) technique, which allows the conversion of the stochastic problem into a set of coupled deterministic partial differential equations (PDEs) by components. The deterministic kernels of the WHE serve as the solution to the stochastic NWS equation by decomposing the stochastic process. The second step involves solving these PDEs using the reduced differential transform (RDT) algorithm, which enables the determination of the deterministic kernels. The final step involves plugging these kernels back into the WHE to derive the series solution of the stochastic NWS equation. The expectation and variance of the solution are calculated and graphically displayed to provide a clear visual representation of the results. We believe that this two-step technique for computing the series solution process can be used to a great extent for stochastic PDEs arising in a variety of sciences.</p></abstract>
Publisher
American Institute of Mathematical Sciences (AIMS)
Reference49 articles.
1. P. Jakobsen, J. Lega, Q. Feng, M. Staley, J. V. Moloney, A. C. Newell, Nonlinear transverse modes of large-aspect-ratio homogeneously broadened lasers: Ⅰ. Analysis and numerical simulation, Phys. Rev. A, 49 (1994), 4189–4200. https://doi.org/10.1103/physreva.49.4189
2. B. A. Malomed, M. I. Tribelskii, Stability of stationary periodic structures for weakly supercritical convection and related problems, J. Exp. Theor. Phys., 65 (1987), 305.
3. L. Segel, Distant side-walls cause slow amplitude modulation of cellular convection, J. Fluid Mech., 38 (1969), 203–224. https://doi.org/10.1017/S0022112069000127
4. A. Hagberg, A. Yochelis, H. Yizhaq, C. Elphick, L. Pismen, E. Meron, Linear and nonlinear front instabilities in bistable systems, Physica D, 217 (2006), 186–192. https://doi.org/10.1016/j.physd.2006.04.005
5. M. Nadeem, J. H. He, C. H. He, H. M. Sedighi, A. H. Shirazi, A numerical solution of nonlinear fractional Newell-Whitehead-Segel equation using natural transform, TWMS J. Pure Appl. Math., 13 (2022), 168–182.