Representation of the solution of a nonlinear molecular beam epitaxy equation

Boubaker Smii; Boubaker Smii

doi:10.3934/math.20241708

AIMS Mathematics

2024, Volume 9, Issue 12: 36012-36030. doi: 10.3934/math.20241708

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Research article

Representation of the solution of a nonlinear molecular beam epitaxy equation

Boubaker Smii ^,

King Fahd University of Petroleum and Minerals, Department of Mathematics, Interdisciplinary Research center for Intelligent Secure System, Dhahran 31261, Saudi Arabia

Received: 06 June 2024 Revised: 31 October 2024 Accepted: 03 December 2024 Published: 26 December 2024
MSC : 05C05, 60G20, 60H15

Stochastic partial differential equations (SPDEs) driven by Lévy noise are extensively employed across various domains such as physics, finance, and engineering to simulate systems experiencing random fluctuations. In this paper, we focus on a specific type of such SPDEs, namely the nonlinear beam epitaxy equation driven by Lévy noise. The Feynman graph formalism emerges as a potent tool for analyzing these SPDEs, particularly in computing their correlation functions, which are essential for understanding the moments of the solution. In this context, the solution to the SPDE and its truncated moments can be expressed as a sum over particular Feynman graphs. Each graph is evaluated according to a set of established rules, providing a systematic method to derive the properties of the solution. Moreover, the study delves into the behavior of the truncated moments for large times. Truncated moments, which capture the statistical properties of the system up to a certain order, are crucial for characterizing the long-term behavior and stability of the solution. The paper will conclude with a discussion on potential applications, highlighting the broader implications of this approach in various scientific and engineering contexts.
- SPDEs,
- Lévy noise,
- Green functions,
- Feynman graphs,
- truncated moments
Citation: Boubaker Smii. Representation of the solution of a nonlinear molecular beam epitaxy equation[J]. AIMS Mathematics, 2024, 9(12): 36012-36030. doi: 10.3934/math.20241708

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Abstract

Stochastic partial differential equations (SPDEs) driven by Lévy noise are extensively employed across various domains such as physics, finance, and engineering to simulate systems experiencing random fluctuations. In this paper, we focus on a specific type of such SPDEs, namely the nonlinear beam epitaxy equation driven by Lévy noise. The Feynman graph formalism emerges as a potent tool for analyzing these SPDEs, particularly in computing their correlation functions, which are essential for understanding the moments of the solution. In this context, the solution to the SPDE and its truncated moments can be expressed as a sum over particular Feynman graphs. Each graph is evaluated according to a set of established rules, providing a systematic method to derive the properties of the solution. Moreover, the study delves into the behavior of the truncated moments for large times. Truncated moments, which capture the statistical properties of the system up to a certain order, are crucial for characterizing the long-term behavior and stability of the solution. The paper will conclude with a discussion on potential applications, highlighting the broader implications of this approach in various scientific and engineering contexts.

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