Στοχαστική μοντελοποίηση προβλημάτων συνοριακών τιμών για την εξίσωση Helmholtz και Navier

Λάλλας, Κωνσταντίνος; Lallas, Konstantinos G.

Stochastic boundary value problems for the Helmholtz and Navier equation

Doctoral Thesis

Author

Λάλλας, Κωνσταντίνος

Lallas, Konstantinos G.

Date

2025-11

Abstract

In the present doctoral dissertation, we study boundary value problems in acoustics and linear elasticity, beginning with the mathematical framework for the Helmholtz and Navier equations. We then analyze plane dyadic-wave scattering in three dimensions and its connection to low-frequency asymptotics. Next, we formulate stochastic counterparts of these models. For the stochastic Helmholtz equation, randomness may appear in the field, the source, and the Dirichlet data; for the Navier equation, randomness enters the field and the forcing while Dirichlet data are deterministic. Using Wiener -- Chaos expansions, we convert each stochastic PDE into an infinite hierarchy of deterministic variational problems and establish well-posedness via a Garding inequality and Fredholm theory. The approach yields existence and uniqueness under standard non-resonance assumptions and clarifies the role of low-frequency approximations in stochastic scattering. In this dissertation, for the acoustic stochastic problem, the wave field, the source of the problem, and the boundary data are all expressed as Wiener chaos expansions. Specifically, we present the appropriate stochastic variational form of the problem and transform our stochastic problem into an infinite hierarchy of deterministic problems, for each of which we formulate its variational form. Subsequently, we prove their well-posedness and present the main objective of our dissertation, which is to connect the resulting deterministic problems with the original stochastic problem and to prove the existence and uniqueness of the weighted Wiener Chaos solution. Finally, for the case of the stochastic problem of linear elasticity, we apply the same methodology where only the wave field and the source of the problem are Wiener chaos expansions. That is, we reformulate the stochastic problem into an infinite hierarchy of deterministic problems and establish the existence and uniqueness of the weighted stochastic solution. For all the above cases of stochastic problems, we present significant conclusions that support the application of the proposed method. We also highlight several observations that provide food for thought and will form our research basis for future work. The doctoral dissertation is structured as follows: Chapter 1 presents the mathematical theory related to acoustic and elastic waves due to Helmholtz and Navier equations, respectively, and the fundamental solutions of these equations are also given. Chapter 2 introduces the reader to the basic mathematical differential equations governing boundary value problems, in 3-dimensional linear elasticity. In addition, we study a scattering problem by dyadic wave field and its linchpin with low frequency approximations and we give analytical formulas for the total differential cross section. Furthermore, in Chapter 2, an introduction to boundary value problems in a stochastic environment, is given. In particular, we present the mathematical theory concerning Wiener chaos expansions, stochastic Hermite polynomials, and the construction of the weighted Wiener chaos space, which are necessary for the stochastic differential equations of our problems. In Chapter 3, we formulate the stochastic problems related to the Helmholtz equation and establish the well-posedness of the stochastic solution. In addition for the above problem we suppose that the Dirichlet boundary condition is a stochastic variable. Finally, in Chapter 4, we prove the existence and uniqueness of the weighted stochastic solution for the stochastic problem of the Navier equation, while also presenting useful research conclusions, observations, and ideas for future research work.

Department

Σχολή Χρηματοοικονομικής και Στατιστικής. Τμήμα Στατιστικής και Ασφαλιστικής Επιστήμης

Number of pages

131

Language

Greek

URI

https://dione.lib.unipi.gr/xmlui/handle/unipi/18802

Collections

Τμήμα Στατιστικής και Ασφαλιστικής Επιστήμης

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Αναφορά Δημιουργού-Μη Εμπορική Χρήση-Όχι Παράγωγα Έργα 3.0 Ελλάδα