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Concept# Boundary value problem

Summary

In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.
Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator.
To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value

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In this work, two problems linked to glacier modeling are investigated. We propose an optimisation method for studying the flow of the ice and we present a numerical study about glacier thermal phenomena. In the first chapter of this thesis, we expose the models of these two problems. On one hand, we note that the boundary conditions on the bedrock are misunderstood, which explains it is difficult to obtain an accurate simulation of the motion of the ice. Also we establish a mathematical model where the bedrock boundary conditions depend on a control parameter. The aim of this study is to minimize a cost functional describing the difference between the computed velocity at the surface and the measure done. We study the cost functional with respect to the control parameter and we detail an optimisation method to solve the optimal control problem. On the other hand, we introduce two thermodynamical model governing the temperature and the water content field. The models correspond to a Stefan problem for the temperature and a convection-diffusion equation for the water content. The second chapter deals with the numerical resolution of the optimisation problem. First, a Finite Element Method (FEM) is described to solve the partial differential equations. Then, the algorithms used for the optimal control problem are detailed. Finally, this techniques are applied on two glaciers : Griesgletcher for 2D and Storglaciaren for 3D. The third chapter deals with the numerical resolution of the temperature and the water content models. A FEM is used for each problem. Concerning the temperature problem, the Stefan problem is numerically solved and the results allow to detect a free surface between the temperated ice and the cold ice. The water content field is also simulated. Numerical results are discussed on the Storglaciaren.

Understanding the mechanical behavior of soft materials is of great importance for many applica-tions, especially for bioengineering and clinical applications. Hertzian theory is frequently used to characterize the mechanical properties of such materials. However, these materials undergo large deformations resulting in non-linear stress-strain response, which cannot be accurately captured by Hertzian contact. These materials are also rate sensitive, where at high velocities, external e˙ects such as air can influence the mechanical response. In this study, we experimentally in-vestigated the contact response of a soft spherical impactor on a rigid substrate and by using the principle of Total Internal reflection, we directly measured the contact radius. The e˙ects of specimen size, loading rate and boundary conditions were also examined and we found that, for the studied velocities, only the boundary conditions a˙ects both the radius-displacement and load-displacement relationships, leading to deviations from the theory at di˙erent indentation depths. We finished by setting up an experiment consisting of a pendulum for studying the e˙ects of high velocities and the influence air.

2021In this thesis we describe a path integral formalism to evaluate approximations to the probability density function for the location and orientation of one end of a continuum polymer chain at thermodynamic equilibrium with a heat bath. We concentrate on those systems for which the associated energy density is at most quadratic in its variables. Our main motivation is to exploit continuum elastic rod models for the approximate computation of DNA looping probabilities. We first re-derive, for a polymer chain system, an expression for the second order correction term due to quadratic fluctuations about a unique minimal energy configuration. The result, originally stated for a quantum mechanical system by G. Papadopoulos (1975), relies on an elegant algebraic argument that carries over to the real-valued path integrals of interest here. The conclusion is that the appropriate expression can be evaluated in terms of the energy of the minimizer and the inverse square root of the determinant of a matrix satisfying a certain non-linear system of differential equations. We then construct a change of variables, which establishes a mapping between the solutions of the aforementioned non-linear Papadopoulos equations and a matrix satisfying an initial value problem for the classic linear system of Jacobi equations associated with the second variation of the energy functional. This conclusion is trivial if no cross-term is present in the second variation, but ceases to be so otherwise. Cross-terms are always present in the application of rod models to DNA. We therefore can conclude that the second order fluctuation correction term to the probability density function for a chain is always given by the inverse square root of the determinant of a matrix of solutions to the Jacobi equations. We believe this conclusion to be original for the real-valued case when the second-variation involves cross-terms. Similar results are known for quantum mechanical systems, and, in this context, a connection between the so called Van-Hove-Morette determinant, which involves partial derivatives of the classical action with respect to the boundary values of the configuration variable, and the Jacobi determinant have also been established. We next apply the formula described above to the specific context of rods, for which the configuration space is that of framed curves, or curves in R3 × SO(3). An immediate application of our theory is possible if the rod model encompasses bend, twist, stretch and shear. However the constrained case, where the rod is considered to be inextensible and unshearable, is more standard in polymer physics. In this last case, our results are more delicate as the Lagrangian description breaks down, and the Hamiltonian formulation must be invoked. It is known that the unconstrained local minimizers approach constrained minimizers as the coefficients in the shear and extension terms of the energy are sent to infinity. Here we observe that the Hamiltonian form of the unconstrained Jacobi system similarly has a limit, so that the fluctuation correction in the path integral can still be expressed as the square root of the determinant of a matrix solution of a set of Jacobi equations appropriate to the constrained problem. As in reality DNA or biological macromolecules are certainly at least slightly shearable and extensible, the limit of the fluctuation correction is undoubtedly physically appropriate. The above theory provides a computationally highly tractable approach to the estimation of the appropriate probability density functions. For application to sequence-dependent models of DNA the associated systems of equations has non-constant coefficients, which is of little consequence for a numerical treatment, but precludes the possibility of finding closed form expressions. On the other hand the theory also applies to simplified homogeneous models. Accordingly, we conclude by applying our approach in a completely analytic and closed-form way to the computation of the approximate probability density function for a uniform, non-isotropic, intrinsically straight and untwisted rod to form a circular loop.

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