>Magister Scientiae - MSc / This thesis reports the results of a twofold study on the recently proposed phenomenon of 'stellar pulsations' in young brown dwarfs by the seminal study of Palla and Baraffe (2005) (PB05, thereafter). The PB05 study presents results of a non-adiabatic linear stability analysis showing that young brown dwarfs should become pulsationally unstable during the deuterium burning phase of their evolution. The PB05 calculations on which this prediction is based have already been applied in a number of ground and space-based observational campaigns aimed at searching for this newly proposed putative class of potential pulsators. However, despite their significance and implications, the theoretical calculations by PB05 have not yet, to date, been subjected to independent verification in a different computational framework. To achieve this, we have generated equilibrium brown dwarf models and performed non-adiabatic linear stability calculations similar to PB05 assuming their 'frozen-in convection' approximation and the relevant input physics. The calculations performed in this thesis show, in overall, that there is a good agreement between the results from our study and those in PB05. However, there seem to be significant differences for very low mass objects as pointed out in our comparative results. We attribute this difference to our different boundary conditions. Our outer boundary condition is equivalent to the Eddington approximation in the 3-D case (e.g see Unno and Spiegel (1966)), while PB05 use a combination of different atmospheric profiles as discussed in Chabriel and Baraffe (2000). The validity of the frozen-in assumption used by PB05, which is based on the argument that the convective time scales calculated for these objects are much less than the pulsation time scales, has not been investigated. In this thesis, we have invoked a time-dependent theory of convection similar to Kuhfuss (1986) and Stellingwerf (1982) which includes turbulent pressure, turbulent diffusion and turbulent viscosity to study the pulsations. We have also investigated the effects of varying a number of free parameters in the above theoretical models. Our results show that turbulent pressure dominates in driving the pulsations in young brown dwarfs yielding growth rates much higher than in the frozen-in scenario. This is a new result that requires further analysis. The perturbation in the convective flux is found to have a damping effect on the acoustic modes. Turbulent viscosity is found to lead to damping which increases with increase in the value of the turbulent viscosity parameter and is found to have very little effect on the fundamental mode pulsation periods. Variation in the turbulent diffusion parameter has a very small effect on the fundamental mode periods and e-folding times. As a side lobe, we have determined theoretical pulsation constants for the fundamental mode and calculated the period ratios for the fundamental mode to those of the first and second harmonics. We find values of pulsation constants falling within the theoretical values calculated for variable stars shown in Cox (1980). This is explained in relation to the terms that go into the theoretical formula discussed later in this thesis. We find a correlation between the period ratios and the BDs mass and argue that such plots of the period ratios vs mass of the BDs could be useful in constraining the masses, given known periods from observations.

The goal of this thesis is to apply the theory of multilinear weighted Fourier estimates to nonlinear dispersive equations in order to tackle problems in regularity, well-posedness, and pointwise convergence of solutions. Dispersion of waves is a ubiquitous physical phenomenon that arises, among others, from problems in shallow-water propagation, nonlinear optics, quantum mechanics, and plasma physics. A natural tool for understanding the related physics is to study waves/signals simultaneously from both physical and spectral perspectives. Specifically, we will treat nonlinearities as multilinear operator perturbations, which (by the method of spacetime Fourier transforms), exhibit smoothing properties in norms defined to reflect the dispersive natures of the solutions. Our model equation is the quantum Zakharov system, which can be viewed as a variation on the cubic nonlinear Schrödinger equation (NLS). We investigate the model in various contexts (adiabatic limits, nonlinear Schrödinger limits, semi-classical limits). We additionally study a variation of Carleson's Fourier convergence problem in the context of pointwise convergence of the full Schrödinger operator with non-zero potential.

The method of steepest descent is used to solve partial differential equations of mixed type. In the main hypothesis for this paper, H, L, and S are Hilbert spaces, T: H -> L and B: H -> S are functions with locally Lipshitz Fréchet derivatives where T represents a differential equation and B represents a boundary condition. Define ∅(u) = 1/2 II T(u) II^2. Steepest descent is applied to the functional ∅. A new smoothing technique is developed and applied to Tricomi type equations (which are of mixed type). Finally, the graphical outputs on some test boundary conditions are presented in the table of illustrations.

No description available.

No / A dynamic system with memory is a system for which knowledge of the equations of motion, together with the state at a given time instant t0 is insufficient to predict the evolution of the state at time instants t>t0. To calculate the response of systems with memory starting from an initial time instant t0, complete knowledge of the history of the system for t<t0 is needed. This is because the state vector does not contain all the information necessary to fully characterize the state of the system, i.e., the state vector of the system is not complete. In this paper, a state space formulation of viscoelastic systems with memory is proposed, which overcomes the concept of memory by enlarging the state vector with a number of internal variables that bear the information about the previous history of the system. The number of these additional internal variables is in some cases finite, in other cases, it would need to be infinite, and an approximated model has to be used with a finite number of internal variables. First a state space representation of the generalized Maxwell model is shown, then a new state space model is presented in which the relaxation function is approximated with Laguerre polynomials. The accuracy of the two models is shown through numerical examples.

Riemann problems are of prime importance in computational fluid dynamics simulations using finite elements or finite volumes discretizations. In some applications, billions of Riemann problems might need to be solved in a single simulation, therefore it is important to have reliable and computationally efficient algorithms to do so. Given the nonlinearity of the flux function in most systems considered in practice, to obtain an exact solution for the Riemann problem explicitly is often not possible, and iterative solvers are required. However, because of issues found with existing iterative solvers like lack of convergence and high computational cost, their use is avoided and approximate solvers are preferred. In this thesis work, motivated by the advances in computer hardware and algorithms in the last years, we revisit the possibility of using iterative solvers to compute the exact solution for Riemann problems. In particular, we focus on the development, implementation, and performance comparison of iterative Riemann solvers for the shallow water and Euler equations. In a one-dimensional homogeneous framework for these systems, we consider several initial guesses and iterative methods for the computation of the Riemann solution. We find that efficient and reliable iterative solvers can be obtained by using recent estimates on the Riemann solution to modify and combine well-known methods. Finally, we consider the application of these solvers in finite volume simulations using the wave propagation algorithms implemented in Clawpack.

The solutions for a class of Neutral Functional Di erential Equations (NFDE) with weakly singular kernels are studied. Using singular expansion techniques, a representation of the solution of the NFDE is obtained by studing an associated Volterra Integral Equation. We study the Collocation Method as a projection method for the approximation of solutions for Volterra Integral Equations. Particulary, the possibility of achieving higher order ap- proximations is discussed. Special attention is given to the choice of the projection space and its relation to the smoothness of the approximated solution. Finally, we study the identification problem for a parameter appearing in the weakly singular operator of the NFDE. / Ph. D.

This work focuses on a comparison of numerical methods for linear quadratic regulator (LQR) problems defined by parabolic partial differential equations. In particular, we study various methods for computing functional gains to boundary control problems for the heat equation. These methods require us to solve various equations including the algebraic Riccati equation, the Riccati partial differential equation and the Chandrasekhar partial differential equations. Numerical results are presented for control of a one-dimensional and a two-dimensional heat equation with Dirichlet or Robin boundary control. / Ph. D.

This thesis is a study of stability of the solutions of differential equations.

In this thesis, we study minimizers of the energy functional 𝐽 (𝑢,Ω) = ∫_Ω |∇𝑢|²/2 + 𝑊(𝑢) 𝑑𝑥 for two different potentials 𝑊(𝑢). In the first part we consider the Allen-Cahn energy, where 𝑊(𝑢) = (1 − 𝑢²)² is a doublewell potential which is relevant in the theory of phase transitions and minimal interfaces. We investigate the rigidity properties of global minimizers in low dimensions. In particular we extend a result of Savin on the De Giorgi’s conjecture to include minimizers that are not necessarily bounded, and that can have subquadratic growth at infinity. In the second part we consider potentials of the type 𝑊(𝑢) = 𝑢⁺ which appear in obstacletype free boundary problems. We establish higher order estimates and the analyticity of the regular part of the free boundary. Our method relies on developing higher order boundary Harnack estimates iteratively and deducing them from Schauder estimates for certain elliptic equations with degenerate weights. Finally we consider similar regularity questions of the free boundary in the Signorini problem which also known as the thin obstacle problem. We develop 𝐶²^𝛼 estimates of the free boundary under sharp assumptions on the coefficients and the data.