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The Nosé-Hoover thermostat is a widely-used technique in molecular-dynamics simulations for maintaining a canonical ensemble at a desired temperature. The associated differential equation, known as the Nosé-Hoover equation, plays a crucial role in understanding the dynamics of the system. This thesis investigates the integrability of the Nosé-Hoover equations using the Darboux theory of integrability. It presents original proofs of the Darboux non-integrability of these equations (thereby identifying and correcting several major mistakes in the previous work by Adam Mahdi and Claudia Valls. “Integrability of the Nosé–Hoover equation”. In: Journal of Geometry and Physics 61.8 (2011), pp. 1348–1352.).

The Nosé-Hoover thermostat is a widely-used technique in molecular-dynamics simulations for maintaining a canonical ensemble at a desired temperature. The associated differential equation, known as the Nosé-Hoover equation, plays a crucial role in understanding the dynamics of the system. This thesis investigates the integrability of the Nosé-Hoover equations using the Darboux theory of integrability. It presents original proofs of the Darboux non-integrability of these equations (thereby identifying and correcting several major mistakes in the previous work by Adam Mahdi and Claudia Valls. “Integrability of the Nosé–Hoover equation”. In: Journal of Geometry and Physics 61.8 (2011), pp. 1348–1352.).

Spectral methods are used to solve partial differential equations numerically. When the solution is analytic, the rate of convergence of the numerical solution is exponential; that is, the error decays exponentially. In time-dependent PDEs, low order finite difference schemes and spectral schemes are traditionally being used for the time and spatial derivatives, respectively. However, applying spectral schemes in both space and time has been thought of recently. These methods have spectral convergence in both spatial and temporal domains. In this thesis, both Chebyshev and Legendre spectral collocation methods are implemented for the Navier–Stokes and Magnetohydrodynamics equations. Numerical solutions for both equations converge exponentially when the solutions are analytic. Moreover, Navier-Stokes and Magnetohydrodynamic equations are implemented for high Reynolds numbers using nonlinear preconditioning methods, ASPIN and RASPEN, which are defined using spectral domain decomposition.

The main theme of this thesis is further scrutinizing classic Hardy inequalities and expanding the study on "Optimal Hardy Inequality for General Elliptic Operators with Improvement". We rediscovered explicit integral form of Hardy inequality with the main focus on its functional aspects, including density of Sobolev space. We discuss our motivation for exploring the main Hardy inequality, with a focus on the quadratic case. The fundamental solution of Laplacian is applied for illustrating the proof of classical Hardy inequality. Furthermore, we conclude the classic Hardy inequalities for a bounded domain and Hardy's boundary inequality for smooth boundary of domain. Then, the inequalities are considered for operators more general than Laplacian and various Hardy inequalities are explored in terms of different boundary weight and interior weight of positive function E. Also The generalization of Cafferelli-Kohn-Nirenberg inequality are examined to find the optimal and not attained constant. Furthermore, the possibility of more general weighted inequalities is investigated. Finally, we consider one common type of improvement for the above-mentioned inequalities.

The main theme of this thesis is further scrutinizing classic Hardy inequalities and expanding the study on "Optimal Hardy Inequality for General Elliptic Operators with Improvement". We rediscovered explicit integral form of Hardy inequality with the main focus on its functional aspects, including density of Sobolev space. We discuss our motivation for exploring the main Hardy inequality, with a focus on the quadratic case. The fundamental solution of Laplacian is applied for illustrating the proof of classical Hardy inequality. Furthermore, we conclude the classic Hardy inequalities for a bounded domain and Hardy's boundary inequality for smooth boundary of domain. Then, the inequalities are considered for operators more general than Laplacian and various Hardy inequalities are explored in terms of different boundary weight and interior weight of positive function E. Also The generalization of Cafferelli-Kohn-Nirenberg inequality are examined to find the optimal and not attained constant. Furthermore, the possibility of more general weighted inequalities is investigated. Finally, we consider one common type of improvement for the above-mentioned inequalities.

n this thesis some Lane-Emden problems of different order are studied. Tackling the issue of existence of a positive solution and regularity of the solutions are of paramount importance for each instance. In addition to discussing the general Lane-Emden equation, the cases of having an advection term to the original problem and investigating some fourth order nonlinear Dirichlet and Navier problems are of considerable interest. While the well-studied general equation points out that for p≥N+2/N−2 and Ω a star-shaped domain in RN there would be no non-trivial solution, some advantageous results regarding the existence of a positive solution and regularity of the solutions on a general bounded domain inRNare addressed for the equations where an advection is involved, as well as some nonlinear fourth order problem with given Dirichlet and Navier boundary conditions.

In this thesis, we present numerical solution of semilinear partial differential equations (PDEs) where the linear differential operator is a self-adjoint. A recent spectral method for self-adjoint operators, based on basis recombination, leads to symmetric definite matrices, which have real spectrum. This allows for developing stable time-stepping algorithm to solve the resulting the ordinary differential equations. The linear part of the discretized problem is usually stiff, which constraints the step size for explicit numerical schemes. We therefore use exponential integrators, a well-known time-stepping methods for solving stiff differential equations. We describe three methods namely, eigen-decomposition, contour integral and Carath\'{e}odory-Fej\'{e}r approximation, for computing the matrix ($\varphi$) functions of the exponential integrators. We perform numerical experiments with some PDEs with different boundary conditions, including time-dependent boundary condition and the numerical results confirm the accuracy of the methods in both space and time.

This thesis investigates some aspects of the integrability problem of a Hamiltonian system. The Hamiltonian system with Hamiltonian function H = Xn i,j=1 1 2gij(x1, . . . , xn)pipj , describes the geodesic flow of a Riemannian metric ds2 = Pn i,j=1 gij(x1, . . . , xn)dxidxj on an n-dimensional manifold. Some results from the research article, Polynomials integrals of magnetic geodesic flows on the 2-torus on several energy levels [3], are studied. In particular, a complex structure on the 2-torus is constructed to prove that if the geodesic flow with non-zero magnetic field on the 2-torus admits an additional cubic-in-momenta first integral on two different energy levels, then the magnetic field and the metric are functions of one variable.

In this thesis, we present numerical solution of semilinear partial differential equations (PDEs) where the linear differential operator is a self-adjoint. A recent spectral method for self-adjoint operators, based on basis recombination, leads to symmetric definite matrices, which have real spectrum. This allows for developing stable time-stepping algorithm to solve the resulting the ordinary differential equations. The linear part of the discretized problem is usually stiff, which constraints the step size for explicit numerical schemes. We therefore use exponential integrators, a well-known time-stepping methods for solving stiff differential equations. We describe three methods namely, eigen-decomposition, contour integral and Carath\'{e}odory-Fej\'{e}r approximation, for computing the matrix ($\varphi$) functions of the exponential integrators. We perform numerical experiments with some PDEs with different boundary conditions, including time-dependent boundary condition and the numerical results confirm the accuracy of the methods in both space and time.

This thesis investigates some aspects of the integrability problem of a Hamiltonian system. The Hamiltonian system with Hamiltonian function H = Xn i,j=1 1 2gij(x1, . . . , xn)pipj , describes the geodesic flow of a Riemannian metric ds2 = Pn i,j=1 gij(x1, . . . , xn)dxidxj on an n-dimensional manifold. Some results from the research article, Polynomials integrals of magnetic geodesic flows on the 2-torus on several energy levels [3], are studied. In particular, a complex structure on the 2-torus is constructed to prove that if the geodesic flow with non-zero magnetic field on the 2-torus admits an additional cubic-in-momenta first integral on two different energy levels, then the magnetic field and the metric are functions of one variable.

The first part of this thesis focuses on quasi-Newton methods. Broyden's method is a quasi-Newton method which is used to solve a system of nonlinear equations. Almost all convergence theory in the literature assumes existence of a root and bounds on the nonlinear function and its derivative in some neighbourhood of the root. All these conditions cannot be checked in practice. The motivation of this work is to derive a convergence theory where all assumptions can be verified, and the existence of a root and its superlinear rate of convergence are consequences of the theory. The theory is simple in the sense that it contains as few constants as possible. The method of Broyden-Fletcher-Goldfarb-Shanno (BFGS) is also a quasi-Newton method for unconstrained minimization. We generalize a convergence theory where all assumptions are verifiable and existence of a minimizer and superlinear convergence of the iteration are conclusions. In a continuation of this part, we consider Perry nonlinear conjugate gradient (NCG) method and scaled memoryless BFGS method. In the second part, space-time spectral methods are considered. For time dependent problems, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for spatial derivatives. Spectral methods which converge spectrally in both space and time have appeared recently. In this thesis, it is shown that a Chebyshev spectral collocation method of Tang and Xu for the heat equation converges exponentially when the solution is analytic. We also derive a condition number estimate of the method. Another space-time Chebyshev collocation scheme which is easier to implement is proposed and analyzed. We also present space-time spectral collocation methods for the Schrodinger, wave, Airy and beam equations. In particular, fully spectral convergence and a condition number estimate are shown for Schrodinger and wave equations. Numerical results verify the theoretical results, and demons

Knowledge about the spectrum of the Uzawa pressure operator is important for solving and performing an error analysis of the Stokes problem. The infimum of the spectrum of the Uzawa pressure operator is significant, for instance, it gives information about the rate of convergence of numerical methods for the Stokes problem. The spectrum of the Uzawa pressure operator is still not known for the case of a square domain. This thesis provides some results related to this problem. It depicts efforts made in estimating the infimum of the spectrum of the Uzawa pressure operator. In 1996, M. Gaultier and M. Lezaun gave an upper bound equal to 0.2260 for the infimum of the spectrum of the Uzawa pressure operator. We have improved it to 0.20164. We conclude this thesis by giving a conjecture that the infimum of the spectrum of the Uzawa pressure operator is equal to 0.181690.