Pratima Sadangi in a Ph.D. thesis submitted to Utkal University proved results on degree of approximation of functions by operators associated with their Fourier series. In this dissertation, we consider degree of approximation of functions in Hα,ρ by different operators. In Chapter 1 we mention basic definitions needed for our work. In Chapter 2 we discuss different methods of summation. In Chapter 3 we define the Hα,ρ metric and present the degree of approximation problem relating to Fourier series and conjugate series of functions in the Hα,ρ metric using Karamata (Κλ) means. In Chapter 4 we present the degree of approximation of an integral associated with the conjugate series by the Euler, Borel and (e,c) means of a series analogous to the Hardy-Littlewood series in the Hα,ρ metric. In Chapter 5 we propose problems to be solved in the future.

Ball and plate balancing control systems are commonly studied due to the complex dynamics associated with the instability of the system in open-loop. For simplicity, mathematical models describing the ball and plate dynamics are often linearized and the effects of complex motion are assumed to be negligible. These assumptions are rarely backed with evidence or explanations validating the simplified form of the dynamical equations of motion. This thesis focuses on developing a nonlinear model that more accurately defines the dynamics of the system, in order to quantify the error of linear and nonlinear models when compared to a Simscape physical system model. To develop the nonlinear model, this thesis considers both Newton-Euler and Lagrangian modeling methods and applies the method best suited for the ball and plate system. A linear state-feedback controller is developed to compare the stable responses of each system model. The response of each plant model in open-loop and closed-loop configurations subject to different inputs, initial conditions, and disturbances are simulated in the Simulink environment. When compared to the physical system, there was less error from the nonlinear model than from the linear model for both initial condition and disturbance responses. The differences in error were as high as 2% compared to 10% for the nonlinear and linear models, respectively. These results show that there are significant differences associated with model simplification. To optimize the performance, it may be advantageous to utilize a nonlinear model, however, the linearized model is still valid to be used in certain applications due to its stable response behavior.

With the renewed interest in the supersonic and hypersonic flight vehicles, new inverse Euler methods are developed in these flow regimes where a space marching numerical technique is valid. In order to get a general understanding for the specification of target pressure distributions, a study of minimum drag body shapes was conducted over a Mach number range from 3 to 12. Numerical results show that the power law bodies result in low drag shapes, where the n=.69 (l/d = 3) or n=.70 (l/d = 5) shapes have lower drag than the previous theoretical results (n=.75 or n=.66 depending on the particular form of the theory). To validate the results, a numerical analysis was made including viscous effects and the effect of gas model. From a detailed numerical examination for the nose regions of the minimum drag bodies, aerodynamic bluntness and sharpness are newly defined. Numerous surface pressure-body geometry rules are examined to obtain an inverse procedure which is robust, yet demonstrates fast convergence. Each rule is analyzed and examined numerically within the inverse calculation routine for supersonic (M_∞= 3) and hypersonic (M_∞ = 6.28) speeds. Based on this analysis, an inverse method for fully three dimensional supersonic and hypersonic bodies is developed using the Euler equations. The method is designed to be easily incorporated into existing analysis codes, and provides the aerodynamic designer with a powerful tool for design of aerodynamic shapes of arbitrary cross section. These shapes can correspond to either "wing like" pressure distributions or to "body like" pressure distributions. Examples are presented illustrating the method for a non-axisymmetric fuselage type pressure distribution and a cambered wing type application. The method performs equally well for both nonlifting and lifting cases. For the three dimensional inverse procedure, the inverse solution existence and uniqueness problem are discussed. Sample calculations demonstrating this problem are also presented. / Ph. D.

Yes / In this paper, an Euler-Lagrange model for computational fluid dynamics was used to model a full-scale gas-mixed anaerobic digester. The flow profiles, local values of non-Newtonian viscosity and average shear rate were analysed. Recommendations to enhance the effectiveness of mixing were given. In particular, the gas mixing input power can be reduced without appreciable detrimental effects on the mixing effectiveness.

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<p> Various analytical methods for studying the behaviour of shallow conical shells on Euler-Winkler elastic foundation are presented. </p> <p> To account for the nature of concrete and the geometric properties of the shallow conical shell, Poisson's ratio and certain radial and circumferential deformations of the middle surface are neglected in deriving the basic differential equation. Analytical methods employed in the solution of this shell problem are the GECKELER and asymptotic types of approximations. </p> <p> The presentations of various methods of analysis are made for a representative case of dimensions and loadings of the conical shell to make them as applicable as possible to the cases of thin conical shell commonly encountered in industry. </p> <p> The shell structure studied is a tank in the form of a rotationally symmetrical cylindrical shell supported by a shallow conical shell foundation. The construction joint between the conical shell and the cylindrical shell is either monolithic or hinged. </p> <p> The analytical results of this water tank supported on Euler-Winkler elastic foundation are compared with the corresponding findings of W. Flügge, who assumed a uniform soil bearing pressure acting on the conical shell structure. </p> The method of analysis which possesses obvious advantages over the other methods studied is selected to examine the effect of different elastic stiffness coefficients of the soil. The validity of simplifying the soil bearing pressure to a uniform distribution by most designers can consequently be studied by comparing it to the bearing pressures of an ideal elastic soil which is postulated to react to its deformation like a bed of independent elastic springs. </p> / Thesis / Master of Engineering (ME)

We look at the Cauchy problem with an accretive Operator on a Banach space. We give an upper bound for the norm of the difference of two solutions of Euler schemes with this accretive operator. This concrete estimate also works for the problem with a non-zero right-hand side in the Cauchy problem and is a generalization of a famous result by Kobayashi. We also show, how this result gives direct proofs for existence, uniqueness, stability and regularity of Euler solutions of the Cauchy problem and also the rate of convergence of solutions of Euler schemes. The results concerning regularity and rate of convergence are generalized for problem data in interpolation sets.:1. Accretive operators 1.1. Thebracket. 1.2. Accretive operators 1.3. The Cauchy problem and Euler solutions 2. A priori estimates for solutions of implicit Euler schemes 2.1. An implicit upper bound 2.2. Properties of the density 2.3. An explicit upper bound 3. Applications 3.1. Wellposedness of the Cauchy problem 3.2. Interpolation theory A. Functions of bounded variation

This study seeks to validate a recently introduced global preconditioning technique for the Euler equations. Energy and enthalpy equations are nondimensionalized by means of a reference enthalpy, resulting in increased numerical accuracy for low-speed flows. A cellbased, finite volume formulation is used, with Roe flux difference splitting and both explicit and implicit time integration schemes. A Newton-linearized iterative implicit algorithm is implemented, with Symmetric Gauss-Seidel (LU/SGS) nested sub-iterations. This choice allows one to retain time accuracy, and eliminates approximate factorization errors, which become dominant at low speed flows. The linearized flux Jacobians are evaluated by numerical differentiation. Higher-order discretization is constructed by means of the MUSCL approach. Locally one-dimensional characteristic variable boundary conditions are implemented at the farfield boundary. The preconditioned scheme is successfully applied to the following traditional test cases used as benchmarks for local preconditioning techniques: point disturbance, flow angle disturbance, and stagnation point arising from the impingement of two identical jets. The flow over a symmetric airfoil and a convergentdivergent nozzle are then simulated for arbitrary Mach numbers. The preconditioned scheme greatly enhances accuracy and convergence rate for low-speed flows (all the way down to M ≈ 10E − 4). Some preliminary tests of fully unsteady flows are also conducted.

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