الفهرس 
chapter 1Only 14 pages are availabe for public view
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Abstract
We develop a transformation free non-uniform high order compact scheme
(HOeS) for solving the steady 3D convection-diffusion equation with variable coefficients. The approach is based on the use of Taylor series expansion, up to the fourth order terms, to approximate the derivatives appearing in the 3D convection
diffusion equation. Then the original convection-diffusion equation is used again to replace the resulting higher order derivative terms. This leads to a higher order scheme on a compact stencil of nineteen points. Effectiveness of this method is seen from the fact that it can handle the singularity perturbed problems by employing a flexible discretized grid that can be adapted to the singularity in the domain. After achieving
the fourth order accuracy to both 2D and 3D convection-diffusion problems, an
operator-based interpolation scheme combined with an extrapolation technique is used to approximate the sixth-order accurate solution for these problems. Due to the singularity perturbed nature of the 3D solved problems, it is important to use efficient solver for the resulting anisotropic algebraic system. In this thesis, we apply an efficient Algebraic Multigrid technique (AMG).
Part two is an application of the HOeS. We present an efficient sixth-order
finite difference discretization for the vibration analysis of a nonlocal Euler-Bernoulli beam embedded in an elastic medium. The Pasternak elastic foundation model is utilized to represent the surrounding elastic medium. Nonlocal differential elasticity of Eringen is exploited to reveal the non-locality effect of nanobeams. Sixth-order
accuracy schemes are developed for discretization of both the governing equation and boundary conditions. Sixth-order accuracy schemes are derived for simply supported,clamped and free boundary conditions. Numerical results include comparison with the exact solutions and with previously published works are presented for the fundamental
frequencies. In addition, numerical results are presented to figure out the effects of nonlocal parameter, slenderness ratios, and boundary conditions on the dynamic acteristics of the beam.