DRBEM applications in fluid dynamics problems and DQM solutions of hyperbolic equations

Pekmen, Bengisen
In this thesis, problems of fluid dynamics defined by the two-dimensional convection-diffusion type partial differential equations (PDEs) are solved using the dual reciprocity boundary element method (DRBEM). The terms other than the Laplacian are treated as inhomogeneous terms in the DRBEM application. Once the both sides are multiplied by the fundamental solution of Laplace equation, and then integrated over the domain, all the domain integrals are transformed to boundary integrals using the Green's identities. The inhomogeneous terms are approximated with radial basis functions, and the space derivatives in convective terms are easily handled by using the DRBEM coordinate matrix constructed from the radial basis functions. The discretization of the boundary is achieved with linear elements. For the solution of unsteady problems, first order Backward-Euler and third order Houbolt time integration schemes are used. The boundary only nature of DRBEM provides one to obtain the results in a small computational cost compared to the domain discretization methods. Incompressible fluid flow in cavities, natural and mixed convection flow in enclosures are simulated when the medium is porous or non-porous, and with or without magnetic effect. The numerical results are visualized for different non-dimensional physical parameters in terms of streamlines, isotherms, vorticity, induced magnetic field lines and current density contours. In the thesis, the differential quadrature method (DQM) is also used for solving especially problems defined by hyperbolic equations and nonlinear in nature. DQM is made use of discretizing both time and space domains, and the solution is obtained at one stroke or blockwise without the need of an iteration. The nonlinearities are handled using an iteration procedure. Accurate results are obtained using considerably small number of Gauss-Chebyshev-Lobatto discretization points at very small expense. Test problems include Klein-Gordon, sine-Gordon equations, hyperbolic telegraph equations, and viscous Burgers' equation.


DRBEM Solution of Incompressible MHD Flow with Magnetic Potential
Pekmen, B.; Tezer, Münevver (2013-12-01)
The dual reciprocity boundary element method (DRBEM) formulation is presented for solving incompressible magnetohydrodynamic (MHD) flow equations. The combination of Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetics through Ohm's law is considered in terms of stream function, vorticity and magnetic potential in 2D. The velocity field and the induced magnetic field can be determined through the relations with stream function and magnetic potential, respectively. The numeri...
DRBEM solution of exterior nonlinear wave problem using FDM and LSM time integrations
Meral, Guelnihal; Tezer, Münevver (Elsevier BV, 2010-06-01)
The nonlinear wave equation is solved numerically in an exterior region For the discretization of the space derivatives dual reciprocity boundary element method (DRBEM) is applied using the fundamental solution of Laplace equation. The time derivative and the nonlinearity are treated as the nonhomogenity. The boundary integrals coming from the far boundary are eliminated using rational and exponential interpolation functions which have decay properties far away from the region of Interest. The resulting sys...
DRBEM solutions of Stokes and Navier-Stokes equations in cavities under point source magnetic field
Senel, P.; Tezer, Münevver (2016-03-01)
This paper describes an iterative dual reciprocity boundary element method (DRBEM) for the solutions of Stokes and Navier-Stokes equations in cavities under the effect of an external point source magnetic field placed very close to the bottom. The fluid is viscous, incompressible and electrically non-conducting but magnetizable, and the flow is steady, laminar and fully developed. Both the Stokes and Navier-Stokes equations are solved in terms of velocity and pressure of the fluid by using DRBEM. Pressure b...
Studies on the perturbation problems in quantum mechanics
Koca, Burcu; Taşeli, Hasan; Department of Mathematics (2004)
In this thesis, the main perturbation problems encountered in quantum mechanics have been studied.Since the special functions and orthogonal polynomials appear very extensively in such problems, we emphasize on those topics as well. In this context, the classical quantum mechanical anharmonic oscillators described mathematically by the one-dimensional Schrodinger equation have been treated perturbatively in both finite and infinite intervals, corresponding to confined and non-confined systems, respectively.
Sturm comparison theory for impulsive differential equations
Özbekler, Abdullah; Ağacık, Zafer; Department of Mathematics (2005)
In this thesis, we investigate Sturmian comparison theory and oscillation for second order impulsive differential equations with fixed moments of impulse actions. It is shown that impulse actions may greatly alter the oscillation behavior of solutions. In chapter two, besides Sturmian type comparison results, we give Leightonian type comparison theorems and obtain Wirtinger type inequalities for linear, half-linear and non-selfadjoint equations. We present analogous results for forced super linear and super...
Citation Formats
B. Pekmen, “DRBEM applications in fluid dynamics problems and DQM solutions of hyperbolic equations,” Ph.D. - Doctoral Program, Middle East Technical University, 2014.