Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
Browse
2 results
Search Results
Master Thesis Numerical Solution Methods for Boundary Value Problems for the Laplace Equation in Semi-Infinite Domains(01. Izmir Institute of Technology, 2023) Plattürk, Sabahat Defne; Tanoğlu, GamzeThe essential purpose of this thesis is to get numerical solutions of the Laplace Equation Boundary Value Problems subject to Dirichlet and Mixed boundary conditions on doubly connected semi-infinite domains, namely the upper half plane and semi-infinite strips, using boundary integral equations. Conformal maps served as a tool to transform the doubly connected semi-infinite domains into a doubly connected bounded domain. Images of boundary conditions are evaluated and the accuracy of the conformal maps are investigated. Then each problem is reduced to a system of linear boundary integral equations by representing the solution to the boundary value problems as combinations of double- and single-layer potentials. In the case of Dirichlet boundary conditions, we used a modification that ensures the unique solvability of the system of Fredholm Integral Equations of the second kind. However, in the case of mixed boundary conditions, such a modification is not needed. After the investigations of uniqueness and existence of solutions to the constructed systems of integral equations of the second kind, the systems of equations are solved by using the Nyström method, based on quadrature rules. For the numerical integration of integral operators with continuous kernels, the trapezoidal rule is used. For the numerical integration of the kernels with logarithmic singularity, we first split off the singularity and apply an extremely accurate quadrature rule for the improper integrals. Error analysis for both numerical integration techniques are given in details and the accuracy of Nyström Method which depend on the quadrature method is explained. Different test cases are considered to check the accuracy of the method and the order of convergence and error results are illustrated by numerical examples.Master Thesis A Compact Finite Difference Method of Lines for Solving Non-Linear Partial Differential Equations(01. Izmir Institute of Technology, 2022) Ismoilov, Shodijon; Tanoğlu, Gamze; Gürarslan, GürhanIn this thesis, an efficient numerical method is proposed for the numerical solution of the chemical reaction-diffusion model governed by a non-linear system of partial differential equations known as a Brusselator model. The method proposed is based on a combination of higher-order Compact Finite Difference schemes and stable time integrator known as an adaptive step-size Runge-Kutta method. The performance of adaptive step-size Runge-Kutta formula of fifth-order accurate in time and Compact Finite Difference scheme of sixth-order in space are investigated. The method is implemented to solve three test problems and reveals that the method is capable of achieving high efficiency, accuracy and reliability. The results obtained are sufficiently accurate compared to some available results in the literature.