Kalıntı metodu kullanımı ile doğrusal ve doğrusal olmayan özdeğer problemlerinin sayısal çözümleri

The aim of this thesis is to find numerical solutions and approximate eigenvalues of linear and nonlinear eigenvalue problems by applying improved residual method. The residual method is based on the construction of the approximate solution by using the Bézier curves. In this thesis, at first, the residual method, which was developed for initial value problems, is improved to find the unknown coefficients of approximate solution explicitly without the need for any system solution. Most significant advantage of the method is finding approximate solutions of nonlinear problems without any linearization or solving any system of equations. Later, the adaptation of improved residual method is given to find approximate eigenvalues of regular, singular Sturm-Liouville and nonlinear eigenvalue problems, such as Euler eigenvalue problem, Paine problem, Laplace tidal wave equation, Dunsch equation, Boyd equation and Bratu problem. Error analysis for regular Sturm-Liouville eigenvalue problems is demonstrated for a special case. For singular problems, according to the location of singularity, different strategies are presented to find the approximate eigenvalues. Comparisons of the obtained numerical results with the theoretical findings and numerical results obtained by using various methods in literature are shown in graphs and tables. Observed orders are demonstrated for regular Sturm-Liouville and Bratu eigenvalue problems in tables, which show that observations are well confirm with theoretical ones. Comparisons and theoretical observations show that the improved and adapted method is very convenient and successful in solving linear and nonlinear eigenvalue problems and finding high index eigenvalues approximately with high accuracy.