Spectral theory of differential operators and their numerical applications

This study has been probed in seven sections. In Section 1, the history of spectral theory (well and ill-posed problem) Sturm-Liouville and Dirac operators and the studies that have been done in this area have been given. In addition to that Adomian Decomposition Method (ADM), Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM) that will be used in the thesis has been discussed in this section. In Section 2, some fundamental definitions and theorems used in spectral theory differential operators have been stated. In Section 3, eigenvalues of regular and singular Sturm-Liouville problems have been explained by means of functions which belongs to Paley-Wiener space. In Section 4, the spectral theory of Dirac operator has been explained. Also, spectral results for regular canonic Dirac equations system have been obtained by means of functions which belongs to Paley-Wiener space. In Section 5, the basic idea of ADM, HAM and HPM, which we use in our study, has been given in details. In Section 6, numerical solutions of equation system have been obtained by being applied HAM and HPM to Dirac equations systems. About sensitivity of these methods have been made comment. Also, convergence analysis of solutions, which have been obtained, have been done. In Section 7, information has been given about spectral theory of Diffusion equations. About the sensitivity of methods for diffusion equation, which has been obtained numerical solutions, with ADM, HAM and HPM have been made comment. Key Words: Eigenvalue, Eigenfunction, Sturm-Liouville Operator, Dirac Equation System, Diffusion Operator, Paley-Wiener Space, Adomian Decomposition Method (ADM), Homotopy Analysis Method (HAM), Homotopy Perturbation Method (HPM).