Numerical solutions of time fractional order Schrodinger equation using Chebyshev collocation method

The first chapter of this thesis, which consists of five chapters, has been organized as an introduction, and the development of fractional order derivative and integral concepts and the literature survey of fractional order Schrödinger equation is given in this chapter. In the second chapter, fundamental definitions and concepts that will be used in the following chapters are given. In this chapter, general information about Gamma and Beta functions, Grünwald-Letnikov, Riemann-Liouville and Caputo approximations used in fractional order derivatives and integral calculations, and numerical approximations to fractional derivatives are given. In the third chapter, Chebyshev polynomials to be selected as basis functions in the study are examined in detail, and general information about the trigonometric definitions of Chebyshev polynomials, orthogonal polynomials and orthogonality of Chebyshev polynomials are given. In addition, in this chapter, information about the fractional order Schrödinger equation with respect to time is given and information on wavelets, Chebyshev wavelets, approach to functions with Chebyshev wavelets, integrals of Chebyshev wavelets and convergence analysis are given. The fourth chapter contains the original parts of the study, and the fractional order Schrödinger and coupled Schrödinger equations with respect to time were investigated by Chebyshev wavelet method. Two different problems are applied for each fractional model. The numerical methods obtained are compared with some of the results available in the literature, and the error norms L2 and L∞ are presented in tables. Also, graphs of numerical results are given in this chapter. The fifth chapter has been arranged as a conclusion section and the results obtained in the study have been evaluated in this chapter. Keywords: Schrödinger Equation, Fractional Derivation, Chebyshev Wavelet Method,Collocation Method