Solutions of fractional order partial differential equations by B-spline finite element methods

The first chapter of this thesis, which is consisting of seven chapters, has been arranged as an introduction chapter and the literature survey has also been given in this chapter. In the second chapter, basic concepts that will be used in later chapters were presented. In this chapter, some information about Gamma and Mittag-Leffler functions, Grünwald-Letnikov, Riemann-Liouville and Caputo approaches which are used in the fractional order derivative and integral calculus, splines and B-spline functions, Galerkin and collocation finite element methods have been presented. Third, fourth, fifth and sixth sections constitute the original parts of this thesis. In the third section, the time fractional order gas equation has been solved using quadratic B-spline Galerkin and cubic B-spline collocation methods. The obtained numerical solutions and error norms $L_{2}$ and $L_{\infty}$ have been presented in tables. Absolute error graphics as well as those of exact and numerical solutions have been given. In the fourth chapter, the time fractional order Burgers equation has been solved by quadratic B-spline Galerkin and cubic B-spline collocation methods. These methods have been applied to three model problems. The obtained numerical solutions and error norms $L_{2}$ and $L_{\infty}$ have been presented in tables. Absolute error graphics as well as those of exact and numerical solutions have been given. In the fifth chapter, the time fractional order telegraph equation has been solved by quadratic B-spline Galerkin and cubic B-spline collocation methods. These methods have been applied to three model problems. The obtained numerical solutions and error norms $L_{2}$ and $L_{\infty}$ have been presented in tables. Absolute error graphics as well as those of exact and numerical solutions have been given. In the sixth chapter, the time fractional order Schrödinger equation has been solved by quadratic B-spline Galerkin and cubic B-spline collocation methods. The obtained numerical solutions and error norms $L_{2}$ and $L_{\infty}$ have been presented in tables. Absolute error graphics as well as those of exact and numerical solutions have been given. Finally, in the seventh chapter, some results have been presented about the numerical solutions of the problems solved by quadratic B-spline Galerkin and cubic B-spline collocation methods.