Numerical solutions of some fractional order partial differential equations via the cubic b-spline finite element method

This thesis consists of six chapters. In the first chapter, a literature review on fractional-order differential equations and the B-spline finite element method is presented. In the second chapter, the fundamental definitions and theorems used in the subsequent chapters of the thesis are provided. In the third chapter, numerical solutions of the anomalous diffusion equation containing fractional derivatives in the Caputo sense are obtained using the cubic B-spline finite element method. An implicit finite difference scheme is proposed, and its stability, convergence, and error analysis are conducted. In the fourth chapter, numerical solutions of the variable coefficient Tricomi-Keldysh equation containing fractional derivatives in the Caputo sense are obtained using the cubic B-spline finite element method. An implicit finite difference scheme is proposed, and its stability and convergence analysis are performed. In the fifth chapter, numerical solutions of the nonlinear Burger equation containing fractional derivatives in the Caputo sense are obtained using the cubic B-spline finite element method. An implicit difference scheme is proposed, and its stability and convergence analysis are carried out. The final chapter consists of the discussion and conclusion sections where the thesis is evaluated.