The numerical solutions of the fractional order diffusion and diffusion wave equations by lucas polynomials

In the first chapter constituting the introduction chapter of this thesis consisting of five chapters some brief information is given about the fractional order derivatives and integrals. In the second chapter, after presenting some special functions used in the thesis, Grünwald-Letniko, Riemann-Liouville and Caputo approximations used in the calculations of fractional order derivative and integral are given. In the third chapter, after giving some fundamental concepts about Lucas polinomial method which is chosen as the method in the thesis, the method is applied to integer-order heat and wave equations and the obtained numerical results together with the error norms L2 and L∞ of are given in tables. In the fourth chapter which forms the basis of this thesis, fractional order diffusion and diffusion wave equations are solved numerically by Lucas polynomial method considering the appropriate initial and boundary conditions. The numerical results obtained with the application of this method to model problems and the comparison of the results with other avaliable studies in the litareture are presented in tablaes. In addition, the results were presented graphically to demonstrate the physical behavior of the problem. The fifth chapter of the thesis is organized as discussion and conclusion chapter. In this chapter, the obtained results are discussed and both the advantages and disadvantages of the Lucas polinomial method are evaluated.