Fibonacci collocation method for numerical solutions of partial differential equations and residual error analysis

In this thesis, Fibonacci collocation method based on matrix operations using Fibonacci polynomials has been developed for some classes of partial differential equations. The Fibonacci collocation method is based on the matrix relation of each term of the equation using the matrix forms of Fibonacci polynomials. The matrices found are transformed into the matrix equation and the collocation points are applied to this equation. The fundamental matrix equation is transformed into a reduced row echelon form using the elementary row operations. Fibonacci polynomial solution is obtained by solving the reduced matrix equation which is found. Fibonacci collocation method has been applied to hyperbolic telegraph equations, one dimensional convection diffusion problem equation, one dimensional variable coefficient fractional diffusion equations and Helmholtz type elliptic boundary value problems. In addition, the Poisson type elliptic boundary value problem, which is a subclass of the Helmholtz type elliptic boundary value problem, has also been applied. Error analysis algorithms based on the residual error function are developed for each of the partial differential equations, and the error functions for the obtained Fibonacci polynomial solutions are found. Fibonacci polynomial solutions have been improved with these estimated error functions. Examples are given for each type of equation in order to demonstrate the accuracy and effectiveness of the method. Fibonacci polynomial solutions for these examples have been compared with the exact solutions of the problems discussed with the aid of figures and tables. In addition, the accuracy of the algorithms is tested by comparing the error functions of the error estimation functions with those of known solutions. The improved solutions found using the estimated error function are presented in figures and tables comparatively. Symbolic programming languages are used for solving and analyzing the examples. Fibonacci collocation method has been seen to be a numerical method that can be easily programmable and give fast results.