Lucas collocation method for solving fractional differantial equations and their systems

In this thesis, a collocation method based on Lucas polynomials is used to solve fractional order linear differential equations and equation systems numerically. In this method, first, an approximate solution to the differential equation is expressed in terms of a finite series of Lucas polynomials with unknown coefficients. Then, using the approximate solution and collocation points, the initial differential equation is transformed into a linear algebraic system, in which Lucas coefficients are the unknowns. Caputo fractional derivative is used to calculate fractional derivatives. Then, using the approximate solution and collocation points, the studied equation is transformed into a linear algebraic system whose unknowns are Lucas coefficients, which can be expressed in matrices. Finally, using matrix arithmetic, approximate solution coefficients are found. By working on numerical examples, the results of the method are compared with the results of other studies in the literature. In this study, gamma function, beta function and fractional derivative definitions are used to solve problems. The examples related to these equations are solved with the code written in Matlab, a multi-paradigm numerical calculation software, and it is observed that the method used has sufficient precision by preparing graphics and tables for the results obtained. In addition, the results are compared with the results of other methods and it is seen from the comparisons that the method gives good results.